17,478 results on '"010101 applied mathematics"'
Search Results
2. Virtual element approximation of two-dimensional parabolic variational inequalities
- Author
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Sundararajan Natarajan, Dibyendu Adak, and Gianmarco Manzini
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Polynomial ,Degrees of freedom (statistics) ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Projection (linear algebra) ,010101 applied mathematics ,Computational Mathematics ,Quadratic equation ,Computational Theory and Mathematics ,Rate of convergence ,Modeling and Simulation ,Variational inequality ,Applied mathematics ,0101 mathematics ,Voronoi diagram ,Mathematics - Abstract
We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.
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- 2022
3. A computationally efficient strategy for time-fractional diffusion-reaction equations
- Author
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Roberto Garrappa and Marina Popolizio
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Scheme (programming language) ,Computation ,Kernel compression scheme ,Pattern formation ,010103 numerical & computational mathematics ,Derivative ,Space (mathematics) ,01 natural sciences ,Kernel (linear algebra) ,Compression (functional analysis) ,Fractional partial differential equations ,Implicit-explicit method ,Matrix equations ,Product integration ,Reaction-diffusion ,Applied mathematics ,0101 mathematics ,Mathematics ,computer.programming_language ,Term (time) ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,computer - Abstract
An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper.
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- 2022
4. Stabilization of the nonconforming virtual element method
- Author
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Gianmarco Manzini, Daniele Prada, Silvia Bertoluzza, and Micol Pennacchio
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Tessellation ,Dual space ,Degrees of freedom (statistics) ,Stability (learning theory) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Bilinear form ,01 natural sciences ,010101 applied mathematics ,Set (abstract data type) ,Computational Mathematics ,Test case ,Computational Theory and Mathematics ,Modeling and Simulation ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Element (category theory) ,Mathematics - Abstract
We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the degrees of freedom. By this approach, we manage to construct different bilinear forms yielding optimal or quasi-optimal stability bounds and error estimates, under weaker assumptions on the tessellation than the ones usually considered in this framework. In particular, we prove optimality under geometrical assumptions allowing a mesh to have a very large number of arbitrarily small edges per element. Finally, we numerically assess the performance of the VEM for several different stabilizations fitting with our new framework on a set of representative test cases.
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- 2022
5. Analytical solution for arbitrary large deflection of geometrically exact beams using the homotopy analysis method
- Author
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Paul M. Weaver and Pedram Khaneh Masjedi
- Subjects
Timoshenko beam theory ,Cantilever ,Applied Mathematics ,Homotopy ,Numerical analysis ,Mathematical analysis ,02 engineering and technology ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Nonlinear system ,020303 mechanical engineering & transports ,Quadratic equation ,0203 mechanical engineering ,Modeling and Simulation ,0101 mathematics ,Homotopy analysis method ,Mathematics - Abstract
Beam-like compliant elements have found wide-ranging application in many fields of engineering and science often where 3D large deflections can be of concern such as soft robotics, DNA mechanics and helicopter/wind turbine rotor blades. The homotopy analysis method (HAM) is used for the first time to obtain a novel analytical solution in converged series form for the arbitrary large deflection of geometrically exact beams subject to both conservative and follower loading scenarios. The homotopy analysis method, which offers desirable characteristics such as being free from small or large parameters, coupled with auxiliary parameters controlling convergence, is applied directly to the intrinsic governing equations of a geometrically exact beam theory. The system of first-order differential governing equations of geometrically exact beams with intrinsic formulation is free from rotation and displacement variables, and offers a low degree of nonlinearity (quadratic at most) and compact mathematical form, making it suitable for analytical solutions. Due to the relatively poor convergence of the original HAM algorithm, the iterative HAM technique is employed which is known to accelerate convergence and to improve the computational efficiency of the homotopy analysis method. The obtained homotopy series offers a number of novel features in the context of the analytical solutions for the large deflection of beams, including (a) the direct calculation of internal forces and moments which is significant for engineering design purposes, (b) being able to capture 3D deflections, (c) considering transverse shear effects which can be important for thicker beams or when the Young’s modulus to shear modulus ratio is significant (such as composite materials) and (d) considering conservative and follower tip and distributed loads, in a unified framework. In order to investigate the efficacy, applicability and accuracy of the proposed method, a number of numerical examples are considered in which a cantilever beam subject to tip or distributed loads undergoes large deflection. Large deflection results for both conservative and follower loads are compared against those of less comprehensive analytical solutions as well as against numerical methods including finite element and Chebyshev collocation methods where good agreement is observed. These results demonstrate the applicability and effectiveness of HAM for the large deflection analysis of geometrically exact beams.
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- 2022
6. A multi-scale Gaussian beam parametrix for the wave equation: The Dirichlet boundary value problem
- Author
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Michele Berra, Maarten V. de Hoop, and José Luis Romero
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010101 applied mathematics ,Mathematics - Analysis of PDEs ,Applied Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,FOS: Mathematics ,35L05, 35L20, 35S05, 42C15 ,0101 mathematics ,01 natural sciences ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We present a construction of a multi-scale Gaussian beam parametrix for the Dirichlet boundary value problem associated with the wave equation, and study its convergence rate to the true solution in the highly oscillatory regime. The construction elaborates on the wave-atom parametrix of Bao, Qian, Ying, and Zhang and extends to a multi-scale setting the technique of Gaussian beam propagation from a boundary of Katchalov, Kurylev and Lassas., Comment: 64 pages, 7 figures, minor update
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- 2022
7. Qualitative property preservation of high-order operator splitting for the SIR model
- Author
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Raymond J. Spiteri and Siqi Wei
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Numerical Analysis ,education.field_of_study ,Correctness ,Applied Mathematics ,Population ,Qualitative property ,010103 numerical & computational mathematics ,Solver ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Operator (computer programming) ,Applied mathematics ,0101 mathematics ,Direct representation ,education ,Epidemic model ,Mathematics ,Physical quantity - Abstract
The susceptible-infected-recovered (SIR) model is perhaps the most basic epidemiological model for the evolution of disease spread within a population. Because of its direct representation of fundamental physical quantities, a true solution to an SIR model possesses a number of qualitative properties, such as conservation of the total population or positivity or monotonicity of its constituent populations, that may only be guaranteed to hold numerically under step-size restrictions on the solver. Operator-splitting methods with order greater than two require backward sub-steps in each operator, and the effects of these backward sub-steps on the step-size restrictions for guarantees of qualitative correctness of numerical solutions are not well studied. In this study, we analyze the impact of backward steps on step-size restrictions for guaranteed qualitative properties by applying third- and fourth-order operator-splitting methods to the SIR epidemic model. We find that it is possible to provide step-size restrictions that guarantee qualitative property preservation of the numerical solution despite the negative sub-steps, but care must be taken in the choice of the method. Results such as this open the door for the design and application of high-order operator-splitting methods to other mathematical models in general for which qualitative property preservation is important.
