Rational Chebyshev functions with new collocation points in semi-infinite domains for solving higher-order linear ordinary differential equations

Mohamed Abdel -Latif Ramadan

doi:10.24297/jam.v11i7.1218

Abstract

The purpose of this paper is to investigate the use of rational Chebyshev (RC) functions for solving higher-order linear ordinary differential equations with variable coefficients on a semi-infinite domain using new rational Chebyshev collocation points. This method transforms the higher-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coefficients. These matrices together with the collocation method are utilized to reduce the solution of higher-order ordinary differential equations to the solution of a system of algebraic equations. The solution is obtained in terms of RC series. Numerical examples are given to demonstrate the validity and applicability of the method. The obtained numerical results are compared with others existing methods and the exact solution where it shown to be very attractive and maintains better accuracy.

Highlights

The well-known Chebyshev polynomial of first kind Tn (x) [1] are orthogonal with respect to the weight-function w x 1/ 1 x2 on the interval [-1, 1], and the recurrence relation isT0 (x) 1, T1(x) x, n 1These polynomials have many applications in numerical analysis, and a lot of studies are devoted to show the merits of them in various ways
One of the applications of Chebyshev polynomials is the solution of ordinary differential equations with boundary conditions [1, 2]
Following the procedure in Section (5), we find the matrix in (18) for N = 2 as: 5407 | P a g e we obtain the R C coefficients as we find the solution y(x) 1 x 1 which is the exact solution of Example 1

Summary

Introduction

The well-known Chebyshev polynomial of first kind Tn (x) [1] are orthogonal with respect to the weight-function w x 1/ 1 x2 on the interval [-1, 1], and the recurrence relation isT0 (x) 1, T1(x) x, n 1These polynomials have many applications in numerical analysis, and a lot of studies are devoted to show the merits of them in various ways. Many studies are considered on the interval [-1, 1] in which Chebyshev polynomials are defined. Parand et al [5, 6], Sezer et al [7, 8] and Ramadan et al [9, 10] successfully applied spectral methods to solve problems on open semi-infinite intervals. In their studies, the basis functions called rational Chebyshev functions Rn (x) and defined by

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Results

Conclusion