Theory and performance of a subroutine for solving Volterra Integral Equations

Abstract

The author considers Volterra Integral Equations of either of the two forms $$u(x) = f(x) + \int\limits_a^x {k(x - t)g(u(t))dt, a \leqslant } x \leqslant b,$$ wheref, k, andg are continuous andg satisfies a local Lipschitz condition, or $$u(x) = f(x) + \int\limits_a^x {\sum\limits_{j = 1}^m {c_j (x)g_j (t,u(t))dt} ,} $$ wheref,c j , andg j ,j=1,2,...,m, are continuous and eachg j satisfies a local Lipschitz condition in its second variable. It is shown that in each case the respective integral equation can be solved by conversion to a system of ordinary differential equations which can be solved by referring to a described FORTRAN subroutine. Subroutine VE1. In the case of the convolution equation, it is shown how VE1 converts the equation via a Chebyshev expansion, and a theorem is proved, and implemented in VE1, wherein the solution error due to truncation of the expansion can be simultaneously computed at the discretion of the user. Some performance data are supplied and a comparison with other standard schemes is made. Detailed performance data and a program listing are available from the author.