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- 2022
8. Matrix transfer technique for anomalous diffusion equation involving fractional Laplacian
- Author
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Vo Anh, Zhengmeng Jin, Fawang Liu, and Minling Zheng
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Numerical Analysis ,Discretization ,Anomalous diffusion ,Applied Mathematics ,Physical system ,010103 numerical & computational mathematics ,Operator theory ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Computational Mathematics ,Matrix (mathematics) ,Lévy flight ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
The fractional Laplacian, ( − △ ) s , s ∈ ( 0 , 1 ) , appears in a wide range of physical systems, including Levy flights, some stochastic interfaces, and theoretical physics in connection to the problem of stability of the matter. In this paper, a matrix transfer technique (MTT) is employed combining with spectral/element method to solve fractional diffusion equations involving the fractional Laplacian. The convergence of the MTT method is analyzed by the abstract operator theory. Our method can be applied to solve various fractional equation involving fractional Laplacian on some complex domains. Numerical results indicate exponential convergence in the spatial discretization which is in good agreement with the theoretical analysis.
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- 2022
9. Semiconcavity and sensitivity analysis in mean-field optimal control and applications
- Author
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Hélène Frankowska, Benoît Bonnet, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Centre National de la Recherche Scientifique (CNRS)
- Subjects
General Mathematics ,30L99, 49K27, 49K40, 49Q12, 49Q22, 58E25 ,Space (mathematics) ,01 natural sciences ,Mean-Field Optimal Control ,Value Function ,Maximum principle ,Bellman equation ,Applied mathematics ,Sensitivity Relations ,Sensitivity (control systems) ,0101 mathematics ,Pontryagin Maximum Principle ,Mathematics - Optimization and Control ,Mathematics ,Probability measure ,Applied Mathematics ,010102 general mathematics ,Optimal control ,010101 applied mathematics ,Mean field theory ,Non-smooth Analysis ,Semiconcavity ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Geometry of Wasserstein Spaces ,Interpolation - Abstract
In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems., Comment: 55 pages
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- 2022
10. A fourth-order block-centered compact difference scheme for nonlinear contaminant transport equations with adsorption
- Author
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Dong Liang, Shusen Xie, and Yilei Shi
- Subjects
Numerical Analysis ,Applied Mathematics ,Flux ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Exact solutions in general relativity ,0103 physical sciences ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Porous medium ,Conservation of mass ,Block (data storage) ,Mathematics - Abstract
Nonlinear contaminant transports through porous media are important in many scientific and engineering applications. In this paper, we develop and analyze fourth-order block-centered compact difference scheme (BCCDS) for the nonlinear contaminant transport equations with adsorption process in porous media. Based on block-centered mesh, a fourth order compact difference scheme of solution and its flux is derived. We prove the mass conservation of the proposed scheme and its unconditional stability. We analyze the convergence and obtain the fourth-order error estimate under the smooth regularity of exact solution. Numerical experiments are presented to show the performance of the schemes.
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- 2022
11. Spreading speed for some cooperative systems with nonlocal diffusion and free boundaries, part 1: Semi-wave and a threshold condition
- Author
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Wenjie Ni and Yihong Du
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Class (set theory) ,Mathematical and theoretical biology ,Series (mathematics) ,West Nile virus ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Space dimension ,medicine.disease_cause ,01 natural sciences ,010101 applied mathematics ,Traveling wave ,medicine ,0101 mathematics ,Diffusion (business) ,Epidemic model ,Analysis ,Mathematics - Abstract
We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. Such a system covers various models arising from mathematical biology, in particular a West Nile virus model and an epidemic model considered recently in [16] and [44] , respectively, where a “spreading-vanishing” dichotomy is known to govern the long time dynamical behaviour, but the question on spreading speed was left open. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and travelling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave. This is Part 1 of a two part series. In Part 2, for some typical classes of kernel functions, we will obtain sharp estimates of the spreading rate for both the finite speed case, and the infinite speed case.
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- 2022
12. Global stability of traveling waves for nonlocal time-delayed degenerate diffusion equation
- Author
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Jiaqi Yang, Changchun Liu, and Ming Mei
- Subjects
Degenerate diffusion ,Applied Mathematics ,Mathematical analysis ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,010101 applied mathematics ,Compact space ,Rate of convergence ,0103 physical sciences ,Initial value problem ,Development (differential geometry) ,0101 mathematics ,Diffusion (business) ,Degeneracy (mathematics) ,Analysis ,Mathematics - Abstract
This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Holder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L 1 -stable, which is the first frame work on L 1 -wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L 1 -weighted energy estimates combining the compactness analysis, but with some new development.
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- 2022
13. Time-optimal control problem for a linear parameter varying system with nonlinear item
- Author
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Yi An, Lei Wang, and Jiao Teng
- Subjects
Equilibrium point ,0209 industrial biotechnology ,Mathematical optimization ,Implicit function ,Computer Networks and Communications ,Computer science ,Applied Mathematics ,Process (computing) ,Particle swarm optimization ,02 engineering and technology ,Optimal control ,01 natural sciences ,Nonlinear programming ,010101 applied mathematics ,Nonlinear system ,020901 industrial engineering & automation ,Control and Systems Engineering ,Signal Processing ,0101 mathematics ,Selection (genetic algorithm) - Abstract
In this paper, we considered a time-optimal control problem for a new type of linear parameter varying (LPV) system which is obtained through data identification in the process of dealing with actual problems. The addition of non-linear terms is compensation for the method that does not require linear expansion at the equilibrium point. Since the objective function is the terminal time which is an implicit function concerning decision variables, it is a non-standard optimal control problem with uncertain terminal time. To find the global optimal solution to this problem, firstly, the control parameterization method is used to transform it into a nonlinear optimization problem of parameter selection, and then the modifed particle swarm optimization (PSO) algorithm is combined to solve the equivalent nonlinear programming problem. Numerical examples are used to illustrate the effectiveness of the proposed algorithm.
