3,161 results
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2. Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities
- Author
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Marius Mitrea and Fritz Gesztesy
- Subjects
Class (set theory) ,Pure mathematics ,Mathematics::Analysis of PDEs ,Boundary (topology) ,Spectral analysis ,Type (model theory) ,01 natural sciences ,Mathematics - Spectral Theory ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Boundary value problem ,0101 mathematics ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Nonlocal Robin Laplacians ,Eigenvalue inequalities ,Mathematics::Spectral Theory ,16. Peace & justice ,Lipschitz continuity ,010101 applied mathematics ,35P15, 47A10 (Primary) 35J25, 47A07 (Secondary) ,Dirichlet laplacian ,Bounded function ,Lipschitz domains ,Analysis ,Analysis of PDEs (math.AP) - Abstract
The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators $\Theta$ which give rise to self-adjoint Laplacians $-\Delta_{\Theta, \Omega}$ in $L^2(\Omega; d^n x)$ with (nonlocal and local) Robin-type boundary conditions on bounded Lipschitz domains $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$. Second, we extend Friedlander's inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains $\Omega$, following an approach introduced by Filonov for this type of problems., Comment: 23 pages, added Remark 5.4
- Published
- 2009
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3. Algebraic bounds on the Rayleigh–Bénard attractor
- Author
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Michael S. Jolly, Edriss S. Titi, Yu Cao, Jared P. Whitehead, Jolly, Michael S [0000-0002-7158-0933], Titi, Edriss S [0000-0002-5004-1746], Apollo - University of Cambridge Repository, Jolly, MS [0000-0002-7158-0933], and Titi, ES [0000-0002-5004-1746]
- Subjects
Paper ,General Mathematics ,General Physics and Astronomy ,global attractor ,Enstrophy ,01 natural sciences ,76F35 ,Attractor ,Periodic boundary conditions ,Boundary value problem ,0101 mathematics ,Algebraic number ,Rayleigh–Bénard convection ,math.AP ,Mathematical Physics ,Mathematics ,Rayleigh-Benard convection ,Plane (geometry) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Statistical and Nonlinear Physics ,76E06 ,Nonlinear Sciences::Chaotic Dynamics ,010101 applied mathematics ,34D06 ,Homogeneous space ,Affine space ,synchronization ,35Q35 - Abstract
Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851, Funder: Einstein Visiting Fellow Program, The Rayleigh–Bénard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the L 2 norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy–palinstrophy plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.
- Published
- 2021
4. A numerical study of different projection-based model reduction techniques applied to computational homogenisation
- Author
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Reza Zabihyan, Julia Mergheim, Dominic Soldner, Benjamin Brands, and Paul Steinmann
- Subjects
Mathematical optimization ,Constitutive equation ,Computational Mechanics ,Ocean Engineering ,010103 numerical & computational mathematics ,01 natural sciences ,Applied mathematics ,Boundary value problem ,Computational homogenisation ,0101 mathematics ,Galerkin method ,Mathematics ,Model order reduction ,Original Paper ,Reduced-order modelling ,Geometrically nonlinear ,Applied Mathematics ,Mechanical Engineering ,Hyperelasticity ,Tangent ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Computational Theory and Mathematics ,Hyperelastic material ,Hyper-reduction - Abstract
Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model.
- Published
- 2017
5. A stabilized finite element method for finite-strain three-field poroelasticity
- Author
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Rafel Bordas, David Kay, Simon Tavener, and Lorenz Berger
- Subjects
Original Paper ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Bandwidth (signal processing) ,Poromechanics ,Fluid flux ,Computational Mechanics ,Ocean Engineering ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Lagrange multiplier ,Finite strain theory ,Compressibility ,symbols ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.
- Published
- 2017
6. An Analysis on the Positive Solutions for a Fractional Configuration of the Caputo Multiterm Semilinear Differential Equation
- Author
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Bouchra Azzaoui, Sina Etemad, Sh. Rezapour, H. P. Masiha, and Brahim Tellab
- Subjects
Article Subject ,Differential equation ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,Exact solutions in general relativity ,QA1-939 ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics ,Analysis - Abstract
In this paper, we consider a multiterm semilinear fractional boundary value problem involving Caputo fractional derivatives and investigate the existence of positive solutions by terms of different given conditions. To do this, we first study the properties of Green’s function, and then by defining two lower and upper control functions and using the wellknown Schauder’s fixed-point theorem, we obtain the desired existence criteria. At the end of the paper, we provide a numerical example based on the given boundary value problem and obtain its upper and lower solutions, and finally, we compare these positive solutions with exact solution graphically.
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- 2021
7. Global bounded weak solutions and asymptotic behavior to a chemotaxis-Stokes model with non-Newtonian filtration slow diffusion
- Author
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Chunhua Jin
- Subjects
Steady state ,Applied Mathematics ,010102 general mathematics ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Bounded function ,Norm (mathematics) ,Convergence (routing) ,Filtration (mathematics) ,Boundary value problem ,0101 mathematics ,Diffusion (business) ,Analysis ,Mathematical physics ,Mathematics - Abstract
In this paper, we deal with the following chemotaxis-Stokes model with non-Newtonian filtration slow diffusion (namely, p > 2 ) { n t + u ⋅ ∇ n = ∇ ⋅ ( | ∇ n | p − 2 ∇ n ) − χ ∇ ⋅ ( n ∇ c ) , c t + u ⋅ ∇ c − Δ c = − c n , u t + ∇ π = Δ u + n ∇ φ , div u = 0 in a bounded domain Ω of R 3 with zero-flux boundary conditions and no-slip boundary condition. Similar to the study for the chemotaxis-Stokes system with porous medium diffusion, it is also a challenging problem to find an optimal p-value ( p ≥ 2 ) which ensures that the solution is global bounded. In particular, the closer the value of p is to 2, the more difficult the study becomes. In the present paper, we prove that global bounded weak solutions exist whenever p > p ⁎ ( ≈ 2.012 ) . It improved the result of [21] , [22] , in which, the authors established the global bounded solutions for p > 23 11 . Moreover, we also consider the large time behavior of solutions, and show that the weak solutions will converge to the spatially homogeneous steady state ( n ‾ 0 , 0 , 0 ) . Comparing with the chemotaxis-fluid system with porous medium diffusion, the present convergence of n is proved in the sense of L ∞ -norm, not only in L p -norm or weak-* topology.
