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The aim of this paper is to introduce a novel approach for estimating the fixed points of multi-valued non-expansive mappings using the AR-iteration scheme in the context of uniformly convex Banach spaces. We establish strong and weak convergence results, providing rigorous analytical proofs and illustrating the results with a detailed example. Our approach showcases the potential of the AR-iteration scheme in solving real-world problems, particularly in addressing two-point boundary value problems. We demonstrate the applicability and effectiveness of the AR-iteration scheme in numerical analysis and computational mathematics. Our results contribute significantly to the advancement of numerical methods in solving boundary value problems, offering new insights and directions for future research in this area. Furthermore, we provide a detailed explanation of Green’s function approach and its implications for various scientific and engineering applications, paving the way for future research in this area. [ABSTRACT FROM AUTHOR]

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for R0<1$$ {\mathcal{R}}_0<1 $$ is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results. [ABSTRACT FROM AUTHOR]

From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method. With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect. [ABSTRACT FROM AUTHOR]

This paper aims to develop an effective theoretical method for analyzing the interaction of interfacial crack and complex defect in functionally graded magnetic-electro-elastic bi-materials with exponential variation under the action of anti-plane incident SH-wave. The boundary value problem of interest is solved by Green's function method. The mechanical model of the cracks is constructed through interface conjunction and crack deviation techniques, thereby simplifying the crack problem to solving a series of the first kind of Fredholm's integral equations, from which the dynamic stress intensity factor (DSIF) at the crack tips of the left crack are expressed. The validity of the present method is verified by comparing with a single crack model in previous work. Through the analyses of numerical cases, it can be concluded that the DSIF is related with 3 factors：the geometry of complex defect and crack, the characteristics of incident wave and the inhomogeneity of materials. Comparing with dual integral equations in dealing with asymmetric defects, the method in this paper is relatively more flexible and applicable, also it comes up with a new way to study fracture problems in functionally graded magnetic-electro-elastic materials with more complex defects. [ABSTRACT FROM AUTHOR]

This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved. A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams. The proposed algorithm, named as two-parameter multiple shooting method, is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions. The impact of the axial force and additional point masses is also taken into account. Due to the fact that the method is based directly on the fourth-order ordinary differential equation, the structures do not have to be divided into many small elements to obtain an accurate enough solution, even though the geometry is very complex. To verify the proposed method, three different examples are considered, i.e., a three-segment non-prismatic beam, a prismatic column subject to non-uniformly distributed compressive loads, and a two-segment beam with an additional point mass. Numerical analyses are carried out with the software MATHEMATICA. The results are compared with the solutions computed by the commercial finite element program SOFiSTiK. Good agreement is achieved, which confirms the correctness and high effectiveness of the formulated algorithm. [ABSTRACT FROM AUTHOR]

This paper deals with the design and analysis of a robust numerical scheme based on an improvised quartic B-spline collocation (IQBSC) method for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The convergence analysis of the method is studied by means of Green's function approach. It should be pointed out that the numerical order of convergence of standard quartic B-spline collocation (SQBC) scheme for second-order boundary value problems (BVPs) is four, however, our proposed IQBSC method is shown to be sixth order convergence. To illustrate the applicability and accuracy of the method, we consider eight test problems. The obtained results are compared to those from some existing numerical schemes in order to show the advantage of present method. It is shown that the rate of convergence of present numerical scheme is higher than that of some of existing numerical methods. The CPU time of the present numerical method is provided. [ABSTRACT FROM AUTHOR]

The paper is focused on the numerical analysis of the two-point inverse stationary heat flow problem, namely the identification of the thermal load. Determination of selected parameters is possible on the basis of temperature measurements, done at specified few locations. This problem may be modelled as the optimization problem, standard numerical analysis of which requires multiple solutions of boundary value problems. However, the novel solution approach, proposed here, is based on the well-known concept of the Monte Carlo method with an appropriate random walk technique. It yields explicit stochastic relations combining computed temperatures and all load parameters. Such relations may be directly applied in most standard optimization algorithms, replacing time-consuming solutions of systems of algebraic equations. Moreover, one may construct the semi-analytical approach, in which the unknown load parameters are obtained explicitly, allowing for the elimination of sensitiveness to initial solutions or requirements for admissible load intervals. The paper is illustrated with results of several benchmark problems with simulated measurement data and various numbers of unknown load parameters. The results comparison, between standard element-free methods with selected optimization algorithms as well as the proposed Monte Carlo solution approach is presented and is especially focused on CPU times. [ABSTRACT FROM AUTHOR]