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- 2022
14. Convergence analysis of the hp-version spectral collocation method for a class of nonlinear variable-order fractional differential equations
- Author
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Xiaohua Ding, Qiang Ma, and Rian Yan
- Subjects
Numerical Analysis ,Polynomial ,Applied Mathematics ,Fixed-point theorem ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Collocation method ,Norm (mathematics) ,Initial value problem ,Applied mathematics ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
In this paper, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H 1 -norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.
- Published
- 2021
15. A continuous semiflow on a space of Lipschitz functions for a differential equation with state-dependent delay from cell biology
- Author
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Philipp Getto, Gergely Röst, and István Balázs
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0303 health sciences ,Differential equation ,Applied Mathematics ,Ode ,State (functional analysis) ,Lipschitz continuity ,01 natural sciences ,Domain (mathematical analysis) ,Cell biology ,010101 applied mathematics ,03 medical and health sciences ,Compact space ,State space ,0101 mathematics ,Invariant (mathematics) ,Analysis ,030304 developmental biology ,Mathematics - Abstract
We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.
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- 2021
16. A picture of the ODE's flow in the torus: From everywhere or almost-everywhere asymptotics to homogenization of transport equations
- Author
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Loïc Hervé and Marc Briane
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Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Absolute continuity ,Lebesgue integration ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,symbols.namesake ,Flow (mathematics) ,symbols ,Almost everywhere ,Invariant measure ,0101 mathematics ,Invariant (mathematics) ,Analysis ,Probability measure ,Mathematics - Abstract
In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.
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- 2021
17. Berry–Esseen bounds and moderate deviations for random walks on GLd(R)
- Author
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Ion Grama, Quansheng Liu, Hui Xiao, Universitat Hildesheim, Institut fur Mathematik and Angewandte Informatik, Hildesheim, Germany, Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), and Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,September 1 ,Spectral radius ,2021. 2010 Mathematics Subject Classification. Primary 60F10 ,Applied Mathematics ,010102 general mathematics ,General linear group ,16. Peace & justice ,Random walk ,01 natural sciences ,Exponential function ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010101 applied mathematics ,Combinatorics ,60J05 ,Modeling and Simulation ,Projective space ,Irreducibility ,0101 mathematics ,Secondary 60B20 ,Operator norm ,Mathematics - Abstract
Let ( g n ) n ⩾ 1 be a sequence of independent and identically distributed random elements of the general linear group G L d ( R ) , with law μ . Consider the random walk G n : = g n … g 1 . Denote respectively by ‖ G n ‖ and ρ ( G n ) the operator norm and the spectral radius of G n . For log ‖ G n ‖ and log ρ ( G n ) , we prove moderate deviation principles under exponential moment and strong irreducibility conditions on μ ; we also establish moderate deviation expansions in the normal range [ 0 , o ( n 1 / 6 ) ] and Berry–Esseen bounds under the additional proximality condition on μ . Similar results are found for the couples ( X n x , log ‖ G n ‖ ) and ( X n x , log ρ ( G n ) ) with target functions, where X n x : = G n ⋅ x is a Markov chain and x is a starting point on the projective space P ( R d ) .
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- 2021
18. The energy-preserving time high-order AVF compact finite difference scheme for nonlinear wave equations in two dimensions
- Author
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Dong Liang and Baohui Hou
- Subjects
Numerical Analysis ,business.industry ,Applied Mathematics ,Operator (physics) ,Compact finite difference ,010103 numerical & computational mathematics ,Computational fluid dynamics ,Lipschitz continuity ,7. Clean energy ,01 natural sciences ,Hamiltonian system ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Norm (mathematics) ,Applied mathematics ,Vector field ,0101 mathematics ,business ,Mathematics - Abstract
In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The L p -norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.
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- 2021
19. Multiplicity and uniqueness for Lane-Emden equations and systems with Hardy potential and measure data
- Author
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Mousomi Bhakta, Debangana Mukherjee, and Phuoc-Tai Nguyen
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Applied Mathematics ,010102 general mathematics ,Multiplicity (mathematics) ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Bounded function ,Domain (ring theory) ,Radon measure ,Boundary value problem ,Uniqueness ,0101 mathematics ,Critical exponent ,Analysis ,Mathematical physics ,Mathematics - Abstract
Let Ω be a C 2 bounded domain in R N ( N ≥ 3 ), δ ( x ) = dist ( x , ∂ Ω ) and C H ( Ω ) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , ( P ρ ) where 0 μ C H ( Ω ) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 p N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem ( P ρ ) admits a positive solution if 0 ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎ . In this paper, we go further in the study of the solution set of ( P ρ ) . We show that the problem admits at least two positive solutions if 0 ρ ρ ⁎ and a unique positive solution if ρ = ρ ⁎ . We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ.
