Two-dimensional Chebyshev Polynomials for Solving Two-dimensional Integro-Differential Equations

Çankaya University Journal of Science and Engineering Volume 12, No. 2 (2015) 001–011 Two-dimensional Chebyshev Polynomials for Solving Two-dimensional Integro-Differential Equations Azim Rivaz1 , Samane Jahan ara1 , Farzaneh Yousefi2 1 2 Department of Applied Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran, 2 Young Researchers Society, Shahid Bahonar University of Kerman, Kerman, Iran, e-mail: , , Abstract: In this paper, we present a new approach to obtain the numerical solution of the linear twodimensional Fredholm and Volterra integro-differential equations (2D-FIDE and 2D-VIDE). First, we introduce the two-dimensional Chebyshev polynomials and construct their operational matrices of integration. Then, both of them, two-dimensional Chebyshev polynomials and their operational matrix of integration, are used to represent the matrix form of 2D-FIDE and 2D-VIDE. The main characteristic of this approach is that it reduces 2D-FIDE and 2D-VIDE to a system of linear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. Keywords: Two-dimensional Fredholm and Volterra integro-differential equations, Two-dimensional Chebyshev polynomials, Operational matrix of integration. 1. Introduction Integral equations have been one of the principal tools in various areas of applied mathematics, physics and engineering. In this paper, we are concerned with two-dimensional integro-differential equations. Scientists have investigated the topic of integro-differential equations through their works in many scientific applications, including heat transfer, diffusion processes, neutron diffusion and biological species coexisting with increasing and decreasing rates of generation. On the other hand, two-dimensional integral equations provide an important tool for modeling numerous problems in engineering and science. These equations appear in electromagnetism, electrodynamics, molecular physics, population in addition to many other fields. One of the main problems is how to solve integro-differential equations in one and two-dimensional space. There are several classical solution techniques to solve some of these equations; it is difficult to obtain the analytical solutions of most of these equations. Therefore, it is important to develop numerical algorithms which have sufficient accuracy. In recent years, numerous works have been focusing on the development of more advanced and efficient methods for integro-differential equations, including the Wavelet-Galerkin method, Lagrange interpolation method, Tau method ISSN 1309 - 6788 c 2015 Çankaya University 2 A. Rivaz et al. and semi-analytical numerical techniques such as Adomians decomposition method and Taylor polynomials [2, 5, 7, 11, 13]. An usual way to solve functional equations is to express the solution as a linear combination of the so-called basis functions. These basis functions can, for instance, be either orthogonal or non-orthogonal bases. Approximation by the orthogonal family of basis functions has found wide application in science and engineering. The most frequently used orthogonal functions are sinecosine functions, block pulse functions, Legendre, Chebyshev and Laguerre polynomials. The main idea of using an orthogonal basis is that the problem under consideration reduces to a system of linear or nonlinear algebraic equations [8, 10, 11]. The main purpose of this paper is to apply the 2D orthogonal Chebyshev polynomials to solve Fredholm and Volterra integro-differential equations. The remainder of this paper is organized as follows: in Section 2, we begin by introducing some necessary definitions. In Section 3, the two-dimensional Chebyshev polynomials and their properties are defined and their integral operational matrices are obtained. Section 4 is devoted to applying the two-dimensional Chebyshev operational matrix of integration to solve two-dimensional linear Fredholm and Volterra integro-differential equations. In Section 5, the proposed method is applied to several examples followed by conclusion in the final section. 2. Preliminaries In this section, we give definitions and properties of Chebyshev polynomials in one-dimensional space. The well known Chebyshev polynomials of the first kind of degree n are defined by [4]: Tn (x) = cos(ncos−1 x), n ≥ 0. (1) Also they are derived from the following recursive formula: T0 (x) = 0, T1 (x) = x, Tn+1 (x) = 2xTn (x) − Tn−1 (x) n = 1, 2, 3, · · · These polynomials are orthogonal on [−1, 1] with respect to the weight function w(x) = √ Z 1 −1 Ti (x)T j (x)w(x)dx = where γi = ( ( 0, π γi , 1, i = 0, 2, i ≥ 1, i 6= j, i = j, 1 1 − x2 : (2) CUJSE 12, No. 2 (2015) Two-dimensional Chebyshev Polynomials 3 Chebyshev polynomials are important in approximation theory and numerical analysis [4, 6]. A function f (x) over [−1, 1] may be represented by Chebyshev polynomials series as: ∞ f (x) = ∑ ai Ti (x). (3) i=0 If the infinite series in (3) is truncated, then (3) can be written as: N f (x) ≃ ∑ ai Ti (x) = T (x)t A, (4) T (x) = [T0 (x), T1 (x), · · · , TN (x)]t , (5) i=0 where A = [a0 , a1 , · · · , aN ]t , γi R 1 f (x)Ti (x)w(x)dx. π −1 Chebyshev polynomials have the following useful property [6]: and ai = Z x −1 TN−1 (s)ds = 1 1 (−1)N−1 TN (x) − TN−2 (x) + T0 (x), N ≥ 3. 2N 2(N − 2) 1 − (N − 1)2 (6) Moreover, for T0 (x) and T1 (x), we have: Rx −1 T0 (s)ds = T0 (x) + T1 (x), (7) Rx 1 −1 −1 T1 (s)ds = 4 T0 (x) + 4 T2 (x). Equations (6) and (7) allow us to write: Z x T (s)ds = PT (x), (8) −1 where P is the (N + 1) × (N + 1) operational matrix:  1 1 0 0 ···    −1 0 14 0 · · ·  4   − 21 0 16 · · ·  − 13  P= .. .. .. .. . .  . . . . .    (−1)N−1   1−(N−1)2 0 0 0 · · ·  (−1)N 0 0 0 ··· 1−N 2 0 0 0 .. . 0 1 − 2N−2 0    0    0  , N ≥ 3. ..   .    1  2N   0 (9) 4 A. Rivaz et al. 3. Two-Dimensional Chebyshev Polynomials In this section, by considering 1D Chebyshev polynomials, we define an (N + 1)2 set of twodimensional Chebyshev polynomials as: Ti j (x,t) = Ti (x)T j (t), i, j = 0, · · · , N. (10) Therefore, the two-dimensional Chebyshev basis vector is as follows: T (x,t) = [T0 (x)T0 (t), · · · , T0 (x)TN (t), T1 (x)T0 (t), · · · , T1 (x)TN (t), · · · , TN (x)TN (t)]t (11) = (CN ⊗ BN )t , in which CN = [T0 (x), T1 (x), · · · , TN (x)], and BN = [T0 (t), T1 (t), · · · , TN (t)] are one dimensional Chebyshev vectors. The orthogonality property for these polynomials with respect to the weight function w(x,t) = 1 √ √ on the interval [−1, 1] × [−1, 1] is: 1 − x2 1 − t 2  2 π   4 , i = k 6= 0, j = l 6= 0     π2  , i = k = 0, j = l 6= 0   Z 1Z 1  2 π2 (Ti, j (x,t), Tk,l (x,t))w(x,t) = Ti, j (x,t)Tk,l (x,t)w(x,t)dxdt = 2 , i = k 6= 0, j = l = 0  −1 −1     π 2 , i = k = 0, j = l = 0      0, else Similarly to the one-dimensional case, a function f (x,t) on [−1, 1] × [−1, 1] can be exp (...truncated)