Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets

doi:10.1016/j.amc.2014.08.091

Applied Mathematics and Computation

Volume 247, 15 November 2014, Pages 353-367

https://doi.org/10.1016/j.amc.2014.08.091 Get rights and content

Abstract

In this paper, a numerical method for solving convection diffusion equations is presented. The method is based upon the second kind Chebyshev wavelets approximation. The second kind Chebyshev wavelets operational matrix of integration is derived and utilized to transform the equation to a system of algebraic equations by combining collocation method. Numerical examples show that the proposed method has good efficiency and precision.

Introduction

Convection diffusion equations are widely used for modeling and simulations of various complex phenomena in science and engineering, such as migration of contaminants in a stream, smoke plume in atmosphere, dispersion of chemicals in reactors, tracer dispersion in a porous medium, etc. Since it is impossible to solve convection diffusion equations analytically for most application problems, efficient numerical algorithms are becoming increasingly important to numerical simulations involving convection diffusion equations. For this model, some authors have studied the numerical techniques such as the finite difference method [1], [2], [3], the Bessel collocation method [4], the finite element method [5], [6], the wavelet-Galerkin method [7], [8], the Crank- Nicholson method [9], the piecewise-analytical method [10] and ADI method [11]. Among these methods, the wavelet method is more attractive. In recent years, wavelets have received considerable attention in different field of science and engineering. Wavelets permit the accurate representation of variety of functions and operators, and establish a connection in dealing with various problems. The main characteristic of this method is that it reduces these problems to those of solving systems of algebraic equations, thus it greatly simplifies the problems.

The interested reader is referred to the recent published paper by Abd-Elhameed, et al. [12], in which a numerical method based on the second kind Chebyshev wavelets operational matrix of differentiation is derived and used to transform differential equation with their initial and boundary conditions to systems of algebraic equations. Another numerical method based on the third and fourth kinds Chebyshev wavelets is fully discussed in Abd-Elhameed, et al. [13].

In this paper, we consider the following convection diffusion equation with variable coefficients: $\frac{\partial μ}{\partial t} + a (x) \frac{\partial μ}{\partial x} = b (x) \frac{\partial^{2} μ}{\partial x^{2}} + g (x, t), 0 ⩽ x ⩽ 1, 0 ⩽ t ⩽ 1,$ with initial condition $μ (x, 0) = f (x), 0 ⩽ x ⩽ 1,$ and boundary conditions $μ (0, t) = g_{0} (t), μ (1, t) = g_{1} (t), 0 ⩽ t ⩽ 1,$ where $a (x)$ and $b (x) (\neq 0)$ are continuous functions. The aim of the present work is to develop Chebyshev wavelets method with the operational matrix of integration to solve the above convection diffusion equation. The rest part of this paper is organized as follows. Section 2 introduces the second kind Chebyshev wavelets and their properties. In Section 3, Chebyshev wavelets operational matrix of integration is derived. In Section 4, the proposed method is applied to approximate the solution of the problem. Section 5 gives some examples to test the proposed method. A conclusion is drawn in Section 6.

Section snippets

Properties of the second kind Chebyshev wavelets

Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets: $ψ_{a, b} (t) = | a |^{- 1 / 2} ψ (\frac{t - b}{a}), a, b \in R, a \neq 0 .$ If we restrict the parameters a and b to discrete values as $a = a_{0}^{- j}, b = {mb}_{0} a_{0}^{- j}$ , where $a_{0} > 1, b_{0} > 0$ , and $j, m$ are positive integers, we have the following family of discrete wavelets: $ψ_{j, m} (t) = | a_{0} |^{j / 2} ψ (a_{0}^{j} t - {mb}_{0}),$

The second kind Chebyshev wavelets operational matrix of integration

In this section, we will derive precise integral of the second kind Chebyshev wavelets functions which play a great role in dealing with the problem of convection diffusion equations. First, we figure out the precise integral of the second kind Chebyshev wavelets functions with $k = 2$ and $M = 3$ . In this case, the six basis functions are given by $\begin{matrix} ψ_{1, 0} (t) = 2 \sqrt{\frac{2}{π}}, \\ ψ_{1, 1} (t) = 2 \sqrt{\frac{2}{π}} (8 t - 2), \\ ψ_{1, 2} (t) = 2 \sqrt{\frac{2}{π}} (64 t^{2} - 32 t + 3) \end{matrix}$ on $0 ⩽ t < \frac{1}{2}$ and $\begin{matrix} ψ_{2, 0} (t) = 2 \sqrt{\frac{2}{π}}, \\ ψ_{2, 1} (t) = 2 \sqrt{\frac{2}{π}} (8 t - 6), \\ ψ_{2, 2} (t) = 2 \sqrt{\frac{2}{π}} (64 t^{2} - 96 t + 35), \end{matrix}$ on $\frac{1}{2} ⩽ t < 1$ . Let $Ψ_{6} (t) = (\begin{matrix} ψ_{1, 0} (t) & ψ_{1, 1} (t) & ψ_{1} \end{matrix})$

Description of the proposed method

In this section, we will use the second kind Chebyshev wavelets operational matrix of integration for solving the initial boundary value problem of convection diffusion equation with variable coefficients. Consider the convection diffusion equation of the following form: $\frac{\partial μ}{\partial t} + a (x) \frac{\partial μ}{\partial x} = b (x) \frac{\partial^{2} μ}{\partial x^{2}} + g (x, t), 0 ⩽ x ⩽ 1, 0 ⩽ t ⩽ 1,$ with initial condition $μ (x, 0) = f (x), 0 ⩽ x ⩽ 1,$ and boundary conditions $μ (0, t) = g_{0} (t), μ (1, t) = g_{1} (t), 0 ⩽ t ⩽ 1,$ where $a (x)$ denotes diffusion coefficient and $b (x)$ denotes convection coefficient, and $g (x,$

Numerical examples

In this section, we will use the proposed method to solve the initial boundary value problem of convection diffusion equation with variable or constant coefficients. The following numerical examples are given to show the effectiveness and practicality of the method and the results have been compared with the exact solution.

Example 1

Consider convection diffusion Eq. (13) with $a (x) = - 0.1, b (x) = 0.01$ and $g (x, t) = 0$ . The boundary conditions are given by $μ (x, 0) = e^{- x}, μ (0, t) = e^{- 0.09 t}, μ (1, t) = e^{- 1 - 0.09 t}$ . The exact

Conclusion

In this paper, we have derived the second kind Chebyshev wavelets operational matrix of integration and proposed a numerical method to approximate the solution of the initial boundary value problem of convection diffusion equation with variable or constant coefficients. The method is computationally efficient and the algorithm can be implemented easily on a computer. The advantage of the method is that only small size operational matrix is required to provide the solution at high accuracy.

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^☆: Supported by the National Natural Science Foundation of China (Grant No. 11326089), the Education Department Youth Science Foundation of Jiangxi Province (Grant No. GJJ14492) and PhD Research Startup Foundation of East China Institute of Technology (Grant No. DHBK2012205).

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Numerical solution of the convection diffusion equations by the second kind Chebyshev wavelets☆

Abstract

Introduction

Section snippets

Properties of the second kind Chebyshev wavelets

The second kind Chebyshev wavelets operational matrix of integration

Description of the proposed method

Numerical examples

Conclusion

J. Comput. Appl. Math.

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

J. Comput. Sci.

J. Comput. Appl. Math.

Appl. Math. Comput.

Appl. Math. Comput.