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- 2021
20. Short proofs of refined sharp Caffarelli-Kohn-Nirenberg inequalities
- Author
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Cristian Cazacu, Nguyen Lam, and Joshua Flynn
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Pure mathematics ,Inequality ,Applied Mathematics ,media_common.quotation_subject ,010102 general mathematics ,81S07, 26D10, 46E35, 26D15 ,Mathematical proof ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,0101 mathematics ,Nirenberg and Matthaei experiment ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics ,media_common - Abstract
This note relies mainly on a refined version of the main results of the paper by F. Catrina and D. Costa (J. Differential Equations 2009). We provide very short and self-contained proofs. Our results are sharp and minimizers are obtained in suitable functional spaces. As main tools we use the so-called \textit{expand of squares} method to establish sharp weighted $L^{2}$-Caffarelli-Kohn-Nirenberg (CKN) inequalities and density arguments., Comment: 13 pages
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- 2021
21. A non-intrusive reduced-order modeling for uncertainty propagation of time-dependent problems using a B-splines Bézier elements-based method and proper orthogonal decomposition: Application to dam-break flows
- Author
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Azzedine Abdedou and Azzeddine Soulaïmani
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FOS: Computer and information sciences ,Propagation of uncertainty ,Polynomial chaos ,Artificial neural network ,Basis (linear algebra) ,Basis function ,Numerical Analysis (math.NA) ,02 engineering and technology ,Parameter space ,01 natural sciences ,Projection (linear algebra) ,Computational Engineering, Finance, and Science (cs.CE) ,010101 applied mathematics ,Computational Mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Computational Theory and Mathematics ,Flow (mathematics) ,Modeling and Simulation ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Computer Science - Computational Engineering, Finance, and Science ,Mathematics - Abstract
A proper orthogonal decomposition-based B-splines B\'ezier elements method (POD-BSBEM) is proposed as a non-intrusive reduced-order model for uncertainty propagation analysis for stochastic time-dependent problems. The method uses a two-step proper orthogonal decomposition (POD) technique to extract the reduced basis from a collection of high-fidelity solutions called snapshots. A third POD level is then applied on the data of the projection coefficients associated with the reduced basis to separate the time-dependent modes from the stochastic parametrized coefficients. These are approximated in the stochastic parameter space using B-splines basis functions defined in the corresponding B\'ezier element. The accuracy and the efficiency of the proposed method are assessed using benchmark steady-state and time-dependent problems and compared to the reduced order model-based artificial neural network (POD-ANN) and to the full-order model-based polynomial chaos expansion (Full-PCE). The POD-BSBEM is then applied to analyze the uncertainty propagation through a flood wave flow stemming from a hypothetical dam-break in a river with a complex bathymetry. The results confirm the ability of the POD-BSBEM to accurately predict the statistical moments of the output quantities of interest with a substantial speed-up for both offline and online stages compared to other techniques., Comment: 45 pages, 15 figures
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- 2021
22. Asymptotic decay of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell systems
- Author
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Shu Wang, Ming Mei, Yue-Hong Feng, and Xin Li
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Electromagnetic field ,Isentropic process ,Applied Mathematics ,010102 general mathematics ,Plasma ,01 natural sciences ,Magnetic field ,010101 applied mathematics ,symbols.namesake ,Maxwell's equations ,Asymptotic decay ,Compressibility ,symbols ,Initial value problem ,0101 mathematics ,Analysis ,Mathematics ,Mathematical physics - Abstract
The initial value problems of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell (CNS-M) systems arising from plasmas in R 3 are studied. The main difficulty of studying the bipolar isentropic/non-isentropic CNS-M systems lies in the appearance of the electromagnetic fields satisfying the hyperbolic Maxwell equations. The large time-decay rates of global smooth solutions with small amplitude in L q ( R 3 ) for 2 ≤ q ≤ ∞ are established. For the bipolar non-isentropic CNS-M system, the difference of velocities of two charged carriers decay at the rate ( 1 + t ) − 3 4 + 1 4 q which is faster than the rate ( 1 + t ) − 3 4 + 1 4 q ( ln ( 3 + t ) ) 1 − 2 q of the bipolar isentropic CNS-M system, meanwhile, the magnetic field decay at the rate ( 1 + t ) − 3 4 + 3 4 q ( ln ( 3 + t ) ) 1 − 2 q which is slower than the rate ( 1 + t ) − 3 4 + 3 4 q for the bipolar isentropic CNS-M system. The approach adopted is the classical energy method but with some new developments, where the techniques of choosing symmetrizers and the spectrum analysis on the linearized homogeneous system play the crucial roles.
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- 2021
23. A numerical scheme for a class of generalized Burgers' equation based on Haar wavelet nonstandard finite difference method
- Author
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Carlo Cattani, Amit K. Verma, and Mukesh Kumar Rawani
- Subjects
Numerical Analysis ,Discretization ,Applied Mathematics ,Numerical analysis ,Finite difference method ,Finite difference ,010103 numerical & computational mathematics ,01 natural sciences ,Haar wavelet ,Burgers' equation ,010101 applied mathematics ,Computational Mathematics ,Wavelet ,Collocation method ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Solving Burgers' equation always posses challenges for a small value of viscosity. Here we present a numerical method based on the Haar wavelet collocation method coupled with a nonstandard finite difference (NSFD) scheme for a class of generalized Burgers' equation. In the solution process, the time derivative is discretized by the NSFD scheme and the spatial derivatives are approximated by the Haar wavelets series. The nonlinear terms are linearized with the help of the quasilinearisation process. We illustrate the efficiency of the proposed method by solving several test problems and report their L 2 -error and L ∞ -error norms. The derived method is quite easy to implement compared to the other methods. Also, the error analysis of the current method is discussed. It is also observed that for the small number of grid points, the current method produces results that are in great agreement with the analytical solutions.
- Published
- 2021
24. A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients
- Author
-
Suayip Toprakseven
- Subjects
Convection ,Numerical Analysis ,Discretization ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Stability (probability) ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Discrete time and continuous time ,Galerkin finite element method ,Reaction–diffusion system ,Applied mathematics ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
In this paper, a weak Galerkin finite element method for solving the time fractional reaction-convection diffusion problem is proposed. We use the well known L 1 discretization in time and a weak Galerkin finite element method on uniform mesh in space. Both continuous and discrete time weak Galerkin finite element method are considered and analyzed. The stability of the discrete time scheme is proved. The error estimates for both schemes are given. Finally, we give some numerical experiments to show the efficiency of the proposed method.
- Published
- 2021
25. Asymptotic stability of homogeneous solutions of incompressible stationary Navier-Stokes equations
- Author
-
Xukai Yan and Yanyan Li
- Subjects
Unit sphere ,Work (thermodynamics) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Rotational symmetry ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Exponential stability ,Dimension (vector space) ,Stability theory ,Compressibility ,0101 mathematics ,Navier–Stokes equations ,Analysis ,Mathematics - Abstract
It was proved by Karch and Pilarczyk that Landau solutions are asymptotically stable under any L 2 -perturbation. In our earlier work with L. Li, we have classified all ( − 1 ) -homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any L 2 -perturbation.
