Variable transformations in the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels; extensions to Fredholm integral equations

Abstract

A number of techniques that use variable transformations in numerical integration have been developed recently (cf. Sidi, Numerical Integration IV, H. Brass, G. Hämmerlin (Eds.), Birkhäuser, Basel, 1993, pp. 359–373; Laurie, J. Comput. Appl. Math. 66 (1996) 337–344.). The use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule. In this paper the application of variable transformation techniques of Sidi and Laurie type to the numerical solution of second kind Volterra integral equations with continuous and weakly singular kernels is considered. Since the transformations are such that the end points of integration need not be used as mesh points, the methods introduced can be used for VIE with both continuous and weakly singular kernel in a uniform way. The methods have also the advantages of simplicity of application and of achieving high order of convergence. The application of the idea to Fredholm integral equations with continuous and weakly singular equations is also considered. Numerical results are included and they verify the expected increased order of convergence. They were obtained by using the trapezoidal formula for the evaluation of the transformed integrals.