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This article reports our explorations for solving second-order hyperbolic interface problems by immersed weak Galerkin (IWG) method on interface independent meshes. The method presented here uses IWG functions for the discretization in spatial variable. The study includes the Newmark algorithm which has been used extensively in applications. The stability analysis based on the energy method is presented for semi-discrete and fully-discrete schemes under some conditions on parameters of the Newmark algorithm. In this work, we carried out a convergence analysis and obtained optimal a priori error estimates in both energy and L 2 norms for the semi-discrete and fully-discrete schemes under piecewise H 2 regularity assumption in space and some conditions on parameters of the Newmark algorithm. We demonstrate that the maximal error in the L 2 -norm error over a finite time interval converges optimally as O (h 2 + τ r (γ)) , where r (γ) = 1 if γ ≠ 1 / 2 , r (1 / 2) = 2 , h and τ are the mesh size and the time step, respectively. Numerical examples are provided to confirm theoretical findings and illustrate the efficiency of the method for standing and traveling waves. [ABSTRACT FROM AUTHOR]

A numerical solution for neutral delay fractional order partial differential equations involving the Caputo fractional derivative is constructed. In line with this goal, the drift term and the time Caputo fractional derivative are discretized by a finite difference approximation. The energy method is used to investigate the rate of convergence and unconditional stability of the temporal discretization. The interpolation of moving Kriging technique is then used to approximate the space derivative, yielding a meshless numerical formulation. We conclude with some numerical experiments that validate the theoretical findings. [ABSTRACT FROM AUTHOR]

The delay PDEs are called partial functional differential equations as their unknown solutions are used in these equations as functional arguments. On the other hand, a neutral delay PDE is an especial case when the equation depends on the derivative(s) of the solution at some past stage(s). The current paper concerns to find an accurate and robust numerical solution for solving neutral delay time-space distributed-order fractional damped diffusion-wave equation based on the Galerkin meshless method. The test and trial functions for the used Galerkin method are constructed from the shape functions of reproducing kernel particle method (RKPM). For this aim, time derivative is approximated by a finite difference formula with convergence order O (τ 3 − α) where 1 < α < 2. The stability and convergence of the time-discrete formulation are studied. For the next stage and to derive a fully-discrete scheme, we employ the Galerkin RKPM method. Moreover, the error estimate of the full-discrete scheme of new technique is discussed. Finally, an example is examined to check the proposed algorithm. [ABSTRACT FROM AUTHOR]

This article is assigned to the numerical analysis of a new weak Galerkin mixed-type finite element method for the diffusion-convection-reaction problem with singular perturbation. The variational form of the considered method compared to the existing methods consists of a single variational equation, where flux is the only unknown. Hence it is sufficient to provide a weak Galerkin approximation for the flux variable, and immediately approximation of the primal unknown can be obtained by a post-processing strategy. We prove the well-posedness, stability and convergence of the weak Galerkin scheme. Eventually, some numerical experiments to support the nice efficiency of the technique and analytical results are presented. [ABSTRACT FROM AUTHOR]

Recently, finding a stable and convergent numerical procedure to simulate the fractional partial differential equations (PDEs) is one of the interesting topics. Meanwhile, the fractional advection‐diffusion equation is a challenge model numerically and analytically. This paper develops a new meshless numerical procedure to simulate the fractional Galilei invariant advection‐diffusion equation. The fractional derivative is the Riemann‐Liouville fractional derivative sense. At the first stage, a difference scheme with the second‐order accuracy has been employed to get a semi‐discrete plan. After this procedure, the unconditional stability has been investigated, analytically. At the second stage, a meshless weak form based upon the interpolating stabilized element‐free Galerkin (ISEFG) method has been used to achieve a full‐discrete scheme. As for the full‐discrete scheme, the order of convergence is O(τ2+rm+1). Two examples are studied, and simulation results are reported to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a discontinuous Galerkin method to solve the generalized Sobolev equation. In this numerical procedure, the temporal variable has been discretized by the Crank–Nicolson idea to get a time-discrete scheme with the second-order accuracy. Then, in the second stage the spatial variable has been discretized by the discontinuous Galerkin finite element method. A prior error estimate has been proposed for the semi-discrete scheme based on the spatial discretization. By applying the Crank–Nicolson idea a full-discrete scheme is driven. Furthermore, an error estimate has been proved to get the convergence order of the developed scheme. Finally, some numerical examples have been presented to show the efficiency and theoretical results of the new numerical procedure. [ABSTRACT FROM AUTHOR]

