Convergence Analysis of Legendre wavelets in numerical solution of
linear weakly singular Volterra integral equation for union of some
intervals with application in heat conduction
Abstract
In this paper we apply the Legendre wavelets basis to solve the linear
weakly singular Volterra integral equation of the second kind. The basis
is defined on [0,1) , and in this work we extend this interval to
[0,n) for some positive integer n. For this aim we solve the problem
on [0,1); then we apply the Legendre wavelets on [1,2) and use the
lag solution on [0,1) to obtain the solution on [0,2) and continue
this procedure. Convergence analysis of Legendre wavelets on [n,n+1),
is considered in Section2. We give a convergence analysis for the
proposed method, established on compactness of operators. In numerical
results we give two sample problems from heat conduction. For this
purpose, in Section 6 we give an equivalent theorem between the proposed
heat conduction problem and an integral equation. Then we solve the
equivalent integral equation by the proposed method on union of some
interval and obtain the solution of the heat conduction problem. As
Tables and Figures of two and three dimensional plots show, accuracy of
the method is reasonable and there is not any propagation of error from
lag intervals. The convergence analysis and these sample problems
demonstrate the accuracy and applicability of the method.