Search

In this paper, we introduce numerical schemes and their analysis based on weak Galerkin finite element framework for solving 2‐D reaction–diffusion systems. Weak Galerkin finite element method (WGFEM) for partial differential equations relies on the concept of weak functions and weak gradients, in which differential operators are approximated by weak forms through the Green's theorem. This method allows the use of totally discontinuous functions in the approximation space. In the current work, the WGFEM solves reaction–diffusion systems to find unknown concentrations (u, v) in element interiors and boundaries in the weak Galerkin finite element space WG(P0, P0, RT0). The WGFEM is used to approximate the spatial variables and the time discretization is made by the backward Euler method. For reaction–diffusion systems, stability analysis and error bounds for semi‐discrete and fully discrete schemes are proved. Accuracy and efficiency of the proposed method successfully tested on several numerical examples and obtained results satisfy the well‐known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point. Acquired numerical results asserted the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the numerical application of radial basis functions (RBFs) approximation in the reconstruction process of well known ENO/WENO schemes. The resulted schemes are employed for approximating the viscosity solution of Hamilton–Jacobi (H–J) equations. The accuracy in the smooth area is enhanced by locally optimizing the shape parameter according to the results. It is revealed that the proposed schemes in this research prepare more accurate reconstructions and sharper solution near singularities by comparing the RBFENO/RBFWENO schemes and the classical ENO/WENO schemes for some benchmark examples. Looking at the several numerical examples in 1D, 2D and 3D illustrate that the proposed schemes in this paper perform better than the traditional ENO/WENO schemes for solving H–J equations. [ABSTRACT FROM AUTHOR]

In this work, we propose a physics-informed solution for a blending PDE model with application in image denoising and edge-preserving. The blending model contains isotropic diffusion (ID) model and the modified Perona-Malik model introduced by Catte et al. Then we solve this model applying a physics-informed approach. Solving the new blending model is done using DeepXDE, which is a deep learning library for solving differential equations, by considering the image as input. Error analysis for neural network approximations of the new PDE model is shown. Applying PSNR criteria for some images, it can be seen that the proposed method can compete with solving denoising PDE models using finite difference method (FDM). [ABSTRACT FROM AUTHOR]

In this paper, a meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions. The moving least squares reproducing kernel particle approximation is employed to construct the shape functions for spatial approximation. Also, the Caputo's time fractional derivatives are approximated by a scheme of order O( τ ), 1< α < 2. Stability and convergence of the proposed scheme are discussed. Some numerical examples are given to confirm the efficiency and reliability of the proposed method. [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 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: A generalization of moving least square reproducing kernel method is presented in this work. The moving least square reproducing kernel method is obtained by using a moving least square scheme but not in the discrete version. The resulted scheme provides a continuous basis which is able to reproduce any - order polynomial, and prepares a scheme that can approximate smooth functions with an optimal accuracy. On the other hand, considering the power of moving least square scheme in meshless approximation for the numerical solution of partial differential equations, the generalized moving least square approximation is able to approximate just in terms of node values where is an arbitrary linear operator. In this paper, a generalization of moving least square reproducing kernel method is presented which employs the generalized version of moving least square method. The method approximates a test functional, based on the values of nodes. The convergence rate of the method is measured in terms of dilation parameter of window function. The method is simpler and faster to implement than the classical ones where it does not use the shape function. Numerical tests are presented to confirm the theoretical results. The numerical results establish the efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: Interest in meshless methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in science and engineering. In this paper, we present the moving least square radial reproducing polynomial (MLSRRP) meshless method as a generalization of the moving least square reproducing kernel particle method (MLSRKPM). The proposed method is established upon the extension of the MLSRKPM basis by using the radial basis functions. Some important properties of the shape functions are discussed. An interpolation error estimate is given to assess the convergence rate of the approximation. Also, for some class of time-dependent partial differential equations, the error estimate is acquired. The efficiency of the present method is examined by several test problems. The studied method is applied to the parabolic two-dimensional transient heat conduction equation and the hyperbolic two-dimensional sine-Gordon equation which are discretized by the aid of the meshless local Petrov–Galerkin (MLPG) method. [Copyright &y& Elsevier]

In the current article, we investigate the RBF solution of second-order two-space dimensional linear hyperbolic telegraph equation. For this purpose, we use a combination of boundary knot method (BKM) and analog equation method (AEM). The BKM is a meshfree, boundary-only and integration-free technique. The BKM is an alternative to the method of fundamental solution to avoid the fictitious boundary and to deal with low accuracy, singular integration and mesh generation. Also, on the basis of the AEM, the governing operator is substituted by an equivalent nonhomogeneous linear one with known fundamental solution under the same boundary conditions. Finally, several numerical results and discussions are demonstrated to show the accuracy and efficiency of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor–corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations. [Copyright &y& Elsevier]