Nonlinear partial differential equations (PDEs) play an important role in the modeling of the natural phenomena as they have great significance in real-world applications. This investigation proposes a new algorithm to find the numerical solution of the non-linear extended Fisher–Kolmogorov equation. Firstly, the time variable is discretized by a second-order finite difference scheme. The rate of convergence and stability of the semi-discrete formulation are studied by the energy method. The existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail. Furthermore, the interpolating element free Galerkin approach based on the interpolation moving least-squares approximation is employed to derive a fully discrete scheme. Finally, the error estimate of the full-discrete plan is proposed and its convergence order is O (τ 2 + δ m + 1) in which τ , δ and m denote the time step, the radius of the weight function and smoothness of the exact solution of the main problem, respectively. [ABSTRACT FROM AUTHOR]

The main objective of the current paper is to propose a meshless weak form to simulate the Oldroyd equation. At the first stage, the Crank-Nicolson method has been utilized to discreet the temporal direction. At the second stage, the meshless element free Galerkin technique has been employed to approximate the spatial direction. The main equation is a non-local model as there is an integral term in the right hand side of the model. The mentioned integral term is approximated by using a difference scheme. The uniqueness and existence of the proposed numerical plan have been proved by using the Browder fixed point theorem. Furthermore, the error estimation of the developed numerical technique based on the stability and convergence order has been studied. Finally, some examples have been investigated to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The current investigation studies a numerical method to solve Volterra integral equations of the second kind arising in the single term fractional differential equations with initial conditions. The proposed method estimates the solution of the mentioned Volterra integral equations using the discrete Galerkin method based on the moving least squares (MLS) approach constructed on scattered points. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least squares polynomial fitting. We compute fractional integrals appeared in the method by a suitable integration rule based on the non-uniform composite Gauss-Legendre quadrature formula. Since the scheme does not need any background meshes, it can be identified as a meshless method. The scheme is simple and effective to solve fractional differential equations and its algorithm can be easily implemented on computers. The error bound and the convergence rate of the presented method are obtained. Finally, numerical examples are included to show the validity and efficiency of the new technique. [ABSTRACT FROM AUTHOR]

Abstract This article describes a numerical scheme to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method estimates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin method results from the numerical integration of all integrals associated with the scheme. In the current work, we employ the composite Gauss-Legendre integration rule to approximate the integrals appearing in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The algorithm of the described scheme is computationally attractive and easy to implement on computers. The error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

Abstract The main aim of this research work is to find the numerical solution based on a meshless technique for both the time-dependent Cahn–Hilliard and tumor growth partial differential equations. The temporal variable is discretized using a second-order method based on semi-implicit backward differential formula, and the stabilized term is added to the considered time discretization. Also, an adaptive time algorithm is used to reduce the number of iterations of the proposed time discretization. Besides, to approximate the spatial variables, the element-free Galerkin method is considered for both mathematical models. The result of full discrete schemes in studied models is solved via Biconjugate gradient stabilized algorithm. Some numerical simulations are reported to show the capability of the numerical scheme presented here. [ABSTRACT FROM AUTHOR]

