Fast methods for the solution of singular integro-differential and differential equations

L F Abd-Elal

doi:10.1155/s0161171281000604

Abstract

Uniform methods based on the use of the Galerkin method and different Chebyshev expansion sets are developed for the numerical solution of linear integrodifferential equations of the first order. These methods take a total solution time0(N2lnN)usingNexpansion functions, and also provide error extimates which are cheap to compute. These methods solve both singular and regular integro-differential equations. The methods are also used in solving differential equations.

Highlights

Chebyshev expansion sets are developed for the numerical solution of linear integrodifferential equations of the first order
The methods we describe in this paper overcome these limitations, and produce the solution at a
L*(N) is not (A.U.D.) and the analysis of Freeman and Delves [8] is not applicable to method I, so as we will see later we do not suggest a value for the truncation error

Summary

Bij i

2(j + i) dx P0(x) Ti(x) Tj+l(X) / i x2 j 0,i i -i dx P0(x)Ti(x) [Tj+l(x)/ 2(j + 17 Tj_l(x)/ 2(j i)]/. The matrix L is said to be "Asymptotically lower diagonal (A.L.D.) of type B(p,r;c)" if constants p,r e 0, c > 0 exist such that. For systems (A.D.) of type B, Freeman and Delves [8] provide estimates of the convergence rate. Method (I): As is clear from equation (2.12) the matrix A (I) is not (A.D.); Aij(1) for j > the elements increase with j. L*(N) is not (A.U.D.) and the analysis of Freeman and Delves [8] is not applicable to method I, so as we will see later we do not suggest a value for the truncation error. From Lemma 1,2 the matrix L*(N) is U.A.D. and the analysis of Freeman and Delves [8] is applicable to methods II, III and a value of the truncation error is suggested as given later. By virtue of Theorems 6 and 7 of Freeman and Delves [8] with normalization we have the following theorem

Theorem leads us to our two main theorems"

The computed errors o N are defined to be