Rational Chebyshev collocation method for solving higher-order linear ordinary differential equations

Abstract

A collocation method to find an approximate solution of higher-order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor-Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9.

On the Exponential Chebyshev Approximation in Unbounded Domains: A Comparison Study for Solving High-Order Ordinary Differential Equations

“…The proposed differential equations and the given conditions were transformed to matrix equation with unknown EC coefficients. This technique is considered to be a modification of the similar presented in [4], [9], [10] and [8]. On the other hand, the EC functions approach deals directly with infinite boundaries without singularities.…”

“…where D is (N + 1) × (N + 1) operational matrix for the derivative given in (10), and B(x) is 1 × (N + 1) row vector which is an actual term to get the equality sign of (16), that was truncated in (9). This added term will improve the obtained approximate solutions as will be shown in the numerical examples in section 6.…”

“…Therefore, this limitation causes a failure of the Chebyshev approach in the problems that are naturally defined on larger domains, especially including infinity. Under a transformation that maps the interval [−1, 1] into a semi-infinite domain [0, ∞), Boyd [2], [3], Parand et al [6], [7], and Sezer [9], [10] successfully applied spectral methods to solve problems on semi-infinite intervals. In their studies the basis functions called rational Chebyshev functions, and defined by…”

Numerical solution of high-order linear integro differential equations with variable coefficients using two proposed schemes for rational Chebyshev functions

“…The concepts of integro-differential equations have motivated a huge size of research work in recent years, several numerical methods were used such as wavelet-Galerkin method [1], Lagrange interpolation method [2], Taylor polynomials [3] and [4], Chebyshev polynomials [5], [6] and [7], Adomian decomposition method [8] and [9], the differential transformation method [10] and [11], Legendre polynomial [12], CAS Wavelet operational matrix [13], Reduced differential transform method [14], Homotopy perturbation method [15]. In our work, we apply rational Chebyshev (RC) collocation method [16] and [17] for solving high-order linear Fredholm integro-differential equations and we will show that convergent rate of RC is more accelerate than other existing method. The organization of this paper, in Section 2, preliminaries introduced while in section 3, properties of the RC functions are presented.…”

A new exponential Chebyshev operational matrix of derivatives for solving high-order ordinary differential equations in unbounded domains

“…Therefore, this limitation causes a failure of the Chebyshev approach in the problems that are defined on larger domains, especially including infinity. Under a transformation that maps the interval [−1, 1] into a semi-infinite domain [0, ∞), the authors of [4,5,6,16,17,24,28,20,21,22,18,19] successfully applied spectral methods to solve problems on semi-infinite intervals. In their studies, the basis functions called rational Chebyshev functions R n (x), and defined by R n (x) = T n x − 1 x + 1 .…”