The dynamic fracture of a crack in a functionally graded one-dimensional (1D) hexagonal piezoelectric quasicrystals (PEQCs) subjected to a time-harmonic elastic SH-wave is studied by using integral transform technique and Copson method. It is assumed that the crack surface is a partially permeable boundary condition and the material properties vary continuously as an exponential function. With the help of the Fourier transform, the boundary value problem of partial differential equation describing fracture problem is formulated to three pairs of dual integral equations, which are numerically solved by Copson method. Explicit expressions for the electroelastic field including phonon and phason stresses and electric field on the crack face are determined. The dynamic intensity factors of the electroelastic field are obtained in closed form, and some special cases of obtained results are discussed. On the basis of theoretical analysis and numerical simulation of the established models, numerical analysis was then conducted to discuss crack length, gradient parameter, electric boundary condition, electric loading, incident angle, amplitude, and wave number on the fracture characteristics of material. The research of this paper will provide a theoretical basis for nondestructive testing, optimal design, and reliability analysis of materials and will enrich the research content of fracture mechanics of multi-field coupling materials. [ABSTRACT FROM AUTHOR]

• The functional relationship between the heat transport coefficients is explored. • The well-posedness of the Guyer–Krumhansl equation is proved. • The uniform stabilization property is proved. • The results are supported by a numerical demonstration. The Guyer–Krumhansl heat equation has numerous important practical applications in heat conduction problems. In recent years, it turned out that the Guyer–Krumhansl model can effectively describe the thermal behavior of macroscale heterogeneous materials. Thus, the Guyer–Krumhansl equation is a promising candidate to be the next standard model in engineering. However, to support the Guyer–Krumhansl equation's introduction into the engineering practice, its mathematical properties must be thoroughly investigated and understood. In the present paper, we show the basic structure of this particular heat equation, focusing on the differences in comparison to the Fourier heat equation obtained when (τ q , μ 2) → (0 , 0). Additionally, we prove the well-posedness of a particular, practically significant initial and boundary value problem. The stability of the solution is also investigated in the discrete space using a finite difference approach. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in finding a numerical solution of a linear BVP for a Fredholm IDE with a degenerate kernel subjected to impulsive actions. By Dzumabaev's parametrization method the original problem is reduced to a multipoint BVP for the system of Fredholm IDEs with additional parameters. For fixed parameters, the special Cauchy problem for the system of FIDEs on subintervals is obtained and by using a solution to this problem, a system of algebraic equations in parameters is constructed. An algorithm for solving the BVP and its computational implementation is developed. In the algorithm, Cauchy problems for ODEs and the calculation of definite integrals are the main auxiliary problems. By using various numerical methods for solving these auxiliary problems, the proposed algorithm can be implemented in different ways. The program codes were written to solve the problem and all calculations are performed on the Matlab 2018 software platform. [ABSTRACT FROM AUTHOR]

In this paper, a fast scheme for solving unsteady nonlinear convection diffusion problems is proposed and analyzed. At each step, we firstly isolate a nonlinear convection subproblem and a linear diffusion subproblem from the original problem by utilizing operator splitting. By Taylor expansion, we explicitly transform the nonlinear convection one into a linear problem with artificial inflow boundary conditions associated with the nonlinear flux. Then a multistep technique is provided to relax the possible stability requirement, which is due to the explicit processing of the convection problem. Since the self-adjointness and coerciveness of diffusion subproblems, there are so many preconditioned iterative solvers to get them solved with high efficiency at each time step. When using the finite element method to discretize all the resulting subproblems, the major stiffness matrices are same at each step, that is the reason why the unsteady nonlinear systems can be computed extremely fast with the present method. Finally, in order to validate the effectiveness of the present scheme, several numerical examples including the Burgers type and Buckley-Leverett type equations, are chosen as the numerical study. [ABSTRACT FROM AUTHOR]