- Published
- 2021
26. A degenerate planar piecewise linear differential system with three zones
- Author
-
Yilei Tang, Hebai Chen, and Man Jia
- Subjects
Hopf bifurcation ,Phase portrait ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Degenerate energy levels ,Bifurcation diagram ,01 natural sciences ,Nonlinear Sciences::Chaotic Dynamics ,010101 applied mathematics ,Piecewise linear function ,symbols.namesake ,Limit cycle ,symbols ,Limit (mathematics) ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Analysis ,Bifurcation ,Mathematics - Abstract
In (Euzebio et al., 2016 [10] ; Chen and Tang, 2020 [8] ), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F ( x ) − y , y ˙ = g ( x ) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincare disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve.
- Published
- 2021
27. Reduced order multirate schemes for coupled differential-algebraic systems
- Author
-
Michael Günther, Angelo Ciccazzo, and M. W. F. M. Bannenberg
- Subjects
Model order reduction ,Numerical Analysis ,Basis (linear algebra) ,Applied Mathematics ,Principle of maximum entropy ,Context (language use) ,010103 numerical & computational mathematics ,Integrated circuit ,01 natural sciences ,law.invention ,010101 applied mathematics ,Reduction (complexity) ,Computational Mathematics ,law ,Control theory ,Convergence (routing) ,0101 mathematics ,Differential (infinitesimal) ,Mathematics - Abstract
In the context of time-domain simulation of integrated circuits, one often encounters large systems of coupled differential-algebraic equations. Simulation costs of these systems can become prohibitively large as the number of components keeps increasing. In an effort to reduce these simulation costs a twofold approach is presented in this paper. We combine maximum entropy snapshot sampling method and a nonlinear model order reduction technique, with multirate time integration. The obtained model order reduction basis is applied using the Gaus-Newton method with approximated tensors reduction. This reduction framework is then integrated using a coupled-slowest-first multirate integration scheme. The convergence of this combined method is verified numerically. Lastly it is shown that the new method results in a reduction of the computational effort without significant loss of accuracy.
- Published
- 2021
28. Variable stepsize SDIMSIMs for ordinary differential equations
- Author
-
A. Jalilian, Gholamreza Hojjati, and Ali Abdi
- Subjects
Numerical Analysis ,Applied Mathematics ,65L05 ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Grid ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,General linear methods ,Rate of convergence ,Ordinary differential equation ,FOS: Mathematics ,Order (group theory) ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Variable (mathematics) ,Second derivative ,Mathematics - Abstract
Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order p and stage order q = p , some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.
- Published
- 2021
29. Superconvergence error estimate of Galerkin method for Sobolev equation with Burgers' type nonlinearity
- Author
-
Huaijun Yang
- Subjects
Numerical Analysis ,Applied Mathematics ,Bilinear interpolation ,010103 numerical & computational mathematics ,Superconvergence ,01 natural sciences ,Backward Euler method ,010101 applied mathematics ,Sobolev space ,Computational Mathematics ,Nonlinear system ,Norm (mathematics) ,Applied mathematics ,Uniqueness ,0101 mathematics ,Galerkin method ,Mathematics - Abstract
In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L ∞ ( H 1 ) -norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.
- Published
- 2021
30. Leaves decompositions in Euclidean spaces
- Author
-
Krzysztof J. Ciosmak
- Subjects
Mathematics - Differential Geometry ,Convex geometry ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Regular polygon ,Metric Geometry (math.MG) ,Context (language use) ,Primary 52A20, 52A40, 28A50, 51F99, Secondary 52A22, 60D05, 49Q20 ,16. Peace & justice ,Isometry (Riemannian geometry) ,01 natural sciences ,Measure (mathematics) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Combinatorics ,Differential Geometry (math.DG) ,Mathematics - Metric Geometry ,FOS: Mathematics ,Partition (number theory) ,0101 mathematics ,Mathematics - Abstract
We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$, $m\leq n$, we define and prove the existence of a partition of $\mathbb{R}^n$, up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension $m$, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag., accepted in Journal de Math\'ematiques Pures et Appliqu\'ees; the present preprint is formed from arXiv:1905.02182, which has been split; 28 pages
- Published
- 2021
31. The double absorbing boundary method for the Helmholtz equation
- Author
-
Symeon Papadimitropoulos and Dan Givoli
- Subjects
Numerical Analysis ,Helmholtz equation ,Discretization ,Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,Domain (mathematical analysis) ,010101 applied mathematics ,Computational Mathematics ,0101 mathematics ,Mathematics - Abstract
The Double Absorbing Boundary (DAB) is a recently proposed absorbing layer used to truncate an unbounded domain with high-order accuracy. While it was originally designed for time-dependent acoustics and elastodynamics, here the DAB construction is adapted and applied to the 2D Helmholtz equation. Both wave-guide and corner configurations are considered. A high-order spectral finite element scheme is used in order to match the discretization accuracy to the accuracy of the DAB. The DAB scheme is analyzed, and numerical experiments demonstrate its performance.
- Published
- 2021
32. Fractional-order Boubaker wavelets method for solving fractional Riccati differential equations
- Author
-
Mohsen Razzaghi and Kobra Rabiei
- Subjects
Numerical Analysis ,Differential equation ,Iterative method ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Algebraic equation ,Operator (computer programming) ,Wavelet ,Collocation method ,Applied mathematics ,Effective method ,0101 mathematics ,Hypergeometric function ,Mathematics - Abstract
We give an effective method for solving fractional Riccati differential equations. We first define the fractional-order Boubaker wavelets (FOBW). Using the hypergeometric functions, we determine the exact values for the Riemann-Liouville fractional integral operator of the FOBW. The properties of FOBW, the exact formula, and the collocation method are used to transform the problem of solving fractional Riccati differential equations to the solution of a set of algebraic equations. These equations are solved via Newton's iterative method. The error estimation for the present method is also determined. The performance of the developed numerical schemes is assessed through several examples. This method yields very accurate results. The given numerical examples support this claim.
