A simple algorithm for numerical solution of nonlinear parabolic partial differential equations
- PDF / 3,406,161 Bytes
- 13 Pages / 595.276 x 790.866 pts Page_size
- 26 Downloads / 212 Views
ORIGINAL ARTICLE
A simple algorithm for numerical solution of nonlinear parabolic partial differential equations Sidra Saleem1 · Imran Aziz2 · Malik Zawwar Hussain1 Received: 9 March 2019 / Accepted: 5 June 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract In this paper, numerical solution of nonlinear two-dimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time derivative is approximated using finite-difference scheme, whereas space derivatives are approximated using Haar wavelet collocation method. The proposed method is developed for semilinear and quasilinear cases; however, it can easily be extended to other types of nonlinearities, as well. The proposed method is also illustrated for nonlinear heat equation and Burgers’ equation. The proposed method is implemented upon five test problems and the numerical results are shown using tables and figures. The numerical results validate the accuracy and efficiency of the proposed method. Keywords Haar wavelet · Parabolic PDE · Nonlinear heat equation · Burgers’ equation
1 Introduction Mathematical modelling of several phenomena in applied mathematics involves Partial Differential Equations (PDEs). The examples include hydrodynamics, quantum mechanics, elasticity, and electromagnetic theory. It is important to obtain analytical solutions of such PDEs; however, in most of the cases, it is not possible. In a few instances, where it is possible to obtain analytical solutions, it requires advanced mathematical techniques. Due to this reason, recently, a significant attention has been devoted to obtain the numerical solutions of such PDEs. Moreover, the availability of the modern digital computers provided further motivation to develop efficient methods for the numerical solution of PDEs. A parabolic PDE is a particular type of PDE which is used to describe a wide variety of time-dependent phenomena such as heat conduction, particle diffusion, ocean acoustic propagation, and pricing of derivative investment instruments [1]. A classical example of parabolic PDEs is * Imran Aziz [email protected] 1
Department of Mathematics, University of the Punjab, Lahore, Pakistan
Department of Mathematics, University of Peshawar, Peshawar, Pakistan
2
the diffusion equation which can be applied to problems in mass diffusion, momentum diffusion, heat diffusion, etc. The problem of finding numerical solution of parabolic PDEs attained much attention during the last 2 decades. As a result, a variety of numerical methods have been established for finding the approximate solutions of such equations. For each method, a computational algorithm is designed which is implemented in a computer software as a computer programme and this programme is run on a high-speed computer to obtain numerical results. Finite-difference method, finite-element method, finite volume method, and Fourier spectral method are more common among these methods. Reviews of numerical methods for nonlinear partial di
Data Loading...