Foundation scour can have a detrimental effect on the performance of bridge piers, inducing a significant reduction of the lateral capacity of the footing and accumulation of permanent settlement and rotation. Although the hydraulic processes responsible for foundation scour are nowadays well known, predicting their mechanical consequences is still challenging. Indeed, its impact on the failure mechanisms developing around the foundation has not been fully investigated. In this paper, numerical simulations are performed to study the vertical and lateral response of a scoured bridge pier founded on a cylindrical caisson foundation embedded in a layer of dense sand. The sand stress–strain behaviour is reproduced by employing the Severn-Trent model. The constitutive model is firstly calibrated on a set of soil element tests, including drained and undrained monotonic triaxial tests and resonant column tests. The calibration procedure is implemented considering the stress and strain nonuniformities within the samples, by simulating the laboratory tests as boundary value problems. The numerical model is then validated against the results of centrifuge tests. The results of the simulations are in good agreement with the experimental results in terms of foundation capacity and settlement accumulation. Moreover, the model can predict the effects of local and general scour. The numerical analyses also highlight the impact of scouring on the failure mechanisms, revealing that the soil resistance depends on the hydraulic scenario. [ABSTRACT FROM AUTHOR]

This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realized through some sequence of wavenumbers; (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss. This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in two dimensions) or two cubes (in three dimensions), whose sides are parallel. We show, via developments of the vector-field/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analyzing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle. [ABSTRACT FROM AUTHOR]

A simple yet effective material model was proposed by Zhao et al. (2019) and demonstrated to be capable of modeling the shape transformations of various planar and three-dimensional material samples programmed with the so-called "hard-magnetic soft materials". Based on the aforementioned material model, this paper aims to further accomplish the following two tasks. First, a detailed analysis is performed to investigate the impact of the multiplicative volumetric-distortional split of the deformation gradient tensor applied to the material magnetic energy. Through a trivial boundary value problem, the impact of the volumetric-distortional split is quantified for the strictly incompressible material and the nearly incompressible material, respectively. Second, a finite element procedure based on the three-field variational principle, or the mixed displacement-Jacobian-pressure formulation (Simo et al., 1985; Simo and Taylor, 1991), is developed for the magnetoelastic materials programmed with complex magnetic patterns. Even though the finite element formulation based on the three-field variational principle is a standard and widely adopted technique in the literature, the introduction of the multiplicative split of the deformation gradient into the magnetic energy makes the derivation of the specific finite element terms less trivial. In this work, the finite element formulation and the consistent linearization of the coupled system are derived in detail for the Newton–Raphson iterations. Through the theoretical analysis and numerical examples, the approach based on the distortional part of the deformation gradient is shown to possess computational advantages over its counterpart in the context of a relatively simple penalty method. Moreover, the convergence behaviors of the finite element simulations and the impact of the finite element spaces on the pressure oscillation are analyzed in detail. [ABSTRACT FROM AUTHOR]

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, W - Rd, d 2 f1, 2, 3g. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aau = f, a 2 (0, 1). Although matrix A 2 RN-N is sparse, symmetric and positive definite (SPD), matrix Aa is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation ra,k(t) of degree k of ta in the interval [0, 1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(N log2 N) can be improved. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear elliptic partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods. [ABSTRACT FROM AUTHOR]

In this paper, an energy-conservation scheme is derived and analysed for solving Hamiltonian wave equations subject to Neumann boundary conditions in two dimensions. The energy-conservation scheme is based on the blend of spatial discretisation by a fourth-order finite difference method and time integration by the Average Vector Field (AVF) approach. The spatial discretisation via the fourth-order finite difference leads to a particular Hamiltonian system of second-order ordinary differential equations. The conservative law of the discrete energy is established, and the stability and convergence of the semi-discrete scheme are analysed. For the time discretisation, the corresponding AVF formula is derived and applied to the particular Hamlitonian ODEs to yield an eficient energy-conservation scheme. The numerical simulation is implemented for various cases including a linear wave equation and two nonlinear sine-Gordon equations. The numerical results demonstrate the spatial accuracy and the remarkable energy-conservation behaviour of the proposed energy-conservation scheme in this paper. [ABSTRACT FROM AUTHOR]

