Numerical solution of ordinary differential equations with impulse solution

Abstract

This paper extends an earlier work [Appl. Math. Comput. 132 (2002) 341] to ordinary differential equations with impulse solution. In this case, the use of spectral methods, generally, dose not work well. Also it is shown that the modified spectral method (mentioned in [Appl. Math. Comput. 132 (2002) 341]) can be useful to solve the ODEs with impulse solution. In addition, with providing some examples, the aforementioned cases are dealt with numerically.

Introduction

Consider the following ODE:

$$\text{Ly=∑}{}_{\mathrm{i=0}}{}^{\mathrm{M}}\text{(f}{}_{\mathrm{M-i}}\text{(x)D}{}^{\mathrm{i}}\text{)y=F(x),}\text{x∈[a,b],}$$

$$\text{Ty=C,}$$

where f_i, i=0,1,…,M, F are known real functions of x, Dⁱ denotes ith order differentiation with respect to x, T is a linear functional of rank N and C∈R^M. The basic of spectral methods to solve this class of equations is to expand the solution function, y, in , as a finite series of very smooth basis functions, as given below

$$\text{y}\text{(x)≈∑}{}_{\mathrm{i=0}}{}^{\mathrm{N}}\text{a}{}_{\mathrm{i}}\text{Φ}{}_{\mathrm{i}}\text{(x)}$$

in which Φ_i represents a family of polynomials which are orthogonal and complete over the interval (a,b) with respect to the nonnegative weight function w(x). In this cases of interest, these are the eigenfunctions of a Sturm–Liouville problem [2], [3], [6]. In this paper, we consider first kind of Chybeshev, {T_k}_k=0^∞, and Legendre, {P_k}_k=0^∞ polynomials [2]. Early in [1], it was shown that straightforward application of spectral methods can be difficult to solve ODE , when the solution function, y, is nonanalytic and alternative treatment is the use of modified spectral method.

In this paper, it will be shown that the use of modified spectral method, compared with spectral methods, can be useful to solve the ODEs with impulse solution.