- Published
- 2021
33. Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems
- Author
-
Shaohong Du, Runchang Lin, and Zhimin Zhang
- Subjects
Numerical Analysis ,Singular perturbation ,Applied Mathematics ,65N15, 65N30, 65J15 ,Petrov–Galerkin method ,Degrees of freedom (statistics) ,Estimator ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Projection (linear algebra) ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,FOS: Mathematics ,A priori and a posteriori ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Dual norm ,Mathematics - Abstract
In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for singularly perturbed convection-diffusion-reaction equations in a new dual norm presented in [Du and Zhang, J. Sci. Comput. (2015)]. The flux is recovered by either local averaging in conforming $H({\rm div})$ spaces or weighted global $L^2$ projection onto conforming $H({\rm div})$ spaces. We further introduce a recovery stabilization procedure, and develop completely robust a posteriori error estimators with respect to the singular perturbation parameter $\varepsilon$. Numerical experiments are reported to support the theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in $\varepsilon$., 20 pages, 14 figures
- Published
- 2021
34. Blow-up criteria for the classical Keller-Segel model of chemotaxis in higher dimensions
- Author
-
Yūki Naito
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Chemotaxis ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Argument ,Applied mathematics ,Initial value problem ,0101 mathematics ,Finite time ,Analysis ,Mathematics - Abstract
We study the simplest parabolic-elliptic model of chemotaxis in space dimensions N ≥ 3 , and show the optimal conditions on the initial data for the finite time blow-up and the global existence of solutions in terms of stationary solutions. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution.
- Published
- 2021
35. Mixed Fourier Legendre spectral Galerkin methods for two-dimensional Fredholm integral equations of the second kind
- Author
-
Bijaya Laxmi Panigrahi
- Subjects
Numerical Analysis ,Applied Mathematics ,Banach space ,010103 numerical & computational mathematics ,01 natural sciences ,Integral equation ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Fourier transform ,Exact solutions in general relativity ,Iterated function ,Kernel (statistics) ,symbols ,Applied mathematics ,0101 mathematics ,Legendre polynomials ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.
- Published
- 2021
36. On mean sensitive tuples
- Author
-
Jie Li and Tao Yu
- Subjects
Applied Mathematics ,010102 general mathematics ,Equicontinuity ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Integer ,Equivalence relation ,Ergodic theory ,0101 mathematics ,Tuple ,Invariant (mathematics) ,Dynamical system (definition) ,Analysis ,Mixing (physics) ,Mathematics - Abstract
In this paper we introduce and study several mean forms of sensitive tuples. It is shown that the topological or measure-theoretical entropy tuples are correspondingly mean sensitive tuples under certain conditions (minimal in the topological setting or ergodic in the measure-theoretical setting). Characterizations of the question when every non-diagonal tuple is mean sensitive are presented. Among other results we show that under minimality assumption a topological dynamical system is weakly mixing if and only if every non-diagonal tuple is mean sensitive and so as a consequence every minimal weakly mixing topological dynamical system is mean n-sensitive for any integer n ≥ 2 . Moreover, the notion of weakly sensitive in the mean tuple is introduced and it turns out that this property has some special lift property. As an application we obtain that the maximal mean equicontinuous factor for any topological dynamical system can be induced by the smallest closed invariant equivalence relation containing all weakly sensitive in the mean pairs.
- Published
- 2021
37. Convergence of the method of reflections for particle suspensions in Stokes flows
- Author
-
Richard M. Höfer
- Subjects
010101 applied mathematics ,Applied Mathematics ,010102 general mathematics ,Volume fraction ,Mathematical analysis ,Convergence (routing) ,Particle ,Boundary value problem ,0101 mathematics ,Suspension (vehicle) ,01 natural sciences ,Analysis ,Mathematics - Abstract
We study the convergence of the method of reflections for the Stokes equations in domains perforated by countably many spherical particles with boundary conditions typical for the suspension of rigid particles. We prove that a relaxed version of the method is always convergent in H ˙ 1 under a mild separation condition on the particles. Moreover, we prove optimal convergence rates of the method in W ˙ 1 , q , 1 q ∞ and in L ∞ in terms of the particle volume fraction under a stronger separation condition of the particles.
- Published
- 2021
38. α-Robust H1-norm convergence analysis of ADI scheme for two-dimensional time-fractional diffusion equation
- Author
-
Hu Chen, Tao Sun, and Yue Wang
- Subjects
Numerical Analysis ,Diffusion equation ,Initial singularity ,Applied Mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,Computational Mathematics ,Scheme (mathematics) ,Norm (mathematics) ,Gronwall's inequality ,Convergence (routing) ,Order (group theory) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.
- Published
- 2021
39. Iterative algorithms for discrete-time periodic Sylvester matrix equations and its application in antilinear periodic system
- Author
-
Wenli Wang and Caiqin Song
- Subjects
Periodic system ,Sylvester matrix ,Numerical Analysis ,Periodic matrix ,Iterative method ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Matrix (mathematics) ,Discrete time and continuous time ,Gradient based algorithm ,Convergence (routing) ,0101 mathematics ,Algorithm ,Mathematics - Abstract
This paper is dedicated to solving the iterative solution to the discrete-time periodic Sylvester matrix equations. Inspired by Jacobi iterative algorithm and hierarchical identification principle, the Jacobi gradient based iterative (JGI) algorithm and the accelerated Jacobi gradient based iterative (AJGI) algorithm are proposed. It is verified that the proposed algorithms are convergent for any initial matrix when the parameter factor μ satisfies certain condition. The necessary and sufficient conditions are provided for the presented new algorithms. Moreover, we also apply the JGI algorithm and AJGI algorithm to study a more generalized discrete-time periodic matrix equations and give the convergence conditions of the algorithms. Finally, two numerical examples are given to illustrate the effectiveness, accuracy and superiority of the proposed algorithms.
- Published
- 2021
40. Global well-posedness of 2D chemotaxis Euler fluid systems
- Author
-
Chongsheng Cao and Hao Kang
- Subjects
Applied Mathematics ,010102 general mathematics ,Dynamics (mechanics) ,Chemotaxis ,01 natural sciences ,Quantitative Biology::Cell Behavior ,Physics::Fluid Dynamics ,010101 applied mathematics ,Coupling (physics) ,symbols.namesake ,Inviscid flow ,Euler's formula ,symbols ,Applied mathematics ,Incompressible euler equations ,Sensitivity (control systems) ,0101 mathematics ,Analysis ,Well posedness ,Mathematics - Abstract
In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity.