- Published
- 2021
8. Determination of the time-dependent convection coefficient in two-dimensional free boundary problems
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M. J. Huntul and Daniel Lesnic
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Work (thermodynamics) ,Discretization ,010102 general mathematics ,General Engineering ,Boundary (topology) ,Inverse problem ,01 natural sciences ,Computer Science Applications ,Nonlinear programming ,010101 applied mathematics ,Tikhonov regularization ,Computational Theory and Mathematics ,Applied mathematics ,Heat equation ,Boundary value problem ,0101 mathematics ,Software ,Mathematics - Abstract
PurposeThe purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.Design/methodologyFor the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.FindingsThe numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueThe inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.
- Published
- 2021
9. Homogenization of enhancing thin layers
- Author
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Zhonggan Huang
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Thin layers ,Trace (linear algebra) ,Scale (ratio) ,Social connectedness ,Applied Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Infinity ,01 natural sciences ,Homogenization (chemistry) ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics ,media_common - Abstract
This paper derives an explicit formula for the effective diffusion tensor by using the solutions to some effective cell problems after homogenizing Road effective boundary conditions (EBCs). The concept of Road EBCs was proposed recently by H. Li and X. Wang, and in this paper, we extend the effective conditions on closed curves to those on patterns, especially on the included nodes. We also prove that homogenization process commutes with the derivation of Road EBCs. By analyzing the effective diffusion tensor, we obtain several rules for maximizing its trace with given Road-effective-diffusivity/scale and length/scale in each cell and define a notion of balanced patterns. Moreover, we give an estimate of the trace of the effective diffusion tensor of patterns satisfying some connectedness as Road-effective-diffusivity/scale goes to infinity.
- Published
- 2021
10. Generalization of the Multipoint meshless FDM application to the nonlinear analysis
- Author
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Irena Jaworska
- Subjects
Geometrically nonlinear ,Generalization ,MathematicsofComputing_NUMERICALANALYSIS ,Finite difference method ,010103 numerical & computational mathematics ,Computer Science::Numerical Analysis ,01 natural sciences ,Collatz conjecture ,Computer Science::Performance ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Computational Theory and Mathematics ,Modeling and Simulation ,Computer Science::Networking and Internet Architecture ,Order (group theory) ,Applied mathematics ,Computer Science::Symbolic Computation ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The paper focuses on the new Multipoint meshless finite difference method, following the original Collatz higher order multipoint concept and the essential ideas of the Meshless FDM. The method was formulated, developed, and tested for various boundary value problems. Generalization of the multipoint method application to nonlinear analysis is the purpose of this research. The first attempt of the multipoint technique application to the geometrically nonlinear problems was successfully done recently. The case of physically nonlinear problem is considered in this paper. Several benefits of the proposed approach are highlighted, numerical algorithm and selected results are presented, and application of the multipoint method to nonlinear analysis is summarized.
- Published
- 2021
11. Nonlinear finite elements: Sub- and supersolutions for the heterogeneous logistic equation
- Author
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D. Aleja and Marcela Molina-Meyer
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Applied Mathematics ,010102 general mathematics ,Boundary (topology) ,01 natural sciences ,Finite element method ,010101 applied mathematics ,symbols.namesake ,Nonlinear system ,Maximum principle ,Jacobian matrix and determinant ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Constant (mathematics) ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its principal eigenvalue and the DMP. Meanwhile a positive subsolution provides the coercivity constant. The numerical simulations show that the nonlinear finite element approximate solutions do not oscillate if the DMP is fulfilled. The characterization of the DMP and the mesh sizes guaranteeing the existence of positive sub- and supersolutions of the nonlinear finite element approximate problem, in the case of variable coefficients and all types of boundary conditions are some of the novelties of this paper. The excellent performance of the method is tested in two examples with boundary layers caused by very small diffusion.
- Published
- 2021
12. Inhomogeneous Vector Riemann Boundary Value Problem and Convolutions Equation on a Finite Interval
- Author
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A. F. Voronin
- Subjects
Work (thermodynamics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Interval (mathematics) ,Wiener algebra ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Riemann problem ,Matrix function ,symbols ,Order (group theory) ,Boundary value problem ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper, we develop a new method for studying the inhomogeneous vector Riemann–Hilbert boundary value problem (which is also called the Riemann boundary value problem) in the Wiener algebra of order two. The method consists in reducing the Riemann problem to a truncated Wiener–Hopf equation (to a convolution equation on a finite interval). The idea of the method was proposed by the author in a previous work. Here the method is applied to the inhomogeneous Riemann boundary value problem and to matrix functions of a more general form. The efficiency of the method is shown in the paper: new sufficient conditions for the existence of a canonical factorization of the matrix function in the Wiener algebra of order two are obtained. In addition, it was established that for the correct solvability of the inhomogeneous vector Riemann boundary value problem, it is necessary and sufficient to prove the uniqueness of the solution to the corresponding truncated homogeneous Wiener–Hopf equation.
- Published
- 2021
13. Numerical Investigation on Improving the Computational Efficiency of the Material Point Method
- Author
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Keyi Chen, Weidong Chen, Shengzhuo Lu, Jingxin Ma, and Yaqin Shi
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Article Subject ,Computer simulation ,Computer science ,General Mathematics ,General Engineering ,Rotational symmetry ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Shock (mechanics) ,Domain (software engineering) ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,QA1-939 ,Boundary value problem ,TA1-2040 ,0101 mathematics ,Algorithm ,Mathematics ,Material point method - Abstract
Based on the basic theory of the material point method (MPM), the factors affecting the computational efficiency are analyzed and discussed, and the problem of improving calculation efficiency is studied. This paper introduces a mirror reflection boundary condition to the MPM to solve axisymmetric problems; to improve the computational efficiency of solving large deformation problems, the concept of "dynamic background domain (DBD)" is also proposed in this paper. Taking the explosion and/or shock problems as an example, the numerical simulation are calculated, and the typical characteristic parameters and the CPU time are compared. The results show that the processing method introducing mirror reflection boundary condition and MPM with DBD can improve the calculation efficiency of the corresponding problems, which, under the premise of ensuring its calculation accuracy, provide useful reference for further promoting the engineering application of this method.
