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In this article, we develop an iterative scheme based on the meshless methods to simulate the solution of one dimensional stochastic evolution equations using radial basis function (RBF) interpolation under the concept of Gaussian random field simulation. We use regularized Kansa collocation to approximate the mean solution at space and the time component is discretized by the global o-weighted method. Karhunen-loève expansion is employed for simulating the Gaussian random field. Statistical tools for numerical analysis are standard deviation, absolute error, and root mean square. In this work, we solve two major problems for showing the convergence, and stability of the presented method on two problems. The first problem is the semilinear stochastic evolution problem, and the second one is stochastic advection-diffusion model with different control values. [ABSTRACT FROM AUTHOR]

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]

In this paper, the distributed-order time fractional sub-diffusion equation on the bounded domains is studied by using the finite-point-type meshless method. The finite point method is a point collocation based method which is truly meshless and computationally efficient. To construct the shape functions of the finite point method, the moving least square reproducing kernel approximation is employed. Two implicit discretisation of order O ⁢ (τ) {O(\tau)} and O ⁢ (τ 1 + 1 2 ⁢ σ ) {O(\tau^{1+\frac{1}{2}\sigma})} are derived, respectively. Stability and L 2 {L^{2}} norm convergence of the obtained difference schemes are proved. Numerical examples are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

A new weak Galerkin finite element method is applied for time dependent Brusselator reaction-diffusion systems by using discrete weak gradient operators over discontinuous weak functions. In this work, we consider the lowest order weak Galerkin finite element space (P0, P0, RT0). Discrete weak gradients are defined in Raviart-Thomas space. Thus we employ this approximate space on triangular mesh for solving unknown concentrations (u, v) in Brusselator reaction-diffusion systems. Based on a weak varitional form, semi-discrete and fully-discrete weak Galerkin finite element scheme are obtained. In addition, the paper presents some numerical results to illustrate the power of proposed method. [ABSTRACT FROM AUTHOR]

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]