Abstract The MHD equation has some applications in physics and engineering. The main aim of the current paper is to propose a new numerical algorithm for solving the MHD equation. At first, the temporal direction has been discretized by the Crank–Nicolson scheme. Also, the unconditional stability and convergence of the time-discrete scheme have been investigated by using the energy method. Then, an improvement of element free Galerkin (EFG) i.e. the interpolating element free Galerkin method has been employed to discrete the spatial direction. Furthermore, an error estimate is presented for the full discrete scheme based on the Crank–Nicolson scheme by using the energy method. We prove that convergence order of the numerical scheme based on the new numerical scheme is O (τ 2 + δ m). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In the present work, we apply a meshless approach based on weak form namely element-free Galerkin for finding the numerical solution of the mathematical modelling of cancer cell invasion of tissue. A semi-implicit finite difference scheme based on backward Euler is also used for estimating the temporal variable. In order to verify this method, we have compared it numerically with another meshless technique i.e. generalized moving least squares approximation. The proposed method depends on a set of scattered points on the domain of the problem. Also, this approach is based on global weak form and requires a background cell for computing the numerical integrations. The studied model consists of a system of time-dependent reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, extracellular matrix and matrix degradation enzymes. At the end of this paper, we present some numerical simulations showing the behavior of cancer cell invasion of tissue at different times using regular and Halton points as well. [ABSTRACT FROM AUTHOR]

The two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277-2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier-Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper presents a computational scheme to solve nonlinear logarithmic singular boundary integral equations. These types of integral equations arise from boundary value problems of Laplace’s equations with nonlinear Robin boundary conditions. The discrete Galerkin method together with the (inverse) multiquadric radial basis functions established on scattered points is utilized to approximate the solution. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals in the method. The proposed scheme uses a special accurate quadrature formula via the nonuniform Gauss–Legendre integration rule to compute logarithm-like singular integrals appeared in the scheme. Since the numerical method developed in the current paper does not require any mesh generations on the boundary of the domain, it is meshless and does not depend to the domain form. We also investigate the error analysis of the proposed method. Illustrative examples show the reliability and efficiency of the new scheme and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

An efficient numerical technique is proposed to solve one- and two-dimensional space fractional tempered fractional diffusion-wave equations. The space fractional is based on the Riemann–Liouville fractional derivative. At first, the temporal direction is discretized using a second-order accurate difference scheme. Then a classic Galerkin finite element is employed to obtain a full-discrete scheme. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique. [ABSTRACT FROM AUTHOR]

One meshless method based on the variational weak form is the element free Galerkin method. The element free Galerkin (EFG) method is similar to the finite element method but the test and trial functions for the EFG method are chosen from moving least squares (MLS) approximations. The shape functions of MLS approximation don't have the δ -Kronecker property thus the essential boundary conditions (Dirichlet boundary conditions) can not be applied, directly. On the other hand, there are some shape functions that have the mentioned property. One of these functions is the radial point interpolation method (RPIM). In the current paper, we employ the shape functions of RPIM as the test and trial functions. We apply the EFG method based on the RPIM (EFG-RPIM) for solving two-dimensional solute transport problems. To reduce the used CPU time, the proper orthogonal decomposition (POD) approach has been combined with the EFG-RPIM technique. Also, the unconditional stability and convergence of POD-EFG-RPIM method are proved by the energy method. Finally, some numerical results have been reported to show the efficiency and computational order of the new method. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to investigate a computational scheme for solving a class of nonlinear boundary integral equations which occurs as a reformulation of boundary value problems of Laplace's equations with nonlinear Robin boundary conditions. The method approximates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least square polynomial fitting. The discrete Galerkin method for solving boundary integral equations results from the numerical integration of all integrals appeared in the method. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule to estimate logarithm-like singular integrals. Since the proposed method is constructed on a set of scattered points, it does not require any background mesh and so we can call it as the meshless local discrete Galerkin (MLDG) method. The scheme is simple and effective to solve boundary integral equations and its algorithm can be easily implemented. We also obtain the error bound and the convergence rate of the presented method. Finally, numerical examples are included to show the validity and efficiency of the new technique and confirm the theoretical error estimates. [ABSTRACT FROM AUTHOR]