- Published
- 2021
14. The treatment of constraints due to standard boundary conditions in the context of the mixed Web-spline finite element method
- Author
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Said El Fakkoussi, Jaouad El-Mekkaoui, Ouadie Koubaiti, Catalin I. Pruncu, Ahmed Elkhalfi, and Hassan Moustachir
- Subjects
Partial differential equation ,Numerical analysis ,Weak solution ,General Engineering ,02 engineering and technology ,Mixed finite element method ,01 natural sciences ,Finite element method ,Computer Science Applications ,010101 applied mathematics ,symbols.namesake ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Computational Theory and Mathematics ,Dirichlet boundary condition ,Lagrange multiplier ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Software ,Mathematics - Abstract
Purpose This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution. Design/methodology/approach In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory. Findings The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity. Originality/value The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.
- Published
- 2021
15. Monotone Iterative Method for Two Types of Integral Boundary Value Problems of a Nonlinear Fractional Differential System with Deviating Arguments
- Author
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Xi Qin and Jungang Chen
- Subjects
Comparison theorem ,Monotone iterative method ,Article Subject ,General Mathematics ,010102 general mathematics ,Differential systems ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,QA1-939 ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Fractional differential ,Mathematics - Abstract
This paper concerns on two types of integral boundary value problems of a nonlinear fractional differential system, i . e ., nonlocal strip integral boundary value problems and coupled integral boundary value problems. With the aid of the monotone iterative method combined with the upper and lower solutions, the existence of extremal system of solutions for the above two types of differential systems is investigated. In addition, a new comparison theorem for fractional differential system is also established, which is crucial for the proof of the main theorem of this paper. At the end, an example explaining how our studies can be used is also given.
- Published
- 2021
16. On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications
- Author
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Huaian Diao, Hongyu Liu, and Xinlin Cao
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,Eigenfunction ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Transmission (telecommunications) ,Inverse scattering problem ,FOS: Mathematics ,Boundary value problem ,Uniqueness ,0101 mathematics ,35Q60, 78A46, 35P25, 78A05, 81U40 ,Electrical conductor ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in [9]. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than $\pi$. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures established for the conductive transmission eigenfunctions in this paper include the results in [9] as a special case. Second, the vanishing property of the conductive transmission eigenfunctions is established for any corner as long as its interior angle is not $\pi$ when the conductive transmission eigenfunctions satisfy certain Herglotz functions approximation properties. That means, as long as the corner singularity is not degenerate, the vanishing property holds if the underlying conductive transmission eigenfunctions can be approximated by a sequence of Herglotz functions under mild approximation rates. Third, the regularity requirements on the interior transmission eigenfunctions in [9] are significantly relaxed in the present study for the conductive transmission eigenfunctions. Finally, as an interesting and practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter.
- Published
- 2021
17. Estimation of boundary condition of two-dimensional nonlinear PDE with application to continuous casting
- Author
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Huaxi (Yulin) Zhang, Yang Yu, Xiaochuan Luo, and Yuan Wang
- Subjects
Optimization problem ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Regularization (mathematics) ,010101 applied mathematics ,Continuous casting ,Computational Mathematics ,Nonlinear system ,Computational Theory and Mathematics ,Modeling and Simulation ,Conjugate gradient method ,Convergence (routing) ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
Solidification heat transfer process of billet is described by nonlinear partial differential equation (PDE). Due to the poor productive environment, the boundary condition of this nonlinear PDE is difficult to be fixed. Therefore, the identification of boundary condition of two-dimensional nonlinear PDE is considered. This paper transforms the identification of boundary condition into a PDE optimization problem. The Lipchitz continuous of the gradient of cost function is proved based on the dual equation. In order to solve this optimization problem, this paper presents a modified conjugate gradient algorithm, and the global convergence of which is analyzed. The results of the simulation experiment show that the modified conjugate gradient algorithm obviously reduces the iterative number and running time. Due to the ill-posedness of the identification of boundary condition, this paper combines regularization method with the modified conjugate gradient algorithm. The simulation experiment illustrates that regularization method can eliminate the ill-posedness of this problem. Finally, the experimental data of a steel plant illustrate the validity of this paper’s method.
- Published
- 2020
18. Global solutions to compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma on exterior domains
- Author
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Tao Luo, Hua Zhong, and Hairong Liu
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Symmetry in biology ,Plasma ,Poisson distribution ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,symbols.namesake ,Exponential stability ,Compressibility ,symbols ,Euler's formula ,Ball (mathematics) ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics - Abstract
The initial boundary value problems for compressible Navier-Stokes-Poisson and Euler-Poisson equations of plasma are considered on exterior domains in this paper. With the radial symmetry assumption, the global existence of solutions to compressible Navier-Stokes-Poisson equations with the large initial data on a domain exterior to a ball in R n ( n ≥ 1 ) is proved. Moreover, without any symmetry assumption, the global existence of smooth solutions near a given constant steady state for both compressible Navier-Stokes-Poisson and Euler-Poisson equations on an exterior domain in R 3 with physical boundary conditions is also established with the exponential stability. A key issue addressed in this paper is on the global-in-time regularity of solutions near physical boundaries. This is in particular so for the 3-D compressible Navier-Stokes-Poisson equations to which global smooth solutions of initial boundary value problems are seldom found in literature to the best of knowledge.