Abstract The paper deals with refinement techniques suitable for application of finite element multigrid methods with cubic trial functions. The 27-refinement strategy by Edelsbrunner and Grayson has not been studied from computational point of view up to now. This refinement strategy is said to be dark red refinement strategy (DRRS) by analogy with the red refinement strategy in the quadratic case. A detail analysis of DRRS is an aim of this paper. The dependence of the DRRS on the numbering of vertex nodes requires an algorithm development. Freudenthal partition of the cube has been widely used by researchers for obtaining hierarchical tetrahedral triangulations. In the cubic case, the refinement of the first Sommerville tetrahedron is of considerable practical importance. Since the DRRS generates one class of similarity only with a particular numbering of nodes, we have developed a detailed algorithm for proper implementation of this numbering. The situation is much more complicated when an arbitrary tetrahedron is refined. A large volume of computations are necessary in order to be established a correct numbering of the vertex nodes. If the best numbering of nodes is not chosen for all elements in all levels, the measure of degeneracy tends to infinity when the number of levels grows up unlimited. To avoid these difficulties a new canonical refinement strategy (CRS) is obtained. The CRS is superior compared to the 27-partition technique with respect to the degeneracy measure. It generates only regular tetrahedra and canonical simplices in all levels and for all mixed domains. These two tetrahedra are the most convenient from a computational point of view. Moreover, for all mixed domains the CRS essentially reduces the number of congruence classes. The new refinement strategy does not need an algorithm for correct numbering of the vertex nodes. The DRRS generates a very complicated refinement tree and a higher measure of degeneracy in combination with the face centered partition of the cube. On the contrary the Z refinement strategy obtained by the authors is superior than the DRRS in regard to the number of congruence classes and the measure of degeneracy. Some results on boundary value problems in changing domains are discussed. The advantages of the CRS are demonstrated by solving an anisotropic diffusion problem in a shrinking domain. The behavior of discretization and truncation errors in approximate finite element solutions is illustrated by varying sequences of finite element triangulations. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of periodic solutions of fractional differential equations with periodic impulses. The first part of the paper is devoted to the uniqueness, existence and asymptotic stability results for periodic solutions of impulsive fractional differential equations with varying lower limits for standard nonlinear cases as well as for cases of weak nonlinearities, equidistant and periodically shifted impulses. We also apply our result to an impulsive fractional Lorenz system. The second part extends the study to periodic impulsive fractional differential equations with fixed lower limit. We show that in general, there are no solutions with long periodic boundary value conditions for the case of bounded nonlinearities. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate mathematically the fundamental mechanism of gas discharge. Townsend discovered that α- and γ-mechanisms are essential for a gas ionization process. Morrow, Degond, and Lucquin-Desreux derived a mathematical model taking these two mechanisms into account. This model consists of nonlinear hyperbolic, parabolic, and elliptic partial differential equations. In this paper, we establish a framework for analyzing this model rigorously, because no mathematical result has been announced. More precisely, we show the unique existence of a time-local solution to an initial-boundary value problem over an unbounded domain. The main difficulty of this problem lies in the fact that the direction of boundary characteristics of a hyperbolic equation degenerates at infinite distance. Generally speaking, the regularity of solutions to linearized hyperbolic equations with degenerate boundary characteristics may be lost near the boundary. This implies the possibility that loss on the derivatives arises at each step of the inductive scheme to solve the nonlinear problem. We first use the weighted Sobolev spaces in the construction of solutions to make clear the direction of characteristics. Furthermore, we reduce the initial-boundary value problem for the hyperbolic equation to an initial value problem by applying several extension operators. This reduction enables us to avoid the loss on the derivatives. [ABSTRACT FROM AUTHOR]