Fractional diffusion equations (FDEs) are a mathematical tool used for describing some special diffusion phenomena arising in many different applications like porous media and computational finance. In this paper, we focus on a two-dimensional space-FDE problem discretized by means of a second order finite difference scheme obtained as combination of the Crank–Nicolson scheme and the so-called weighted and shifted Grünwald formula. By fully exploiting the Toeplitz-like structure of the resulting linear system, we provide a detailed spectral analysis of the coefficient matrix at each time step, both in the case of constant and variable diffusion coefficients. Such a spectral analysis has a very crucial role, since it can be used for designing fast and robust iterative solvers. In particular, we employ the obtained spectral information to define a Galerkin multigrid method based on the classical linear interpolation as grid transfer operator and damped-Jacobi as smoother, and to prove the linear convergence rate of the corresponding two-grid method. The theoretical analysis suggests that the proposed grid transfer operator is strong enough for working also with the V-cycle method and the geometric multigrid. On this basis, we introduce two computationally favourable variants of the proposed multigrid method and we use them as preconditioners for Krylov methods. Several numerical results confirm that the resulting preconditioning strategies still keep a linear convergence rate. [ABSTRACT FROM AUTHOR]

In this research paper, we introduce a numerical approach to solve a particular type of nonlinear integro-differential equations derived from Volterra's population model. This model characterizes the growth of a biological species in a closed system and includes an integral term to consider the influence of toxin accumulation on the species, along with the conventional terms found in the logistic equation. The proposed technique estimates the solution of integro-differential equations utilizing the discrete Galerkin scheme using the moving least squares (MLS) algorithm. The locally weighted least squares polynomial fitting, known as the MLS method, is a valuable approach for approximating unknown functions. Since the offered scheme does not require any cell structures, it can be known as a meshless local discrete Galerkin method. Moreover, we obtain the error estimate of the proposed approach. The validity and efficiency of the newly developed technique are assessed over several nonlinear integro-differential equations. [ABSTRACT FROM AUTHOR]

This paper is intended to provide exponentially accurate Galerkin, Petrov–Galerkin and pseudo-spectral methods for fractional differential equations on a semi-infinite interval. We start our discussion by introducing two new non-classical Lagrange basis functions: NLBFs-1 and NLBFs-2 which are based on the two new families of the associated Laguerre polynomials: GALFs-1 and GALFs-2 obtained recently by the authors in [28] . With respect to the NLBFs-1 and NLBFs-2, two new non-classical interpolants based on the associated- Laguerre–Gauss and Laguerre–Gauss–Radau points are introduced and then fractional (pseudo-spectral) differentiation (and integration) matrices are derived. Convergence and stability of the new interpolants are proved in detail. Several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives (and integrals) of some functions. Moreover, the pseudo-spectral, Galerkin and Petrov–Galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones. Some useful comments from the numerical point of view on Galerkin and Petrov–Galerkin methods are listed at the end. [ABSTRACT FROM AUTHOR]