- Published
- 2020
19. Sharp bounds for the resolvent of linearized Navier Stokes equations in the half space around a shear profile
- Author
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Toan T. Nguyen and Emmanuel Grenier
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Vorticity ,01 natural sciences ,Euler equations ,Physics::Fluid Dynamics ,010101 applied mathematics ,Boundary layer ,symbols.namesake ,Mathematics - Analysis of PDEs ,Inviscid flow ,Dirichlet boundary condition ,FOS: Mathematics ,symbols ,Boundary value problem ,0101 mathematics ,Navier–Stokes equations ,Analysis ,Analysis of PDEs (math.AP) ,Resolvent ,Mathematics - Abstract
In this paper, we derive sharp bounds on the semigroup of the linearized incompressible Navier-Stokes equations near a stationary shear layer in the half plane and in the half space ($\mathbb{R}_+^2$ or $\mathbb{R}_+^3$), with Dirichlet boundary conditions, assuming that this shear layer in spectrally unstable for Euler equations. In the inviscid limit, due to the prescribed no-slip boundary conditions, vorticity becomes unbounded near the boundary. The novelty of this paper is to introduce boundary layer norms that capture the unbounded vorticity and to derive sharp estimates on this vorticity that are uniform in the inviscid limit., Comment: this greatly revised and shortened the previous version
- Published
- 2020
20. Nodal solutions of weighted indefinite problems
- Author
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Julián López-Gómez and Martin Fencl
- Subjects
Class (set theory) ,Superlinear indefinite problems ,Nodal solutions ,Finite-difference scheme ,Structure (category theory) ,34B15, 34B08, 34L16 ,Global components ,01 natural sciences ,Set (abstract data type) ,Mathematics - Analysis of PDEs ,Mathematics (miscellaneous) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,Eigencurves ,Pseudo-spectral methods ,0101 mathematics ,Positive solutions ,Bifurcation ,Eigenvalues and eigenvectors ,Mathematics ,Concavity ,010102 general mathematics ,Numerical Analysis (math.NA) ,Path-following ,010101 applied mathematics ,Weighted problems ,NODAL ,Analysis of PDEs (math.AP) ,Sign (mathematics) - Abstract
This paper analyzes the structure of the set of nodal solutions of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite astonishingly, the associated high order eigenvalues might not be concave as it is the lowest one. As a consequence, in many circumstances the nodal solutions can bifurcate from three or even four bifurcation points from the trivial solution. This paper combines analytical and numerical tools. The analysis carried over on it is a paradigm of how mathematical analysis aids the numerical study of a problem, whereas simultaneously the numerical study confirms and illuminate the analysis., 19 pages, 29 figures
- Published
- 2020
21. A novel approach for the solution of BVPs via Green’s function and fixed point iterative method
- Author
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Faeem Ali, Javid Ali, and Izhar Uddin
- Subjects
Iterative method ,Applied Mathematics ,Function (mathematics) ,Fixed point ,01 natural sciences ,010305 fluids & plasmas ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Operator (computer programming) ,Rate of convergence ,Green's function ,0103 physical sciences ,Convergence (routing) ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In the present paper, a new fixed point iterative method is introduced based on Green’s function and it’s successfully applied to approximate the solution of boundary value problems. A strong convergence result is proved for the integral operator by using the proposed method. It is also showed that the newly defined iterative method has a better rate of convergence than the Picard–Green’s, Mann–Green’s and Ishikawa–Green’s iterative methods. Some illustrative numerical examples are presented for the validity, applicability and high efficiency of the proposed iterative method. The results of this paper extend and generalize the corresponding results in the literature and particularly in Khuri and Louhichi (Appl Math Lett 82:50–57, 2018).
- Published
- 2020
22. Steklov approximations of Green’s functions for Laplace equations
- Author
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Manki Cho
- Subjects
Laplace transform ,Applied Mathematics ,010102 general mathematics ,Boundary (topology) ,Harmonic (mathematics) ,01 natural sciences ,Orthogonal basis ,Computer Science Applications ,010101 applied mathematics ,Computational Theory and Mathematics ,Harmonic function ,Fundamental solution ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Electrical and Electronic Engineering ,Laplace operator ,Mathematics - Abstract
Purpose This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator. Design/methodology/approach The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary. Findings The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary. Originality/value This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.
- Published
- 2020
23. Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems
- Author
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Ying Xu, Fengxin Sun, and Jufeng Wang
- Subjects
Correctness ,Article Subject ,General Mathematics ,Mathematical analysis ,General Engineering ,Boundary (topology) ,02 engineering and technology ,Radius ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,QA1-939 ,Boundary value problem ,TA1-2040 ,0101 mathematics ,Coefficient matrix ,Boundary element method ,Condition number ,Mathematics ,Interpolation - Abstract
The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.
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- 2020
24. Fractional Hybrid Differential Equations and Coupled Fixed-Point Results for <math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <mi>α</mi> </math>-Admissible <math xmlns='http://www.w3.org/1998/Math/MathML' id='M2'> <mi>F</mi> <mfenced open='(' close=')' separators='|'> <mrow> <msub> <mrow> <mi>ψ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mi>ψ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </mfenced> <mo>−</mo> </math>Contractions in <math xmlns='http://www.w3.org/1998/Math/MathML' id='M3'> <mi>M</mi> <mo>−</mo> </math>Metric Spaces
- Author
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Shimaa I. Moustafa, Erdal Karapınar, Ravi P. Agarwal, and Ayman Shehata
- Subjects
010101 applied mathematics ,Metric space ,Differential equation ,Modeling and Simulation ,010102 general mathematics ,Mathematical analysis ,Uniqueness ,Boundary value problem ,0101 mathematics ,Fixed point ,01 natural sciences ,Mathematics - Abstract
In this paper, we investigate the existence of a unique coupled fixed point for α − admissible mapping which is of F ψ 1 , ψ 2 − contraction in the context of M − metric space. We have also shown that the results presented in this paper would extend many recent results appearing in the literature. Furthermore, we apply our results to develop sufficient conditions for the existence and uniqueness of a solution for a coupled system of fractional hybrid differential equations with linear perturbations of second type and with three-point boundary conditions.
- Published
- 2020
25. On strong solutions of viscoplasticity without safe-load conditions
- Author
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Konrad Kisiel and Krzysztof Chełmiński
- Subjects
Pointwise ,Polynomial ,Work (thermodynamics) ,Viscoplasticity ,Applied Mathematics ,010102 general mathematics ,Function (mathematics) ,Type (model theory) ,Lipschitz continuity ,01 natural sciences ,010101 applied mathematics ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper we discuss existence of pointwise solutions for dynamical models of viscoplasticity. Among other things, this work answers the question about necessity of safe-load conditions in case of viscoplasticity, which arise in the paper of K. Chelminski (2001) [11] . We proved that solutions can be obtained without assuming any kind of safe-load conditions. Moreover, in the manuscript we consider much more general model than in the above mentioned paper. Namely, we consider the model with mixed boundary conditions and we allow a possible disturbance of the inelastic constitutive function by a globally Lipschitz function. Presented approach shows that via the same methods one can prove existence of pointwise solutions for: coercive models, self-controlling models, models with polynomial growth (not necessary of single valued) and monotone-gradient type models of viscoplasticity.
