RBFs Based Meshfree Methods for Simulation of Some Parabolic Partial Differential Equations

Abstract

This study covers the analysis and computations of some parabolic partial differ ential equations (PDEs) with the help of radial basis functions based differential quadrature method. Parabolic PDEs contains the time dependent process, includ ing heat conduction equation, Black–Scholes equation, diffusion equation, nonlinear acoustics, traffic flow, shallow-water waves and financial Mathematics. Some linear PDEshave the analytic solution present in the literature. The non-linear PDEs plays an important role to explore the model perfectly. The analysis and computational work of PDEs with non-linear terms is a tough job. Therefore, approximate solution is one of the best way to explore the model. In this work, we consider some parabolic non-linear PDEs and try to approximate their solutions using meshfree radial basis functions based differential quadrature method and also establish stability analysis of proposed method. A soliton type analytic solution is also derived via tanh-coth method. Chapter 1 includes the basis concepts regarding differential quadrature method (DQM) and radial basis functions (RBFs). A detailed literature survey related to these methods is presented. In chapter 2, we consider Burgers’ types problems. Two meshfree algorithms based on multiquadric radial basis functions (RBFs) and differential quadrature (DQ) tech niques are developed for numerical simulation and to capture the shock wave be havior of Burgers’ types problems. In the development of algorithms, global and local radial basis functions with DQ method are used. The algorithms convert the v vi problems into respective system of ordinary differential equations which are solved by explicit Runge-Kutta method. The stability of each algorithms is discussed with eigenvalue method. Numerical results is done to check the efficiency of the algo rithms and some shock wave behaviors of the problems are presented. The proposed algorithms are found to be accurate, simple and fast. In chapter 3, a general reaction diffusion Brusselator model is considered. The main focus of this chapter is to capture the patterns of reaction-diffusion Brusse lator model arising in chemical processes such as enzymatic reaction, formation of turing patterns on animal skin, formation of ozone by atomic oxygen through a triple collision. For this purpose, a meshfree algorithm is developed based on ra dial basis multiquadratic functions and differential quadrature (DQ) technique. The algorithm is more general than algorithms presented in literature due to meshfree and C∞ properties of radial basis functions. Numerical results section support the accuracy and efficiency of the algorithm. The computed results satisfy the theory of Brusselator model which says for small values of diffusion coefficient, the steady state solution converges to equilibrium point (α, β α) if 1 − β + α2 > 0. In chapter 4, we consider non-linear Schro¨dinger equation. In this work, we pro posed a meshfree approach for simulation of non-linear Schro¨dinger equation with constant and variable coefficients. Schro¨dinger equation is classical field equation whose principal applications are to the propagation of light in nonlinear optical f ibers and planar waveguides and in in Quantum mechanics. First of all, spatial derivatives are discretized by using local radial basis functions based on differential quadrature method (LRBF-DQM) and subsequently, obtained system of non-linear ordinary differential equations (ODEs) is solved by fourth order Runge-Kutta (RK 4) method. Stability analysis of the proposed approach is discussed by the matrix method. Numerical results ensure that the proposed approach is accurate and com putationally efficient. vii In chapter 5, the general form of the Regularized Long Wave (RLW) model is con sidered. The dark and bright soliton solutions of 1D and 2D regularized long wave (RLW) models have been derived. The RLW model occurred in various fields such as shallow-water waves, plasma drift waves, longitudinal dispersive waves in elastic rods, rotating flow down a tube, and the anharmonic lattice and pressure waves in liquid-gas bubble mixtures. First of all, the tanh-coth method is applied to ob tain the soliton solutions of RLW equations, and thereafter, the approximation of f inite domain interval is done by truncating the infinite domain interval. For com putational modeling of the problems, a meshfree method based on local radial basis functions (LRBFs) and differential quadrature technique (DQT) is developed. The meshfree method converts the RLW model into a system of non-linear ordinary dif ferential equations (ODEs), then the obtained system of ODEs is simulated by the Runge-Kutta method. Further, the stability of the proposed method is discussed by the matrix technique. Finally, in numerical results, some problems are considered to check the competence and chastity of the developed method. In chapter 6, the non-linear extended Fisher-Kolmogorov (EFK) model is consid ered. This chapter offers two radial basis functions (RBF) based meshfree schemes for the numerical simulation of non-linear extended Fisher-Kolmogorov model. In the development of the first scheme, first of all, time derivative is discretized by for ward finite difference and then stability and convergence of the semi-discrete model is analyzed in L2 and H2 0 spaces. After that, RBF-differential quadrature method (RBF-DQM)is applied for fully discretization. In the second numerical scheme, RBF-DQM and RK4 method is applied for spatial and fully discretization respec tively. Also, the stability of the scheme is discussed via matrix method. Some examples of the model are considered to examine the reliability and chastity of the proposed schemes and found accurate results.