Numerical solution of ordinary differential equations with impulse solution
Introduction
Consider the following ODE:where fi, 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∈RM. 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 belowin which Φi 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, {Tk}k=0∞, and Legendre, {Pk}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.
Section snippets
The ODEs with impulse solution
We begin this section by an interesting example. Consider for −1⩽t⩽1where F(x) is chosen such that the exact solution is y(x)=cotg(x2+0.1). The above problem seems be very simple, because the coefficient functions are constant and the solution function is belong to C∞[−1,1]. But spectral methods are not successful to solve the above problem, see Table 1. To discover the reason of this occurrence, first consider the graph of the solution function (Fig. 1).
Fig. 1
The modified spectral method
In this section, the modified spectral method for numerical solution of ordinary differential equations is introduced. For this reason, first, consider the following differential equation:It must be noted that the discussion in this section can simply extend to the general form , . First, for an arbitrary natural number N, we suppose that the approximate solution of Eqs. , , iswhere a=(a0,a1,…,aN)∈RN+1 and {Tk}k=0∞ is the sequence of
Numerical examples
This section deals with two numerical tests on simple, but interesting, problems. These problems are solved by modified spectral, , and pseudospectral, , (the best method between spectral methods for these problems, see [1]) methods. As will be seen, the results are shown the advantages of modified spectral method, proposed in Section 3. Here, maximum errors are approximately obtained through their graphs and the presented algorithms in this article are performed using Maple V with 20
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