- Published
- 2021
41. Ghost-point based radial basis function collocation methods with variable shape parameters
- Author
-
Shin-Ruei Lin, Chuin-Shan Chen, and D.L. Young
- Subjects
Partial differential equation ,Applied Mathematics ,General Engineering ,Boundary (topology) ,02 engineering and technology ,Interval (mathematics) ,Collocation (remote sensing) ,01 natural sciences ,Shape parameter ,010101 applied mathematics ,Computational Mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Applied mathematics ,Radial basis function ,Point (geometry) ,0101 mathematics ,Analysis ,Mathematics ,Variable (mathematics) - Abstract
In this study, a strategy was proposed to determine the interval of the variable shape parameter for the ghost point method using radial basis functions. The determination of a suitable interval for the variable shape parameter remains a challenge. The modified Franke formula was used as an initial predictor of the center of the interval of the variable shape parameter in this study. After extensive tests, a numerical procedure was found for the determination of a suitable interval. The improvement from the imposition of the partial differential equation on the boundary points using the ghost point method was also investigated. To demonstrate the effectiveness of the proposed approach, four numerical examples are presented, including second and fourth order partial differential equations in 2D and 3D.
- Published
- 2021
42. Stability of a class of problems for time-space fractional pseudo-parabolic equation with datum measured at terminal time
- Author
-
Zakia Hammouch, Vo Viet Tri, Tran Bao Ngoc, and Nguyen Huu Can
- Subjects
Numerical Analysis ,Class (set theory) ,Terminal time ,Laplace transform ,Applied Mathematics ,Mathematical analysis ,Geodetic datum ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Terminal value ,Computational Mathematics ,Order (group theory) ,0101 mathematics ,Mathematics ,Resolvent - Abstract
We consider terminal value problems for time-space fractional pseudo-parabolic equation subjected to a final/terminal value condition. In fact, fractional orders α , s are not exactly known in modeling. These are determined experimentally. The main purpose is to investigate the continuity of the solution with respect to the fractional order α ∈ ( 0 , 1 ) , which accordingly answer the question: does ρ α n → ρ α in an appropriate sense as α n → α ? Firstly, a formulation for integral solutions has been established, which based on Laplace transform and spectral expansion of the Mittag-Leffler operators. Then, the desired continuity will be obtained by making use of resolvent representations of the Mittag-Leffler operators on Hankel's contour. Finally, we present some numerical examples to illustrate the proposed theory.
- Published
- 2021
43. The Cahn–Hilliard equation with a nonlinear source term
- Author
-
Alain Miranville
- Subjects
Logarithm ,Applied Mathematics ,Weak solution ,010102 general mathematics ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Nonlinear system ,Scheme (mathematics) ,Applied mathematics ,0101 mathematics ,Finite time ,Cahn–Hilliard equation ,Analysis ,Mathematics - Abstract
Our aim in this paper is to prove the existence of solutions to the Cahn–Hilliard equation with a general nonlinear source term. An essential difficulty is to obtain a global in time solution. Indeed, due to the presence of the source term, one cannot exclude the possibility of blow up in finite time when considering regular nonlinear terms and when considering an approximated scheme. Considering instead logarithmic nonlinear terms, we give sufficient conditions on the source term which ensure the existence of a global in time weak solution. These conditions are satisfied by several important models and applications which can be found in the literature.
- Published
- 2021
44. Improved geometric modeling using the method of fundamental solutions
- Author
-
Huiqing Zhu, Ching-Shyang Chen, Kwesi Acheampong, and Lionel Amuzu
- Subjects
Partial differential equation ,Computer science ,Applied Mathematics ,Numerical analysis ,General Engineering ,Boundary (topology) ,02 engineering and technology ,Iterative reconstruction ,01 natural sciences ,010101 applied mathematics ,Computer graphics ,Computational Mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Simple (abstract algebra) ,Method of fundamental solutions ,Applied mathematics ,0101 mathematics ,Geometric modeling ,Analysis - Abstract
In this paper, we propose a new geometric model that includes a fourth-order partial differential equation (PDE) for reconstructing 2D curves. For instance, we use this model to reproduce letters in Time Roman font. The method of fundamental solutions (MFS), which is a simple and easily implemented meshless method, is employed for solving the proposed PDEs. In addition, no fictitious boundary is required for the proposed MFS formulation, which further simplifies the implementation of the numerical method. Three examples of 2D curve reproduction are presented to demonstrate the effectiveness of the proposed model.
- Published
- 2021
45. A nonlocal strain gradient isogeometric nonlinear analysis of nanoporous metal foam plates
- Author
-
Chien H. Thai, P. Phung-Van, Hung Nguyen-Xuan, and António Ferreira
- Subjects
Length scale ,Materials science ,Nanoporous ,Applied Mathematics ,General Engineering ,02 engineering and technology ,Metal foam ,Mechanics ,Isogeometric analysis ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Deflection (engineering) ,Plate theory ,Virtual work ,0101 mathematics ,Analysis - Abstract
We investigate the nonlinear bending behavior of nanoporous metal foam plates within the framework of isogeometric analysis (IGA) and higher-order plate theory. The nonlocal strain gradient theory (NSGT) taking into account the length scale and nonlocal parameters has been adopted to establish a scale dependent model of metal foam nanoscale plates. Von Karman nonlinear strains are then used to take up the geometric nonlinearity. Different pore dispersions, namely uniform, symmetric and asymmetric, are confirmed. By using the principle of virtual work, nonlinear governing equations are derived and then solved by using an isogeometric analysis and iterative Newton-Raphson method. Influences of the length scale parameter, porosity distributions, nonlocal parameter and nanoporous coefficient on the nonlinear deflection of the plate are numerically experimented in detail. Some findings would play an important role for designing metal foam structures.