Drift rate is a very important parameter in the evolution of stock price, which has significant impact on the corresponding option pricing. This paper deals with an inverse problem of recovering the drift function by current market prices of options. Different from the usual inverse volatility problem, our mathematical model does not tend to zero at infinity, which may bring great trouble to theoretical analysis and numerical calculation. To overcome this difficulty, we use an artificial boundary and homogenization technique to transform the original problem into a homogeneous initial boundary value problem on a bounded domain. Then, based on the optimal control framework, we construct the corresponding optimization problem and strictly prove the well-posedness of the minimizer. Finally, we design an iterative algorithm to obtain the numerical solution. We give some typical examples to verify the validity of our method. [ABSTRACT FROM AUTHOR]

This paper presents a new numerical method for the solution of ordinary differential equations (ODE). The linear second-order equations considered herein are solved using operational matrices of Wang-Ball Polynomials. By the improvement of the operational matrix, the singularity of the ODE is removed, hence ensuring that a solution is obtained. In order to show the employability of the method, several problems were considered. The results indicate that the method is suitable to obtain accurate solutions. [ABSTRACT FROM AUTHOR]

• A new numerical scheme for solving p -Laplacian fractional boundary value problem. • Convergence analysis for the numerical scheme when p > 2 in p -Laplacian operator. • Convergence analysis for the numerical scheme when 1 < p < 2 in p -Laplacian operator. In this paper, we propose the numerical method for positive solutions to the m -point boundary value problems of fractional differential equations with p -Laplacian operator and prove the convergence for our scheme. The operational matrix of fractional integration, based on the accurate calculation of fractional integrals of Haar wavelet functions, is combined with the collocation method and the fixed point iterative method to introduce the function approximation and obtain the approximate solution to the given problem. Also, we present some numerical examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation lawsut+fux=0,ut=0=u0x,where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form∑tjinteraction timeσαj−σαj′αjαj′αj+αj′≤CfTot.Var.u02,where αj and αj′ are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form).The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which:• all the main ideas of the construction are presented;• all the technicalities of the proof in the general setting [8] are avoided. [ABSTRACT FROM AUTHOR]

Throughout this paper, we outline some aspects of fractional calculus in Banach spaces. Some examples are demonstrated. In our investigations, the integrals and the derivatives are understood as Pettis integrals and the corresponding derivatives. Our results here extended all previous contributions in this context and therefore are new. To encompass the full scope of our paper, we show that a weakly continuous solution of a fractional order integral equation, which is modeled off some fractional order boundary value problem (where the derivatives are taken in the usual definition of the Caputo fractional weak derivative), may not solve the problem. [ABSTRACT FROM AUTHOR]

The flow of dense water along continental slopes is considered. There is a large literature on the topic based on observations and laboratory experiments. In addition, there are many analytical and numerical studies of dense water flows. In particular, there is a sequence of numerical investigations using the dynamics of overflow mixing and entrainment (DOME) setup. In these papers, the sensitivity of the solutions to numerical parameters such as grid size and numerical viscosity coefficients and to the choices of methods and models is investigated. In earlier DOME studies, three different bottom boundary conditions and a range of vertical grid sizes are applied. In other parts of the literature on numerical studies of oceanic gravity currents, there are statements that appear to contradict choices made on bottom boundary conditions in some of the DOME papers. In the present study, we therefore address the effects of the bottom boundary condition and vertical resolution in numerical investigations of dense water cascading on a slope. The main finding of the present paper is that it is feasible to capture the bottom Ekman layer dynamics adequately and cost efficiently by using a terrain-following model system using a quadratic drag law with a drag coefficient computed to give near-bottom velocity profiles in agreement with the logarithmic law of the wall. Many studies of dense water flows are performed with a quadratic bottom drag law and a constant drag coefficient. It is shown that when using this bottom boundary condition, Ekman drainage will not be adequately represented. In other studies of gravity flow, a no-slip bottom boundary condition is applied. With no-slip and a very fine resolution near the seabed, the solutions are essentially equal to the solutions obtained with a quadratic drag law and a drag coefficient computed to produce velocity profiles matching the logarithmic law of the wall. However, with coarser resolution near the seabed, there may be a substantial artificial blocking effect when using no-slip. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. The paper concludes with the study of multi-term fractional integro-differential equations supplemented with multi-point boundary conditions. Our results are new and contribute significantly to the existing literature on the topic. [ABSTRACT FROM AUTHOR]