In the current decade, the meshless methods have been developed for solving partial differential equations. The meshless methods may be classified in two basic parts: The element-free Galerkin (EFG) method is a meshless method based on the global weak form. The test and trial functions in element-free Galerkin are shape functions of moving least squares (MLS) approximation. Also, the traditional MLS shape functions have not the δ-Kronecker property. Recently, a new class of MLS shape functions has been presented. These are well-known as the interpolating MLS (IMLS) shape functions. The IMLS shape functions have the δ-Kronecker property; thus the essential boundary conditions can be applied directly. The main aim of this paper is to combine the alternating direction implicit approach with the IEFG method. To this end, we apply the mentioned technique on the distributed order time-fractional diffusion-wave equation. For comparing the numerical results, we propose three schemes based on the trapezoidal, Simpson, and Gauss-Legendre quadrature techniques. Also, we investigate the uniqueness, existence and stability analysis of the new schemes and we obtain an error estimate for the full-discrete schemes. The time-fractional derivative has been described in Caputo's sense. Numerical examples demonstrate the theoretical results and the efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, the element-free Galerkin (EFG) meshless method and moving Kriging collocation meshless technique are applied for finding the numerical solution of a class of two-dimensional (2D) nonlinear time fractional partial differential equations.The Klein–Gordon, sine-Gordon, diffusion wave and Cattaneo equations with Neumann boundary conditionare studied. The time fractional derivative has been described in the Caputo’s sense. Firstly, we use a semi-implicit finite difference scheme of convergence order, and then for obtaining a full discrete scheme, the space derivative is discretized with the EFG and moving Kriging collocation techniques. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method, test and trial functions are moving least squares approximation (MLS) shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular, or other types of meshes. The EFG method is a global method while finite element method is a local one. The EFG method is not a truly meshless method and for integration uses a background mesh. We prove the unconditional stability and obtain an error bound for the EFG method using the energy method. Numerical examples are reported which support the theoretical results and the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The traditional element free Galerkin (EFG) approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares (MLS) approximation. In the current paper, we propose a new version of the EFG method based on the shape functions of reproducing kernel particle method (RKPM). In other words, based on the developed approach in Han and Meng (2001) the fractional Tricomi-type equation will be solved using the new technique. The fractional derivative has been introduced in the Caputo’s sense and is approximated by a finite difference plan of order O ( τ 3 − α ) , 1 < α < 2 . We use the EFG-RKPM to discrete the spatial direction. We illustrate some numerical results on non-rectangular domains. The unconditional stability and convergence of the new technique have been proved. Numerical examples display the theoretical results and the efficiency of the proposed approach. Also, the numerical results are compared with the finite element method (FEM) and EFG-MLS procedure. [ABSTRACT FROM AUTHOR]

A new adaptive moving least squares (MLS) method with variable radius of influence is presented to improve the accuracy of Meshless Local Petrov–Galerkin (MLPG) methods and to minimize the computational cost for the numerical solution of singularly perturbed boundary value problems. An error indicator based on a posteriori error estimation, accurately captures the regions of the domain with insufficient resolution and adaptively determines the new nodes location. The effectiveness of the new method is demonstrated on some singularly perturbed problems involving boundary layers. [ABSTRACT FROM AUTHOR]

Recently, because of simplicity of meshless methods, they have been employed for solving many partial differential equations. One of the meshless methods is element free Galerkin technique. The element free Galerkin is very similar to the finite element method with this difference that the test and trial spaces of EFG procedure are shape functions of moving least squares approximation. Using the mentioned shape functions, solving a problem on a complex domain is very simple. Some researchers proposed several modifications and enriched approaches for improving the element free Galerkin method that one of them is variational multiscale element free Galerkin procedure. Up to the best of authors’ knowledge, the element free Galerkin method based on the shape functions of moving least squares approximation needs more CPU time than the element free Galerkin method based on the shape functions of moving Kriging interpolation. Thus, in the current paper, we employ the variational multiscale element free Galerkin based on the shape functions of moving Kriging interpolation. Also, for reducing the CPU time of the presented numerical scheme, we use the proper orthogonal decomposition (POD) approach. Therefore, in the current paper, we propose the proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) method for solving time-dependent incompressible Navier–Stokes equation. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The element free Galerkin technique is a meshless method based on the variational weak form in which the test and trial functions are the shape functions of moving least squares approximation. Since the shape functions of moving least squares approximation do not have the Kronecker property thus the Dirichlet boundary condition can not be applied directly and also in this case obtaining an error estimate is not simple. The main aim of the current paper is to propose an error estimate for the extracted numerical scheme from the element free Galerkin method. To this end, we select the fractional cable equation with Dirichlet boundary condition. Firstly, we obtain a time-discrete scheme based on a finite difference formula with convergence order O ( τ 1 + min ⁡ { α , β } ) , then we use the meshless element free Galerkin method, to discrete the space direction and obtain a full-discrete scheme. Also, for calculating the appeared integrals over the boundary and the domain of problem the Gauss–Legendre quadrature rule has been used. In the next, we change the main problem with Dirichlet boundary condition to a new problem with Robin boundary condition. Then, we show that the new technique is unconditionally stable and convergent using the energy method. We show convergence orders of the time discrete scheme and the full discrete scheme are O ( τ 1 + min ⁡ { α , β } ) and O ( r p + 1 + τ 1 + min ⁡ { α , β } ) , respectively. So, we can say that the main aim of this paper is as follows, (1) Transferring the main problem with Dirichlet boundary condition (old problem) to a problem with Robin boundary condition (new problem), (2) Showing that with special condition (when σ → + ∞ ) the solution of the new problem is convergent to the solution of the old problem, (3) Obtaining an error estimate for the new problem. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge–Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on three-dimensional PDEs, we select some equations and system of PDEs such as Allen–Cahn, Gray–Scott, Ginzburg–Landau, Brusselator models, predator–prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under artificial smoothness assumption on the solution. In this study, we prove that IWG-FEM converges optimally in energy norm under natural smoothness assumption on solution. Furthermore, we show that IWG-FEM converges optimally in the L 2 norm which did not present in Mu and Zhang (2019) because of the artificial H 3 smoothness requirement. A series of numerical experiments are conducted and reported to verify the theoretical finding. [ABSTRACT FROM AUTHOR]

