Numerical Solution of the One-Dimensional Heat Equation by Using Chebyshev Wavelets Method

In this paper we study the physical problem of heat conduction in a rod of length L. This problem first studied by Fourier at the beginning of the 19th century in his celebrated volume on the analytical theory of heat, has become during the intervening century and a half the paradigm for the very extensive study of parabolic partial differential equations (PDEs), linear and nonlinear [1]. The temperature of a rod is governed by a PDE that is often defined by [2]:


Introduction
In this paper we study the physical problem of heat conduction in a rod of length L. This problem first studied by Fourier at the beginning of the 19th century in his celebrated volume on the analytical theory of heat, has become during the intervening century and a half the paradigm for the very extensive study of parabolic partial differential equations (PDEs), linear and nonlinear [1]. The temperature of a rod is governed by a PDE that is often defined by [2]: u ku x L t (1) where ≡ ( , ) u u x t represents the temperature of the rod at the position x at time t, and k is the thermal diffusivity of the material that measures the rod ability to heat conduction.
The domain of the solution is a semi-infinite strip of width L that continues indefinitely in time. In a practical computation, the solution is obtained only for a finite time, say = max T t . Solution to equation 1 requires specification of initial condition at t = 0 and boundary conditions at x = 0 and x = L. Simple initial and boundary conditions (IBCs) are: The initial condition in equation 2 describes the initial temperature u at time t=0 and the given boundary conditions in equation 3 and equation 4 indicate that the temperature of rod ends are functions of t. Other boundary conditions, e.g. gradient (Neumann) or mixed conditions, can be specified. In this article, only the conditions in equation 3 and 4 are considered. The existence and uniqueness property of this case problem have been studied by J. R. Cannon in [1]. It is of interest to note that the PDE in equation 1 arises in two different types, namely:

Homogeneous heat equation
Further, heat equation with a lateral heat loss is formally derived as a homogeneous PDE of the form: where c is a positive constant.

Inhomogeneous heat equation
This type of equations is often given by where g(x; t) is called the heat source.
Heat equation mainly in one-dimension had been studied by many authors [3]. A comparative study between the traditional separation of variables method and Adomian method for heat equation had been examined by Gorguis and Chan [4]. Dehghan [5] considered the use of second-order finite difference scheme to solve the two-dimensional heat equation. After that, Mohebbi and Dehghan [6] presented a fourthorder compact finite difference approximation and cubic C1-spline collocation method for the solution with fourth-order accuracy in both space and time variables. Recently Dabral et al. [3], propose B-spline finite element method to get numerical solutions of one dimensional heat Equation.
Wavelet methods have been applied for solving PDEs from beginning of the early 1990s [7]. In the last two decades this problem has attracted great attention and numerous papers about this topic have been published. Due to this fact we must confine somewhat our analysis; in the following only PDEs of mathematical physics (elliptic, parabolic and hyperbolic equations) and of elastostatics are considered. From the first field of investigation the papers [8][9][10][11][12][13] can be cited. As to the elasticity problems we refer to the papers [14][15][16][17][18][19][20]. In all these papers different wavelet families have been applied. In most cases the wavelet coefficients were calculated by the Galerkin or collocation method, by it we have to evaluate integrals of some combinations of the wavelet functions (called also connection coefficients).
The aim of the present work is to develop Chebyshev wavelet method with operational matrices of integration for solution the onedimensional heat equation with Dirichlet boundary conditions, which is fast, mathematically simple and guarantees the necessary accuracy for a relative small number of grid points. The outline of this article is as follows. In Properties of Chebyshev wavelets section; we describe properties of Chebyshev wavelet. In Description section; the proposed method is used to approximate the solution of the problem. After description section; the numerical examples of applying the method of this article are presented. Finally a conclusion is drawn in last section.

Wavelets and Chebyshev wavelets
Wavelets constitute a family of functions constructed from dilations and translations of a single function called the mother wavelet ψ ( ) t .
When the dilation parameter a and the translation parameter b varies continuously we have the following family of continuous wavelets as [21]: If we restrict the parameters a and b to discrete values as  (8) where ψ , ( ) k n t forms a wavelet basis for Where,

T t T t t T t tT t T T m
We should note that in dealing with Chebyshev polynomials the weight function have to be dilated and translated as to get orthogonal wavelets.

Function approximation
An arbitrary function pq pq p q f t c t (12) where coefficients pq pq c f t t in which (,) denotes the inner product. If the infinite series in (12) is truncated, then (12) can be written as where C and For simplicity, we write (13) as where ψ ψ The index i, is determined by the relation Therefore we have Similarly, an arbitrary function of two variables 1), may be expanded into Chebyshev wavelets basis as, where U=[u ij ] and ψ ψ = ( ( ),( ( , ), ( ))).
Taking the collocation points

The operational matrices of Chebyshev wavelet
Chen and Hsiao [22], introduced the concept of operational matrix in 1975, and Kilicman and Al Zhour [23] investigated the generalized integral operational matrix, that is, the integral of the matrix Ψ( ) t defined in (15) can be approximated as follows:  [24] proposed a uniform method to obtain the corresponding integral operational matrix of different basis. For example, the operational matrix of Ψ( ) t can be expressed as: (20) and in general the operational matrix P n can be expressed as follows: where, ξ

Description of Numerical Method
In this section, we will use the Chebyshev wavelet operational matrices for solving the heat equation. Let us consider the inhomogeneous one-dimensional heat equation with lateral heat loss as: And inhomogeneous Dirichlet boundary conditions: Where ×

U=[u ]
ij m m is an unknown matrix which should be found and Ψ( ) t is the vector that defined in (15). By integrating of (27) one time with respect to t and considering (24) we obtain: Also by integrating of (27) two times with respect to x we get: By putting x = 1 into (29) and considering (25) and (26), we have: .

T T u x P t x t h t x h t h t t (30)
Now by integrating of (30) one times with respect to t we get: Now by replacing (28), (30), and (31) into (23) we get: Equation (33)

Example 2
Consider the heat equation (23)

Example 3
In this example we consider the heat equation (23) with = 1 k , c = 0 and nonhomogeneous term = + 2 ( , ) (2 ) sin( ) g x t t t x the initial and boundary conditions are given by

Example 4
Finally consider the heat equation (23) with = 1 k and coefficient lateral heat c=2 and g(x; t)=0. The initial and boundary conditions are given by

Conclusion
This paper presents a numerical method by combining wavelet function with operational matrices of integration to approximate numerical solutions of well-known one-dimensional heat equation. In the proposed method already a small number of grids points guarantee the necessary accuracy. The method is very convenient for solving boundary value problems, since the boundary condition are taken into account automatically. Also the proposed method is very simple in implementation and as the numerical results show the method is very efficient for numerical solution of mentioned problem and can be used for other partial differential equations.