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- 2020
26. Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions
- Author
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Ya-Hong Zhao, Jian-Ping Sun, and Min Li
- Subjects
Algebra and Number Theory ,Functional analysis ,Applied Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Fixed-point theorem ,Existence ,lcsh:QA1-939 ,01 natural sciences ,Fractional differential equation ,Fractional calculus ,010101 applied mathematics ,Nonlinear fractional differential equations ,Combinatorics ,Positive solution ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics ,Integral boundary condition - Abstract
This paper is concerned with the following boundary value problem of nonlinear fractional differential equation with integral boundary conditions: $$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+f(t,u(t))=0,\quad t\in [0,1], \\ u^{\prime \prime }(0)=0, \\ \alpha u(0)-\beta u^{\prime }(0)=\int _{0}^{1}h_{1}(s)u(s)\,ds, \\ \gamma u(1)+\delta ({}^{C}D_{0+}^{\sigma }u)(1) =\int _{0}^{1}h_{2}(s)u(s)\,ds, \end{cases} $${(CD0+qu)(t)+f(t,u(t))=0,t∈[0,1],u″(0)=0,αu(0)−βu′(0)=∫01h1(s)u(s)ds,γu(1)+δ(CD0+σu)(1)=∫01h2(s)u(s)ds, where $2< q\leq 3$2
0$β>0 satisfying $00<ρ:=(α+β)γ+αδΓ(2−σ)<β[γ+δΓ(q)Γ(q−σ)]. ${}^{C}D_{0+}^{q}$D0+qC denotes the standard Caputo fractional derivative. First, Green’s function of the corresponding linear boundary value problem is constructed. Next, some useful properties of the Green’s function are obtained. Finally, existence results of at least one positive solution for the above problem are established by imposing some suitable conditions on f and $h_{i}$hi ($i=1,2$i=1,2). The method employed is Guo–Krasnoselskii’s fixed point theorem. An example is also included to illustrate the main results of this paper.
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- 2020
27. EXISTENCE AND MULTIPLICITY OF WEAK SOLUTIONS FOR A CLASS OF FRACTIONAL STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS WITH IMPULSIVE CONDITIONS
- Author
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Yan Qiao, Fangqi Chen, and Yukun An
- Subjects
Class (set theory) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Sturm–Liouville theory ,Multiplicity (mathematics) ,Mathematics::Spectral Theory ,01 natural sciences ,Boundary values ,010101 applied mathematics ,Homogeneous ,Critical point (thermodynamics) ,Boundary value problem ,0101 mathematics ,Fractional differential ,Mathematics - Abstract
In this paper, we consider the existence and multiplicity of weak solutions for a class of fractional differential equations with non-homogeneous Sturm-Liouville conditions and impulsive conditions by using the critical point theory. In addition, at the end of this paper, we also give the existence results of infinite weak solutions of fractional differential equations under homogeneous Sturm-Liouville boundary value conditions. Finally, several examples are given to illustrate our main results.
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- 2020
28. A family of measures of noncompactness in the Hölder space Cn,γ(R+) and its application to some fractional differential equations and numerical methods
- Author
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Hojjatollah Amiri Kayvanloo, Mahnaz Khanehgir, and Reza Allahyari
- Subjects
Pure mathematics ,Applied Mathematics ,Numerical analysis ,Hölder condition ,Fixed-point theorem ,010103 numerical & computational mathematics ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Shaping ,Boundary value problem ,0101 mathematics ,Fractional differential ,Mathematics - Abstract
In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u ( t ) = f ( t , u ( t ) ) , α ∈ ( n , n + 1 ) , 0 ≤ t + ∞ , u ( 0 ) = 0 , u ′ ′ ( 0 ) = 0 , … , u ( n ) ( 0 ) = 0 , lim t → + ∞ c D α − 1 u ( t ) = β u ( ξ ) . The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ ( R + ) satisfying the Holder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy.
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- 2020
29. EXISTENCE OF SOLUTIONS FOR DUAL SINGULAR INTEGRAL EQUATIONS WITH CONVOLUTION KERNELS IN CASE OF NON-NORMAL TYPE
- Author
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Pingrun Li
- Subjects
General Mathematics ,010102 general mathematics ,Singular integral ,Type (model theory) ,01 natural sciences ,Integral equation ,Dual (category theory) ,Convolution ,010101 applied mathematics ,Riemann hypothesis ,symbols.namesake ,Fourier transform ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.
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- 2020
30. Numerical solution of non-Newtonian fluid flow and heat transfer problems in ducts with sharp corners by the modified method of fundamental solutions and radial basis function collocation
- Author
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Jakub Krzysztof Grabski
- Subjects
Applied Mathematics ,Mathematical analysis ,General Engineering ,02 engineering and technology ,Kansa method ,01 natural sciences ,Non-Newtonian fluid ,Physics::Fluid Dynamics ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Fixed-point iteration ,Heat transfer ,Fluid dynamics ,Method of fundamental solutions ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, non-Newtonian fluid flow and heat transfer in an internally finned duct is considered. The problems are governed by nonlinear equations with linear boundary conditions. In this paper, the solution is obtained with two Picard iteration processes, separately for the fluid flow and heat transfer problems. Furthermore, in both problems the solutions consist of the general and particular solutions. The method of fundamental solutions is employed in order to obtain the general solution while radial basis function approximations are used for obtaining the general solution by two possible methods. The results obtained using both approaches give similar results.
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- 2019
31. Homogenization of a nonlinear monotone problem with a big nonlinear Signorini boundary interaction in a domain with highly rough boundary
- Author
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Taras A. Mel'nyk, Antonio Gaudiello, Gaudiello, Antonio, and Mel'Nyk, Taras
- Subjects
Chemical activity ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,A domain ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,01 natural sciences ,Homogenization (chemistry) ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Monotone polygon ,Homogeneous ,Dirichlet boundary condition ,symbols ,Boundary value problem ,0101 mathematics ,Mathematical Physics ,Mathematics - Abstract
This paper is devoted to study the asymptotic behavior, as vanishes, of a nonlinear monotone Signorini boundary value problem modelizing chemical activity in an -periodic structure of thin cylindrical absorbers, like a comb in 2D a or a brush in 3D. The novelty of this paper is the presence of a perturbed coefficient , with , in the nonlinear Signorini boundary conditions (the case was previously studied by the same authors). It is shown that the limit problem is the same as what one would get by replacing the Signorini boundary conditions with the homogeneous Dirichlet boundary condition in the original problem.
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- 2019
32. Boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach
- Author
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Wolfgang L. Wendland and Mirela Kohr
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Weak solution ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Fixed-point theorem ,Riemannian manifold ,Lipschitz continuity ,01 natural sciences ,Dirichlet distribution ,Physics::Fluid Dynamics ,010101 applied mathematics ,Sobolev space ,Nonlinear system ,symbols.namesake ,symbols ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to show well-posedness results in L 2 -based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m ≥ 2 . The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L ∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L ∞ coefficients and its mixed variational counterpart, and then the well posedness in L 2 -based Sobolev spaces by using the Necas-Babuska-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L ∞ coefficients in L 2 -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m ∈ { 2 , 3 } .