In this paper, the vibrational characteristics of hetero-junction carbon nanotubes (HJCNTs) are investigated using the well-known molecular mechanics approach. The paper contains two main novel parts. In the first part, the influence of connecting region location on the first five natural frequencies of perfect HJCNTs is investigated. In the second part, the effects of some common defects on fundamental frequencies are studied. The study is performed for different boundary conditions and different structures of HJCNTs. The results show that the frequencies and mode shapes are effectively influenced by changing the location of connecting region. Interestingly, it is found that the frequencies of HJCNTs can be even higher than those of their constituent CNTs for a specific location of connecting region. In the second part, it is shown that the fundamental frequency of HJCNTs decreases with introducing the defects. Furthermore, the frequency shift of defective structures with lower aspect ratios is more affected by the level of imperfections. Finally, the obtained numerical results are adopted to develop a predictive equation for calculating the fundamental frequency shift of defective straight HJCNTs based on the kind and level of defect, aspect ratio as well as the type of boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of a nonlinear first-order singularly perturbed differential equation with integral boundary condition is considered. The discrete method is generated by a backward Euler formula and the grid is obtained by equidistributing a monitor function based on arc-length. We first give a rigorous error analysis for the numerical method of this problem on a grid that is constructed adaptively from a knowledge of the exact solution. A first-order rate of convergence, independent of the perturbation parameter, is established. Then, an a posteriori error bound and the corresponding convergence result are derived for the presented numerical scheme on an adaptive grid, which is constructed adaptively from a discrete approximation of the exact solution. At last, numerical experiments are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem. Design/methodology/approach: The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh. Findings: The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature. Originality/value: A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates. [ABSTRACT FROM AUTHOR]

It is well-known that stationary localized patterns involving a periodic stripe core can undergo a process that is known as \homoclinic snaking" where patterns are added to the stripe core as a bifurcation parameter is varied. The parameter region where homoclinic snaking takes place usually occupies a small region in the bistability region between the stripes and quiescent state. Outside the homoclinic snaking region, the localized patterns invade or retreat where stripes are either added or removed from the core forming depinning fronts. It remains an open problem to carry out a numerical bifurcation analysis of depinning fronts. In this paper, we carry out a numerical bifurcation analysis of depinning of fronts near the homoclinic snaking region, involving a spatial stripe cellular pattern embedded in a quiescent state, in the two-dimensional Swift{Hohenberg equation with either a quadratic-cubic or cubic-quintic nonlinearity. We focus on depinning fronts involving stripes that are orientated either parallel, oblique, or perpendicular to the front interface, and almost planar depinning fronts. We show that invading parallel depinning fronts select both a far-field wavenumber and a propagation wavespeed, whereas retreating parallel depinning fronts come in families where the wavespeed is a function of the far-field wavenumber. Employing a far-field core decomposition, we propose a boundary value problem for the invading depinning fronts which we numerically solve and use path-following routines to trace out bifurcation diagrams. We then carry out a thorough numerical investigation of the parallel, oblique, perpendicular stripe and almost planar invasion fronts. We find that almost planar invasion fronts in the cubic-quintic Swift{Hohenberg equation bifurcate off parallel invasion fronts and co-exist close to the homoclinic snaking region. Sufficiently far from the one-dimensional homoclinic snaking region, no almost planar invasion fronts exist and we find that parallel invasion stripe fronts may regain transverse stability if they propagate above a critical speed. Finally, we show that depinning fronts shed light on the time simulations of fully localized patches of stripes on the plane. The numerical algorithms detailed have wider application to general modulated fronts in reaction-diffusion systems. [ABSTRACT FROM AUTHOR]