The finite element method (FEM) is one of the basic methods for solving deterministic and stochastic partial differential equations. This method is proposed in the 19 decade and after years several modifications for this well-known technique such as mortal FEM, discontinuous Galerkin FEM, extended FEM, least squares FEM, spectral FEM, mixed FEM, immersed FEM, adaptive FEM, etc. have been proposed. Also, for improving the accuracy and for considering complex domain, shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique. The name of this improved method is element free Galerkin (EFG) method which is classified in the category of meshless methods. The EFG method has been applied for solving many problems in engineering. In the current paper, we select two numerical procedures that are extracted from FEM. The discontinuous Galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity. It should be noted that many enriched ideas have been explained for improving accuracy of EFG method. In the current paper, the local discontinuous Galerkin and variational multiscale EFG methods are applied for obtaining numerical solution of two-dimensional Brusselator system with and without cross-diffusion. The Brusselator system is a theoretical model for a type of autocatalytic reaction. Up to best of authors’ knowledge, the accuracy of the obtained numerical solutions using local discontinuous Galerkin method is very related to the used time-discrete scheme. Therefore, fourth-order exponential time differencing method has been employed for discretizing the time variable. Also, to achieve a full discretization scheme, the local discontinuous Galerkin finite element method is used. Moreover, several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper we apply a high order difference scheme and Galerkin spectral technique for the numerical solution of multi-term time fractional partial differential equations. The proposed methods are based on a finite difference scheme in time. The time fractional derivatives which have been described in Caputo’s sense are approximated by a scheme of order O ( τ 3 − α ) , 1 < α < 2 and the space derivative is discretized with a fourth-order compact finite difference procedure and Galerkin spectral method. We prove the unconditional stability of the compact procedure by coefficient matrix property. The L ∞ -convergence of the compact finite difference method has been proved by the energy method. Also we obtain an error estimate for Galerkin spectral method. Numerical results are provided to verify the accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