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- 2019
33. An accurate a posteriori error estimator for the Steklov eigenvalue problem and its applications
- Author
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Fei Xu and Qiumei Huang
- Subjects
Series (mathematics) ,General Mathematics ,Estimator ,010103 numerical & computational mathematics ,Mathematics::Spectral Theory ,Space (mathematics) ,01 natural sciences ,Finite element method ,010101 applied mathematics ,A priori and a posteriori ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Eigenvalues and eigenvectors ,Smoothing ,Mathematics - Abstract
In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.
- Published
- 2019
34. Convergence of boundary layers for the Keller–Segel system with singular sensitivity in the half-plane
- Author
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Qianqian Hou and Zhi-An Wang
- Subjects
Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Prandtl number ,Boundary (topology) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Boundary layer ,symbols.namesake ,symbols ,Boundary value problem ,0101 mathematics ,Layer (object-oriented design) ,Degeneracy (mathematics) ,Mathematics - Abstract
Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63] ), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by e ≥ 0 . Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with e = 0 ) plus the inner layer as e → 0 , where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63] , and open a new window in the future theoretical study of chemotaxis models.
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- 2019
35. Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings
- Author
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Mohammed Said Souid, Mohammed K. A. Kaabar, Zailan Siri, Shahram Rezapour, Francisco Martínez, Sina Etemad, and Ahmed Refice
- Subjects
Ulam–Hyers–Rassias stability ,Mathematics::Functional Analysis ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,variable-order operators ,implicit problem ,01 natural sciences ,Stability (probability) ,fixed point theorems ,010101 applied mathematics ,Nonlinear fractional differential equations ,piecewise constant functions ,Nonlinear system ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,Order (group theory) ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,Variable (mathematics) - Abstract
In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.
- Published
- 2021
36. On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations
- Author
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Ashwini D. Mali, Kishor D. Kucche, and José Vanterler da Costa Sousa
- Subjects
Applied Mathematics ,010102 general mathematics ,Computational Mechanics ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Mechanics of Materials ,Modeling and Simulation ,Applied mathematics ,Initial value problem ,Boundary value problem ,0101 mathematics ,Fractional differential ,Engineering (miscellaneous) ,Mathematics - Abstract
This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results.
- Published
- 2021
37. k-Version of Finite Element Method for BVPs and IVPs
- Author
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Sri Sai Charan Mathi, Karan S. Surana, and Celso H. Carranza
- Subjects
General Mathematics ,finite element method ,MathematicsofComputing_GENERAL ,higher order global differentiability ,02 engineering and technology ,Isogeometric analysis ,01 natural sciences ,k-version ,0203 mechanical engineering ,Convergence (routing) ,QA1-939 ,Computer Science (miscellaneous) ,Applied mathematics ,Initial value problem ,isogeometric ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,tensor product ,higher order spaces ,Differential operator ,IVPs ,Finite element method ,variational consistency ,010101 applied mathematics ,BVPs ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,020303 mechanical engineering & transports ,Tensor product ,Self-adjoint operator - Abstract
The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work.
- Published
- 2021
- Full Text
- View/download PDF
38. Existence and Regularity of Weak Solutions for $$\psi $$-Hilfer Fractional Boundary Value Problem
- Author
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J. Vanterler da C. Sousa, E. Capelas de Oliveira, and M. Aurora P. Pulido
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,symbols ,High Energy Physics::Experiment ,Integration by parts ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In the present paper, we investigate the existence and regularity of weak solutions for $$\psi $$ -Hilfer fractional boundary value problem in $$\mathbb {C}^{\alpha ,\beta ;\psi }_{2}$$ and $$\mathcal {H}$$ (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for $$\psi $$ -Riemann–Liouville fractional integral and $$\psi $$ -Hilfer fractional derivative.
- Published
- 2021
39. Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?
- Author
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B. Cano and Nuria Reguera
- Subjects
Order reduction ,General Mathematics ,Krylov methods ,010103 numerical & computational mathematics ,Exponential integrator ,01 natural sciences ,Exponential function ,010101 applied mathematics ,efficiency ,QA1-939 ,Computer Science (miscellaneous) ,Spite ,Applied mathematics ,avoiding order reduction ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.
- Published
- 2021
- Full Text
- View/download PDF
40. An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems
- Author
-
Mahboub Baccouch
- Subjects
Discretization ,Adaptive mesh refinement ,Applied Mathematics ,Estimator ,010103 numerical & computational mathematics ,Superconvergence ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Rate of convergence ,Discontinuous Galerkin method ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form $u^{\prime \prime }=f(x,u),\ x\in [a,b]$ subject to some suitable boundary conditions at the endpoint of the interval [a, b]. We first use the superconvergence results proved in the first part of this paper as reported by Baccouch (Numer. Algorithm. 79(3), 697–718 2018) to show that the significant parts of the local discretization errors are proportional to (p + 1)-degree Radau polynomials, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed a posteriori error estimates converge to the exact errors in the L2-norm with order of convergence p + 3/2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p ≥ 1. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.
