Ali Shokri

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.

A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions

In this paper, we propose a numerical scheme to solve the two-dimensional (2D) time-dependent Schrodinger equation using collocation points and approximating the solution using multiquadrics (MQ) and the Thin Plate Splines (TPS) Radial Basis Function (RBF). The scheme works in a similar fashion as finite-difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.

A Not-a-Knot meshless method using radial basis functions and predictor-corrector scheme to the numerical solution of improved Boussinesq equation

Abstract A numerical simulation of the improved Boussinesq (IBq) equation is obtained using collocation and approximating the solution by radial basis functions (RBFs) based on the third-order time discretization. To avoid solving the nonlinear system, a predictor–corrector scheme is proposed and the Not-a-Knot method is used to improve the accuracy in the boundary. The method is tested on two problems taken from the literature: propagation of a solitary wave and interaction of two solitary waves. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the new scheme presented in this paper.

A MESHLESS METHOD FOR NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL WAVE EQUATION WITH AN INTEGRAL CONDITION USING RADIAL BASIS FUNCTIONS

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional hyperbolic equation that combines classical and integral boundary conditions using collocation points and approximating the solution using radial basis functions (RBFs). The results of numerical experiments are presented, and are compared with analytical solution and finite difference method to confirm the validity and applicability of the presented scheme.

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

Abstract In this article, a high-order compact alternating direction implicit (HOC-ADI) finite difference scheme is applied to numerical solution of the complex Ginzburg–Landau (GL) equation in two spatial dimensions with periodical boundary conditions. The GL equation has been used as a mathematical model for various pattern formation systems in mechanics, physics, and chemistry. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. To avoid solving the nonlinear system and to increase the accuracy and efficiency of the method, we proposed the predictor–corrector scheme. Validation of the present numerical solutions has been conducted by comparing with the exact and other methods results and evidenced a good agreement.

On the first-and second-order strongly monotone dynamical systems and minimization problems

Motivated by application to Tikhonov regularization in convex minimization where the objective functions are strongly convex, we study the rate of convergence of the first- and second-order evolution equations associated with a maximal strongly monotone operator on a real Hilbert space. We show that the convergence rate of solutions to the second-order evolution equation of monotone type to a zero of the monotone operator (or a minimum point of a convex function) is faster than the first-order one when the maximal monotone operator is strongly monotone. Since the bounded solutions of the second-order evolution equation on the half line are not directly computable, we have used the computation of solutions to the corresponding second-order boundary value problem of monotone type. By using the numerical methods and approximation of solutions to the second-order boundary value problem associated with a gradient of a convex function with at least a minimum point, we approximate a minimum of the convex function. Finally, with a simple concrete example a comparison between this method and the steepest descent method is presented.