A new numerical algorithm has been investigated for solving time fractional reaction–subdiffusion equation. The fractional derivative of the considered equation is described in the Riemann–Liouville sense. Firstly, we discrete the temporal dimension of the considered model using a finite difference scheme. A central difference scheme has been applied to discrete the first time derivative and then for discretizing the fractional integral term a difference scheme has been employed with convergence order O ( τ 1 + γ ) . Moreover, to achieve a full discretization scheme a type of meshless method has been improved that is known as element free Galerkin (EFG) method. The EFG method for integration uses a background mesh. This method is based on the Galerkin weak form in which the test and trial functions are shape functions of moving least squares (MLS) approximation. Since the shape functions of traditional MLS lack the Kronecker δ property, essential boundary conditions of a boundary value problem can not be directly computed and other methods must be employed for this issue. To this end, a new class of MLS shape functions has been applied that is called shape functions of interpolating MLS. The new shape function has the mentioned property. In the EFG method, calculating the appeared two-dimensional integrals is a basic issue. In this research work, the alternating direction implicit approach is combined with the element free Galerkin method. Then, using the new proposed method, the two-dimensional integrals on rectangular domain will be changed to simple one-dimensional integrals. We prove that the new numerical algorithm is unconditionally stable and also we obtain an error bound for the new procedure using the energy method. Numerical examples are reported which demonstrate the theoretical results and the efficiency of proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite element method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O ( τ ) . Since the shape functions of moving least squares approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed. An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this work, we design, analyze, and test the multiwavelets Galerkin method to solve the two‐dimensional Burgers equation. Using Crank–Nicolson scheme, time is discretized and a PDE is obtained for each time step. We use the multiwavelets Galerkin method for solving these PDEs. Multiwavelets Galerkin method reduces these PDEs to sparse systems of algebraic equations. The cost of this method is proportional to the number of nonzero coefficients at each time step. The results illustrate, by selecting the appropriate threshold while the number of nonzero coefficients reduces, the error will not be less than a certain amount. The L2 stability and convergence of the scheme have been investigated by the energy method. Illustrative examples are provided to verify the efficiency and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction–subdiffusion equation. Firstly, we obtain a time discrete scheme based on a finite difference scheme, then we use the meshless Galerkin method, to approximate the spatial derivatives and obtain a full discrete scheme. In the proposed scheme, some integrals appear over the boundary and the domain of problem which will be approximated using Gauss–Legendre quadrature rule. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method. We show convergence order of the time discrete scheme is O ( τ γ ) . The aim of this paper is to obtain an error estimate and to show convergence for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the meshless local Petrov–Galerkin (MLPG) method is employed to solve the 2-D time-dependent Maxwell equations. The MLPG method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces. In the current work, the moving least square reproducing kernel (MLSRK) scheme is chosen to be the trial function. The method is applied for the unsteady Maxwell equations in different media. In the local weak form, by employing the difference operator for evolution in time and simultaneously in time and space, the semi-discrete and fully discrete schemes are obtained respectively. The error estimation is discussed for both the semi-discrete and fully-discrete numerical schemes for modelling the time-dependent Maxwell equations. We show that provided that the time step size is sufficiently small, the proposed scheme yields an error of in the norm for the square of error. The new scheme is implemented and the numerical results are provided to justify our theoretical analysis. [Copyright &y& Elsevier]

In recent years, there have been extensive efforts to find the numerical methods for solving problems with interface. The main idea of this work is to introduce an efficient truly meshless method based on the weak form for interface problems. The proposed method combines the direct meshless local Petrov-Galerkin method with the visibility criterion technique to solve the interface problems. It is well-known in the classical meshless local Petrov-Galerkin method, the numerical integration of local weak form based on the moving least squares shape function is computationally expensive. The direct meshless local Petrov-Galerkin method is a newly developed modification of the meshless local Petrov-Galerkin method that any linear functional of moving least squares approximation will be only done on its basis functions. It is done by using a generalized moving least squares approximation, when approximating a test functional in terms of nodes without employing shape functions. The direct meshless local Petrov-Galerkin method can be a very attractive scheme for computer modeling and simulation of problems in engineering and sciences, as it significantly uses less computational time in comparison with the classical meshless local Petrov-Galerkin method. To create the appropriate generalized moving least squares approximation in the vicinity of an interface, we choose the visibility criterion technique that modifies the support of the weight (or kernel) function. This technique, by truncating the support of the weight function, ignores the nodes on the other side of the interface and leads to a simple and efficient computational procedure for the solution of closed interface problems. In the proposed method, the essential boundary conditions and the jump conditions are directly imposed by substituting the corresponding terms in the system of equations. Also, the Heaviside step function is applied as the test function in the weak form on the local subdomains. Some numerical tests are given including weak and strong discontinuities in the Poisson interface problem. To demonstrate the application of these problems, linearized Poisson-Boltzmann and linear elasticity problems with two phases are studied. The proposed method is compared with analytical solution and the meshless local Petrov-Galerkin method on several test problems taken from the literature. The numerical results confirm the effectiveness of the proposed method for the interface problems and also provide significant savings in computational time rather than the classical meshless local Petrov-Galerkin method. [ABSTRACT FROM AUTHOR]