- Published
- 2019
41. A robust incompressible Navier-Stokes solver for high density ratio multiphase flows
- Author
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Neelesh A. Patankar, Boyce E. Griffith, Amneet Pal Singh Bhalla, and Nishant Nangia
- Subjects
Mass flux ,Physics and Astronomy (miscellaneous) ,Discretization ,FOS: Physical sciences ,01 natural sciences ,010305 fluids & plasmas ,Momentum ,Total variation diminishing ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,0101 mathematics ,Mathematics ,Numerical Analysis ,Adaptive mesh refinement ,Applied Mathematics ,Mathematical analysis ,Fluid Dynamics (physics.flu-dyn) ,Order of accuracy ,Physics - Fluid Dynamics ,Numerical Analysis (math.NA) ,Computational Physics (physics.comp-ph) ,Solver ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,Physics - Computational Physics - Abstract
This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The algorithm combines the interface capturing level set method with a variable-coefficient incompressible Navier-Stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude. The discretization approach ensures second-order pointwise accuracy for both velocity and pressure with several physical boundary treatments, including velocity and traction boundary conditions. The paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver. To ensure the stability of the numerical scheme in the presence of high density and viscosity ratios, we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations. This consistency is achieved by solving an additional mass balance equation, which we approximate via a strong stability preserving Runga-Kutta time integrator and by employing the same mass flux (obtained from the mass equation) in the discrete momentum equation. The scheme uses higher-order total variation diminishing (TVD) and convection-boundedness criterion (CBC) satisfying limiter to avoid numerical fluctuations in the transported density field. The high-order bounded convective transport is done on a dimension-by-dimension basis, which makes the scheme simple to implement. We also demonstrate through several test cases that the lack of consistent mass and momentum transport in non-conservative formulations, which are commonly used in practice, or the use of non-CBC satisfying limiters can yield very large numerical error and very poor accuracy for convection-dominant high density ratio flows., Comment: Figures are compressed to comply with arXiv size requirements
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- 2019
42. Unique iterative positive solutions for a singular p-Laplacian fractional differential equation system with infinite-point boundary conditions
- Author
-
Lishan Liu and Limin Guo
- Subjects
Sequence ,Algebra and Number Theory ,Partial differential equation ,010102 general mathematics ,Mathematical analysis ,lcsh:QA299.6-433 ,Monotonic function ,Function (mathematics) ,Fractional differential equation system ,lcsh:Analysis ,01 natural sciences ,010101 applied mathematics ,Rate of convergence ,Singular p-Laplacian ,Ordinary differential equation ,p-Laplacian ,Infinite-point ,Boundary value problem ,0101 mathematics ,Iterative positive solution ,Analysis ,Mathematics - Abstract
By using the method of mixed monotone operator, a unique positive solution is obtained for a singular p-Laplacian boundary value system with infinite-point boundary conditions in this paper. Green’s function is derived and some useful properties of the Green’s function are obtained. Based upon these new properties and by using mixed monotone operator, the existence results of the positive solutions for the boundary value problem are established. Moreover, the unique positive solution that we obtained in this paper is dependent on $\lambda ,\mu $ , and an iterative sequence and convergence rate, which are important for practical application, are given. An example is given to demonstrate the application of our main results.
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- 2019
43. Euler type linear and half-linear differential equations and their non-oscillation in the critical oscillation case
- Author
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Michal Veselý, Zuzana Došlá, Serena Matucci, and Petr Hasil
- Subjects
Oscillation theory ,Differential equation ,Non-oscillation criterion ,01 natural sciences ,Euler type equations ,symbols.namesake ,Linear differential equation ,Half-linear equations ,Oscillation criterion ,Discrete Mathematics and Combinatorics ,Boundary value problem ,0101 mathematics ,Mathematics ,Oscillation ,Applied Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Mathematical analysis ,lcsh:QA1-939 ,010101 applied mathematics ,Linear equations ,Oscillation constant ,p-Laplacian ,Euler's formula ,symbols ,Analysis ,Linear equation - Abstract
This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.
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- 2019
44. Trefftz method in solving the pennes’ and single-phase-lag heat conduction problems with perfusion in the skin
- Author
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Krzysztof Grysa and Artur Maciag
- Subjects
Laplace's equation ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,02 engineering and technology ,Thermal conduction ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Mechanics of Materials ,Trefftz method ,Heat transfer ,Bioheat transfer ,Boundary value problem ,0101 mathematics ,Hyperbolic partial differential equation ,Condition number ,Mathematics - Abstract
Purpose The purpose of this paper is to derive the Trefftz functions (T-functions) for the Pennes’ equation and for the single-phase-lag (SPL) model (hyperbolic equation) with perfusion and then comparing field of temperature in a flat slab made of skin in the case when perfusion is taken into account, with the situation when a Fourier model is considered. When considering the process of heat conduction in the skin, one needs to take into account the average values of its thermal properties. When in biological bodies relaxation time is of the order of 20 s, the thermal wave propagation appears. The initial-boundary problems for Pennes’ model and SPL with perfusion model are considered to investigate the effect of the finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. As a reference model, the solution of the classic Fourier heat transfer equation for the considered problems is calculated. A heat flux has direction perpendicular to the surface of skin, considered as a flat slab. Therefore, the equations depend only on time and one spatial variable. Design/methodology/approach First of all the T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. Then, an approximate solutions of the problems are expressed in the form of a linear combination of the T-functions. The T-functions satisfy the equation modeling the problem under consideration. Therefore, approximating a solution of a problem with a linear combination of n T-functions one obtains a function that satisfies the equation. The unknown coefficients of the linear combination are obtained as a result of minimization of the functional that describes an inaccuracy of satisfying the initial and boundary conditions in a mean-square sense. Findings The sets of T-functions for the Pennes’ equation and for the SPL model with perfusion are derived. An infinite set of these functions is a complete set of functions and stands for a base functions layout for the space of solutions for the equation used to generate them. Then, an approximate solutions of the initial-boundary problem have been found and compared to find out the effect of finite velocity of heat in the skin, perfusion and thickness of the slab on the rate of the thermal wave attenuation. Research limitations/implications The methods used in the literature to find an approximate solution of any bioheat transfer problems are more complicated than the one used in the presented paper. However, it should be pointed out that there is some limitation concerning the T-function method, namely, the greater number of T-function is used, the greater condition number becomes. This limitation usually can be overcome using symbolic calculations or conducting calculations with a large number of significant digits. Originality/value The T-functions for the Pennes’ equation and for the SPL equation with perfusion have been reported in this paper for the first time. In the literature, the T-functions are known for other linear partial differential equations (e.g. harmonic functions for Laplace equation), but for the first time they have been derived for the two aforementioned equations. The results are discussed with respect to practical applications.
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- 2019
45. Dynamical behavior of integro-differential boundary value problem arising in nano-structures via Cellular Nanoscale Network approach
- Author
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Grigory Agranovich, Angela Slavova, and Elena Litsyn
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Partial differential equation ,Field (physics) ,Applied Mathematics ,Describing function ,Fracture mechanics ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Stress (mechanics) ,Computational Mathematics ,Nano ,Boundary value problem ,Statistical physics ,0101 mathematics ,Differential (mathematics) ,Mathematics - Abstract
Piezoelectrical solid with nano-holes under time-harmonic anti-plane load is studied in the paper. The model is defined by the system of two partial differential equations and the boundary conditions which define the generalized stress. We reduce the model into an integro-differential boundary value problem and construct Cellular Nanoscale Network (CNN) architecture. Dynamical behavior of the obtained CNN model is studied by means of describing function technique. The simulations are provided which illustrate the theoretical results. The results of this paper are applicable in the field of non-destructive testing and fracture mechanics of multi-functional materials and structural elements based on them.