Abstract: This article investigates the numerical solution of the nonlinear integro-differential equations. The numerical scheme developed in the current paper is based on the moving least square method. The moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. It consists of a local weighted least square fitting, valid on a small neighborhood of a point and only based on the information provided by its closet points. Hence the method is a meshless method and does not need any background mesh or cell structures. The error analysis of the proposed method is provided. The validity and efficiency of the new method are demonstrated through several tests. [Copyright &y& Elsevier]

Abstract: In this research, we study the problem of finding the approximate solution of a class of Hamilton–Jacobi equations, namely the Eikonal equation. We employ the Legendre pseudospectral viscosity method to solve this problem. This method basically consists of adding a spectral viscosity to the equation. This spectral viscosity, which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme. Several test problems are considered and the numerical results are given to show the efficiency of the proposed method. [Copyright &y& Elsevier]

In this paper, the variational iteration method (VIM) and the Adomian decomposition method (ADM) are applied to solve the Eikonal equation. In addition, we apply the VIM using He's polynomials to find a semi-analytic solution of the studied model. The VIM is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. On the other hand, in the ADM the approximate solution is considered as an infinite series usually converging to the accurate solution. In addition, the results obtained using the two methods are presented. Comparison of the results reveals that the present methods are very efficient and convenient. Copyright© 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: A wide range of applications requires an accurate solution of a particular Hamilton–Jacobi (H–J) equation known as the Eikonal equation. In this paper, we employ the Chebyshev pseudospectral viscosity method to solve this equation. This method essentially consists of adding a spectral viscosity to the equation for high wave numbers of the numerical solution. This spectral viscosity, which is sufficiently small to retain the formal spectral accuracy is large enough to stabilize the numerical scheme. Here the method is described in detail and the numerical results for several examples are provided which reveals the efficiency of the proposed method. [Copyright &y& Elsevier]

• We consider weak Galerkin finite element method for 2D Keller-Segel chemotaxis models including the blow-up problem in square domains, two-species chemotaxis blow-up problem, chemotactic bacteria pattern formation in a liquid medium, and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue. • The weak Galerkin finite element method (WGFEM) is applied to approximate the spatial variables in the chemotaxis system to capture the rapidly growing solutions in small neighborhoods of concentration points (blow-up). • The proposed method produced numerical results that are nonnegative, oscillation-free, and suitable for solving models that have a blow-up of the cell density. • Error bounds for cell density and chemoattractant equation are obtained. • Numerical simulations for haptotaxis models are obtained for showing the process of tumor cell invasion. • For examples 6.1 and 6.2 in the Keller-Segel model CPU times are computed. (Response to comment 5) • We consider the linear instability criterion around a steady-state solution. (Response to comment 8) • We construct weak Galerkin finite element space WG(P 1 , P 0 , [RT 0 , ] 2) for solving the Keller-Segel system. The errors measured in the L. norm converge with the rateO(h 2). (Response to comment 1) In this paper, we introduce a weak Galerkin finite element method for 2D Keller-Segel chemotaxis models and closely related haptotaxis models to simulate tumor invasion into surrounding healthy tissue. The chemotaxis and haptotaxis models consist of a system of time-dependent nonlinear reaction-diffusion-taxis partial differential equations. It is well known that solutions of chemotaxis systems may blow-up in finite time in the cell density. This blow-up is a mathematical description of a cell concentration phenomenon, which mathematically results in rapidly growing solutions in small neighborhoods of concentration points or curves. Capturing such singular solutions numerically is a challenging problem. We applied the weak Galerkin finite element method (WGFEM) to approximate the spatial variables. Weak Galerkin finite element method can be considered as an extension of the standard finite element method where classical derivatives are replaced in the variational equation by the weakly defined derivatives on discontinuous weak functions. The weak derivatives of weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the WGFEM is impacted by the selection of such polynomials. We show that the proposed method produced numerical results that are nonnegative, oscillation-free and suitable for solving models that have a blow-up of the cell density. Error bounds for cell density and chemoattractant equation are obtained. The proposed method is applied to several Keller-Segel chemotaxis models and extended to the haptotaxis model for simulating the behavior of cancer cell invasion of surrounding healthy tissue. Numerical results demonstrate the accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]