In the current paper, we use backward difference and local discontinuous Galerkin (BD-LDG) methods for temporal and spatial discretization of fourth-order partial integro-differential equations (PIDEs) with memory terms containing weakly singular kernels. This work provides a stability analysis of the proposed method, and at the end, by presenting some numerical experiments, we demonstrate the stability of the resulting scheme and show numerically that the optimal convergence rate is O (h k + 1) in the discrete L 2 norm. [ABSTRACT FROM AUTHOR]

Abstract: The spectral method is proposed for the partial integro-differential equations with a weakly singular kernel on irregular domains. The space discretization is based on the nodal spectral element method using the Lagrange polynomials basis associated with the Gauss–Lobatto–Legendre quadrature nodes. Also the model is discretized in time with the Legendre spectral Galerkin method. The discretization leads to conversion of the problem to a Sylvester matrix equation which can be solved efficiently by the QZ algorithm (Gardiner et al., 1992). The convergence of the method is proven by providing a priori -error estimate. Numerical results illustrate the efficiency and spectral accuracy of the proposed method. [Copyright &y& Elsevier]

The current paper proposes a technique for the numerical solution of Burgers equations. The method is based on finite difference formula combined with the Galerkin method, which uses the interpolating scaling functions. Several test problems are given, and the numerical results are reported to show the accuracy and efficiency of the new algorithm. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper a meshfree weak-strong (MWS) form method is considered to solve the coupled equations in velocity and magnetic field for the unsteady magnetohydrodynamic flow throFor this modified estimaFor this modified estimaFor this modified estimaugh a pipe of rectangular and circular sections having arbitrary conducting walls. Computations have been performed for various Hartman numbers and wall conductivity at different time levels. The MWS method is based on applying a meshfree collocation method in strong form for interior nodes and nodes on the essential boundaries and a meshless local Petrov–Galerkin method in weak form for nodes on the natural boundary of the domain. In this paper, we employ the moving least square reproducing kernel particle approximation to construct the shape functions. The numerical results for sample problems compare very well with steady state solution and other numerical methods. [ABSTRACT FROM AUTHOR]

Abstract: In the last three decades, Sinc numerical methods have been extensively used for solving differential equations, not only because of their exponential convergence rate, but also due to their desirable behavior toward problems with singularities. This paper illustrates the application of Sinc-collocation and Sinc-Galerkin methods to the approximate solution of the two-dimensional time dependent Schrödinger equation with nonhomogeneous boundary conditions. Some numerical examples are presented and the proposed methods are compared with each other. [Copyright &y& Elsevier]

Meshless local Petrov-Galerkin (MLPG) method is discussed for solving 2D, nonlinear, elliptic p-Laplace or p-harmonic equation in this article. The problem is transferred to corresponding local boundary integral equation (LBIE) using Divergence theorem. The analyzed domain is divided into small circular sub-domains to which the LBIE is applied. To approximate the unknown physical quantities, nodal points spread over the analyzed domain and MLS approximation, are utilized. The method is a meshless method, since it does not require any background interpolation and integration cells and it dose not depend on geometry of domain. The proposed scheme is simple and computationally attractive. Applications are demonstrated through illustrative examples. [ABSTRACT FROM AUTHOR]

Abstract: During the past few years, the idea of using meshless methods for numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. The meshless local Petrov–Galerkin (MLPG) method is one of the “truly meshless” methods since it does not require any background integration cells. The integrations are carried out locally over small sub-domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. In this paper the MLPG method for numerically solving the non-linear two-dimensional sine-Gordon (SG) equation is developed. A time-stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non-linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of proposed method to deal with the unsteady non-linear problems in large domains. [Copyright &y& Elsevier]

Abstract: Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi''s numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius'' equation to the solution of a system of algebraic equations. [Copyright &y& Elsevier]