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- 2019
46. A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method
- Author
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Sławomir Milewski
- Subjects
Discretization ,business.industry ,Applied Mathematics ,Numerical analysis ,Monte Carlo method ,010103 numerical & computational mathematics ,System of linear equations ,Random walk ,01 natural sciences ,010101 applied mathematics ,Software ,Applied mathematics ,Meshfree methods ,Boundary value problem ,0101 mathematics ,business ,Mathematics - Abstract
This paper is devoted to the development of an innovative Matlab software, dedicated to the numerical analysis of two-dimensional elliptic problems, by means of the probabilistic approach. This approach combines features of the Monte Carlo random walk method with discretization and approximation techniques, typical for meshless methods. It allows for determination of an approximate solution of elliptic equations at the specified point (or group of points), without a necessity to generate large system of equations for the entire problem domain. While the procedure is simple and fast, the final solution may suffer from both stochastic and discretization errors. The attached Matlab software is based on several original author’s concepts. It permits the use of arbitrarily irregular clouds of nodes, non-homogeneous right-hand side functions, mixed type of boundary conditions as well as variable material coefficients (of anisotropic materials). The paper is illustrated with results of analysis of selected elliptic problems, obtained by means of this software.
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- 2019
47. Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations
- Author
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Ching-Shyang Chen, Chia-Ming Fan, Shyh-Rong Kuo, and Yi-Ling Huang
- Subjects
Laplace's equation ,Laplace transform ,Applied Mathematics ,General Engineering ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Algebraic equation ,Matrix (mathematics) ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Biharmonic equation ,Method of fundamental solutions ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics - Abstract
The localized method of fundamental solutions (LMFS) is proposed in this paper for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains. Traditionally, the method of fundamental solutions (MFS) is a global method and the resultant matrix is dense and ill-conditioned. In this paper, it is the first time that the LMFS, the localized version of the MFS, is proposed. In the LMFS, the solutions at every interior node are expressed as linear combinations of solutions at some nearby nodes, while the numerical procedures of MFS are implemented within every local subdomain. The satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes can yield a sparse system of linear algebraic equations, so the numerical solutions can be efficiently acquired by solving the resultant sparse system. Six numerical examples are given to demonstrate the effectiveness of the proposed LMFS.
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- 2019
48. NURBS-enhanced line integration boundary element method for 2D elasticity problems with body forces
- Author
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Gang Ma, Qiao Wang, Yonggang Cheng, Xiaolin Chang, and Wei Zhou
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Body force ,Mathematical analysis ,Dirac delta function ,Basis function ,010103 numerical & computational mathematics ,Singular integral ,Parameter space ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Modeling and Simulation ,symbols ,Boundary value problem ,0101 mathematics ,Boundary element method ,Mathematics ,Parametric statistics - Abstract
A NURBS-enhanced boundary element method for 2D elasticity problems with body forces is proposed in this paper. The non-uniform rational B-spline (NURBS) basis functions are applied to construct the geometry and the model can be reproduced exactly at all stages since the refinement will not change the shape of the boundary. Both open curves and closed curves are considered. The fields are approximated by the traditional Lagrangian basis functions in parameter space, rather than by the same NURBS basis functions for geometry approximation. The parametric boundary elements and collocation nodes are defined from the knot vector of the curve and the refinement of the NURBS curve is easy. Boundary conditions can be imposed directly since the Lagrangian basis functions have the property of delta function. In addition, most methods for the treatment of singular integrals in traditional boundary element method can be applied in the proposed method. To overcome the difficulty for evaluation of the domain integrals in problems with body forces, a line integration method is further applied in this paper to compute the domain integrals without additional volume discretizations. Numerical examples have shown the accuracy of the proposed method.
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- 2019
49. Bounds on the effective response for gradient crystal inelasticity based on homogenization and virtual testing
- Author
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Fredrik Larsson, Kenneth Runesson, and Kristoffer Carlsson
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Numerical Analysis ,Applied Mathematics ,Computation ,Ergodicity ,General Engineering ,02 engineering and technology ,01 natural sciences ,Homogenization (chemistry) ,Upper and lower bounds ,Dirichlet distribution ,Global variable ,010101 applied mathematics ,symbols.namesake ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Saddle point ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
This paper presents the application of variationally consistent selective homogenization applied to a polycrystal with a subscale model of gradient-enhanced crystal inelasticity. Although the full gradient problem is solved on Statistical Volume Elements (SVEs), the resulting macroscale problem has the formal character of a standard local continuum. A semi-dual format of gradient inelasticity is exploited, whereby the unknown global variables are the displacements and the energetic micro-stresses on each slip-system. The corresponding time-discrete variational formulation of the SVE-problem defines a saddle point of an associated incremental potential. Focus is placed on the computation of statistical bounds on the effective energy, based on virtual testing on SVEs and an argument of ergodicity. As it turns out, suitable combinations of Dirichlet and Neumann conditions pertinent to the standard equilibrium and the micro-force balance, respectively, will have to be imposed. Whereas arguments leading to the upper bound are quite straightforward, those leading to the lower bound are significantly more involved; hence, a viable approximation of the lower bound is computed in this paper. Numerical evaluations of the effective strain energy confirm the theoretical predictions. Furthermore, heuristic arguments for the resulting macroscale stress-strain relations are numerically confirmed.
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- 2019
50. Stokes system with local Coulomb’s slip boundary conditions: Analysis of discretized models and implementation
- Author
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Jaroslav Haslinger, Václav Šátek, and Radek Kučera
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Discretization ,Mathematical analysis ,Slip coefficient ,010103 numerical & computational mathematics ,Slip (materials science) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Coulomb ,Boundary value problem ,Uniqueness ,0101 mathematics ,Interior point method ,Mathematics - Abstract
The theoretical part of the paper analyzes discretized Stokes systems with local Coulomb’s slip boundary conditions. Solutions to discrete models are defined by means of fixed-points of an appropriate mapping. We prove the existence of a fixed-point, establish conditions guaranteeing its uniqueness and examine how they depend on the discretization parameter h and the slip coefficient κ . The second part of the paper is devoted to computational aspects. Numerical experiments are presented.
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- 2019
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