Trotter's theorem forms the theoretical basis of most modern molecular dynamics. In essence this theorem states that a time displacement operator (a Lie operator) constructed by exponentiating a sum of noncommuting operators can be approximated by a product of single operators provided the time interval is 'very small.' In theory 'very small' implies infinitesimally small (at which point the approximate product becomes exact), while in practical analysis a finite time interval is divided into several small subintervals or steps. It follows, therefore, that the larger the number of steps the better the approximation to the exact time displacement operator. The question therefore arises: How many steps are sufficient? For bounded operators, standard theorems are available to provide the answer. In this paper we show that a very simple combinatorial formula can be derived which allows the computation of the global differences (as a function of the number of steps) between the Taylor coefficients of the exact time displacement operator and an approximate one constructed by using a finite number of steps. The formula holds for both bounded and nonbounded operators and shows, quantitatively, what is qualitatively expected-that the error decreases with increasing number of steps. Furthermore, the formula applies irrespective of the complexity of the system, boundary conditions, or the thermodynamic ensemble employed for averaging the initial conditions. The analysis yields explicit expressions for the Taylor coefficients which are then used to compute the errors. In the case of the algorithmically based practical numerical simulations in which fixed, albeit small, steps are repeatedly applied, the rise in the number of steps does not reduce the size of the steps but increases the total time of interest. The combinatorial formula shows that, here, the errors diverge. Furthermore, this work can be used to supplement other efforts such as the use of shadow Hamiltonians where the truncation of the series expansion of the latter will produce errors in the higher order propagator moments. [ABSTRACT FROM AUTHOR]

• Stability analysis of a multi-span finite length beam resting on periodic rigid and elastic supports and compressed at its edges. • A critical force beyond which the hinged beam buckling does not depend on the number of supports is found. • For infinite beam the smallest value of critical force coincides with critical force of the hinged beam. • Critical force's analytical equations are provided for beam with clamped edges and for beam with clamped-hinged edges. • The stiffness of elastic support significantly increases the critical force values of multi-span compressed beam. The stability of a multi-span finite length beam resting on periodic rigid and elastic supports is studied in this paper. The beam is compressed by axial forces applied at the edges. The stability analysis of the compressed beam is carried out using the transfer matrix method in conjunction with Bloch-Floquet's approach. Based on analytical and numerical analysis several boundary value stability problems are considered and comparison with the infinite beam model is carry out. It is shown that the model of the periodic infinite beam is in good agreement with the results of the finite length beam with the equidistant rigid and elastically supports when the number of spans is more than six. It is also shown that elastic supports significantly increase the multi-span beam stability. [ABSTRACT FROM AUTHOR]

Current constant pressure molecular-dynamics (MD) algorithms are not consistent with the recent reformulation of the isothermal-isobaric (NpT) ensemble. The NpT ensemble partition function requires the use of a “shell” molecule to identify uniquely the volume of the system, thereby avoiding the redundant counting of configurations [e.g., G. J. M. Koper and H. Reiss, J. Phys. Chem. 100, 422 (1996); D. S. Corti, Phys. Rev. E, 64, 016128 (2001)]. So far, only the NpT Monte Carlo method has been updated to allow the system volume to be defined by a shell particle [D. S. Corti, Mol. Phys. 100, 1887 (2002)]. A shell particle has yet to be incorporated into MD simulations. The proper modification of the NpT MD algorithm is therefore the subject of this paper. Unlike Andersen’s method [H. C. Andersen, J. Chem. Phys. 72, 2384 (1980)] where a piston of unknown mass serves to control the response time of volume fluctuations, the newly proposed equations of motion impose a constant external pressure via the introduction of a shell particle of known mass. Hence, the system itself sets the time scales for pressure and volume fluctuations. The new algorithm is subject to a number of fundamentally rigorous tests to ensure that the equations of motion sample phase space correctly. We also show that the Hoover NpT algorithm [W. G. Hoover, Phys. Rev. A. 31, 1695 (1985); 34, 2499 (1986)] does sample phase correctly, but only when periodic boundary conditions are employed. [ABSTRACT FROM AUTHOR]

