Wavelet Method for Numerical Solution of Parabolic Equations

We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions. 1. Introduction In this paper, we seek a numerical solution of linear parabolic partial differential equation (PDE) in one space dimension given by with initial condition and with boundary conditions of Dirichlet, mixed, or Neumann type, where , , and are functions of and , in general. Here we assume that for and . PDE (1) is classified as follows.(i)If and , PDE (1) is called diffusion (or heat) equation.(ii)If and , PDE (1) is called diffusion-reaction equation.(iii)If and , PDE (1) is called convection-diffusion equation.(iv)If and , PDE (1) is called convection-diffusion-reaction equation. These equations have applications in a number of fields, for example, heat transfer, fluid mechanics, ground water pollutants, financial mathematics, and so on. Several methods exist for the solution of parabolic problems, for example, [1–7]. But still there is a need for modification of the solution methodology in case of (i) Neumann and mixed boundary conditions, (ii) time-dependent boundary conditions, and (iii) time-dependent source term . The algorithm in this paper is suitable for such situations. In the present paper, we apply semidiscrete approximation for the solution of initial-boundary value problem (IBVP) governed by the PDE (1). Here we use wavelet-Galerkin (variational) method to discretize the space direction and the time variable is discretized by using various classical finite difference schemes. Wavelets in consideration here are Daubechies compactly supported wavelets [8] which are differentiable. Wavelet applications to the solution of PDE problems are relatively new. Some recent applications are [5, 6, 9–12] among many more. To discretize a PDE problem by wavelet-Galerkin method, the Galerkin bases are constructed from orthonormal bases of compactly supported wavelets. This can be done in a number of ways. If wavelets