- Published
- 2021
46. Non-cooperative finite element games
- Author
-
Dong Liu, Liang Chen, Danny Smyl, and Li Lai
- Subjects
TheoryofComputation_MISCELLANEOUS ,Computer Science::Computer Science and Game Theory ,Numerical Analysis ,Work (thermodynamics) ,Discretization ,Applied Mathematics ,Linear elasticity ,ComputingMilieux_PERSONALCOMPUTING ,TheoryofComputation_GENERAL ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Nash equilibrium ,symbols ,Applied mathematics ,0101 mathematics ,Element (category theory) ,Game theory ,Mathematics - Abstract
This work proposes an approach to using the non-cooperative game theory for solving finite element problems. For this, the concept of generalized Nash equilibrium is applied to finite elements implying that each element is treated as a non-cooperative “player” in a larger finite element “game”. The aim of the approach is for all players to reach a Nash equilibrium ensuring that the entire discretization is at a minimum with respect to the decision variables considered. The approach is numerically demonstrated by investigating a nonlinear elasticity problem formulated as a finite element game. It is shown that the approach matches analytical solutions in linear elasticity and is convergent to a prescribed precision for two-player nonlinear problems.
- Published
- 2021
47. Topology optimization in fluid mechanics using continuous adjoint and the cut-cell method
- Author
-
K. D. Samouchos, Panayiotis Yiannis Vrionis, and Kyriakos C. Giannakoglou
- Subjects
Topology optimization ,Fluid mechanics ,010103 numerical & computational mathematics ,01 natural sciences ,law.invention ,Physics::Fluid Dynamics ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Incompressible flow ,law ,Modeling and Simulation ,Applied mathematics ,Polygon mesh ,Cartesian coordinate system ,Sensitivity (control systems) ,Boundary value problem ,0101 mathematics ,Topology (chemistry) ,Mathematics - Abstract
A topology optimization method for steady-state flows of incompressible fluids which is capable of imposing accurate boundary conditions along the solid walls of the sought fluid paths is presented. In each topology optimization cycle, body-conforming Cartesian meshes are generated around shapes of any complexity by tracing the fluid-solid interfaces and a cut-cell flow solver is implemented together with its adjoint. Sensitivity derivatives are computed along the fluid-solid interfaces via the continuous adjoint method. Changes in topology are caused by expressing the computed sensitivity derivatives w.r.t. an auxiliary background material distribution, that helps updating the fluid-solid interfaces. The proposed method performance is assessed on three 2D benchmark examples and a 3D case. Two out of the three 2D examples are also solved using a porosity-based topology optimization approach in which impermeable regions are penalized by a Brinkman term and useful conclusions are drawn. For a fair comparison, designs optimized using the porosity-based method are re-evaluated after extracting fluid-solid interfaces from the computed porosity fields.
- Published
- 2021
48. Stable periodic orbits for the Mackey–Glass equation
- Author
-
Alexandra Vígh, Ferenc Bartha, and Tibor Krisztin
- Subjects
010101 applied mathematics ,Applied Mathematics ,Stability theory ,010102 general mathematics ,Periodic orbits ,Delay differential equation ,Limiting ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics ,Mathematical physics - Abstract
We study the classical Mackey–Glass delay differential equation x ′ ( t ) = − a x ( t ) + b f n ( x ( t − 1 ) ) where a , b , n are positive reals, and f n ( ξ ) = ξ / [ 1 + ξ n ] for ξ ≥ 0 . As a limiting ( n → ∞ ) case we also consider the discontinuous equation x ′ ( t ) = − a x ( t ) + b f ( x ( t − 1 ) ) where f ( ξ ) = ξ for ξ ∈ [ 0 , 1 ) , f ( 1 ) = 1 / 2 , and f ( ξ ) = 0 for ξ > 1 . First, for certain parameter values b > a > 0 , an orbitally asymptotically stable periodic orbit is constructed for the discontinuous equation. Then it is shown that for large values of n, and with the same parameters a , b , the Mackey–Glass equation also has an orbitally asymptotically stable periodic orbit near to the periodic orbit of the discontinuous equation. Although the obtained periodic orbits are stable, their projections R ∋ t ↦ ( x ( t ) , ( x ( t − 1 ) ) ) ∈ R 2 can be complicated.
- Published
- 2021
49. A fast multipole boundary element method based on higher order elements for analyzing 2-D elastostatic problems
- Author
-
Hu Zongjun, Niu Zhong-rong, Hu Bin, and Li Cong
- Subjects
Applied Mathematics ,General Engineering ,02 engineering and technology ,Singular integral ,Elasticity (physics) ,System of linear equations ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,020303 mechanical engineering & transports ,Quadratic equation ,0203 mechanical engineering ,symbols ,Gaussian quadrature ,Applied mathematics ,0101 mathematics ,Multipole expansion ,Constant (mathematics) ,Boundary element method ,Analysis ,Mathematics - Abstract
A new fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D elastostatic problems by using linear and three-node quadratic elements. The use of higher-order elements in BEM analysis results in more complex forms of the integrands, in which the direct Gaussian quadrature is difficult to calculate the singular and nearly singular integrals. Herein, the complex notation is first introduced to simplify all integral formulations (including the near-field integrals) in FM-BEM for 2-D elasticity. In direct evaluation of the near-field integrals, the nearly singular integrals on linear elements are calculated by the analytic scheme, and those on quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the present method possesses higher accuracy than the FM-BEM with constant elements. The computed efficiency of FM-BEM with higher order elements for analyzing large scale problems is still O(N), where N is the number of linear system of equations. In particular, the proposed FM-BEM is available for solving thin structures.
- Published
- 2021
50. Gradient Hölder regularity for parabolic normalized p(x,t)-Laplace equation
- Author
-
Chao Zhang and Yuzhou Fang
- Subjects
Laplace's equation ,Spacetime ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Differential game ,FOS: Mathematics ,0101 mathematics ,Viscosity solution ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We consider the interior Holder regularity of spatial gradient of viscosity solution to the parabolic normalized p ( x , t ) -Laplace equation u t = ( δ i j + ( p ( x , t ) − 2 ) u i u j | D u | 2 ) u i j with some suitable assumptions on p ( x , t ) , which arises naturally from a two-player zero-sum stochastic differential game with probabilities depending on space and time.
- Published
- 2021
Catalog
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