Thermal expansion behaviors of carbon fiber/epoxy plain woven composites were experimentally and numerically studied in this paper. The thermal strains and linear coefficient of thermal expansion (CTE) of plain woven composites were measured by a classic dilatometer. Based on the periodical displacement and temperature boundary conditions, two-scale finite element models, i.e. the micro-scale and meso-scale representative volume elements (RVEs) were established to analyze the thermal expansion behaviors of carbon fiber yarns and plain woven composites, respectively. In addition, a neat epoxy resin (NER) model with the same geometrical shape as the filled epoxy resin (FER) was presented to compare their thermal expansion differences. From the results it could be found that the glass transition temperatures of epoxy resin and carbon fiber yarns had significant effects on the thermal expansion behaviors of plain woven composites. The interlacing network of yarns could effectively restrict the FER to make further thermal expansion. The special structure effects of plain woven composites led to nonuniform distributions of thermal stress/strain, which mainly showed that the regions with higher fiber volume fraction produced higher thermal stress/strain. The analyses methods this paper presented can be also used for thermal expansion researches of other complex structure composites. [ABSTRACT FROM AUTHOR]

Isogeometric collocation has emerged as an efficient numerical technique for solving boundary value problems and is a potential alternative for Galerkin based Isogeometric methods. In this investigation, Isogeometric collocation has been proposed for the linear static bending analysis of laminated composite plates governed by Reissner–Mindlin theory. Three formulations are presented in this paper namely, standard primal formulation, mixed formulation and a locking-free primal formulation. The standard primal formulation adopts displacements and rotations as unknown field variables. Mixed formulation considers displacements, rotations and transverse shear forces as the unknown field variables. Locking-free primal formulation is a rotation-free formulation with displacements and transverse shear strains as the unknown field variables. Results for benchmark problems on bending of rectangular laminated composite plates are obtained and compared with the ones existing in the literature. The three formulations of Isogeometric collocation presented in this paper are assessed in terms of accuracy and the computational time required to assemble and solve the problem. [ABSTRACT FROM AUTHOR]

Gaussian elimination with no pivoting and block Gaussian elimination are attractive alternatives to the customary but communication intensive Gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely , that is, run into neither division by 0 nor numerical problems. Empirically, safety and numerical safety of GENP have been consistently observed in a number of papers where an input matrix was pre-processed with various structured multipliers chosen ad hoc. Our present paper provides missing formal support for this empirical observation and explains why it was elusive so far. Namely we prove that GENP is numerically unsafe for a specific class of input matrices in spite of its pre-processing with some well-known and well-tested structured multipliers, but we also prove that GENP and BGE are safe and numerically safe for the average input matrix pre-processed with any nonsingular and well-conditioned multiplier. This should embolden search for sparse and structured multipliers, and we list and test some new classes of them. We also seek randomized pre-processing that universally (that is, for all input matrices) supports (i) safe GENP and BGE with probability 1 and/or (ii) numerically safe GENP and BGE with a probability close to 1. We achieve goal (i) with a Gaussian structured multiplier and goal (ii) with a Gaussian unstructured multiplier and alternatively with Gaussian structured augmentation. We consistently confirm all these formal results with our tests of GENP for benchmark inputs. We have extended our approach to other fundamental matrix computations and keep working on further extensions. 1 Hereafter we use the acronyms GENP , BGE , and GEPP . [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe concisely the recent theoretical and numerical developments concerningabsorbing boundary conditions and perfectly matched layers for solving classical and relativistic quantum waves problems. The equations considered in this paper are the Schrödinger, Klein–Gordon and Dirac equations. [ABSTRACT FROM AUTHOR]

This paper presents an approach for co-state initialization which is a critical step in solving minimum-time low-thrust trajectory optimization problems using indirect optimal control numerical methods. Indirect methods used in determining the optimal space trajectories typically result in two-point boundary-value problems and are solved by single- or multiple-shooting numerical methods. Accurate initialization of the co-state variables facilitates the numerical convergence of iterative boundary value problem solvers. In this paper, we propose a method which exploits the trajectory generated by the so-called pseudo-equinoctial and three-dimensional finite Fourier series shape-based methods to estimate the initial values of the co-states. The performance of the approach for two interplanetary rendezvous missions from Earth to Mars and from Earth to asteroid Dionysus is compared against three other approaches which, respectively, exploit random initialization of co-states, adjoint-control transformation and a standard genetic algorithm. The results indicate that by using our proposed approach the percent of the converged cases is higher for trajectories with higher number of revolutions while the computation time is lower. These features are advantageous for broad trajectory search in the preliminary phase of mission designs. [ABSTRACT FROM AUTHOR]