Numerical solution of integral equations by means of the Sinc collocation method based on the double exponential transformation

Abstract

Numerical solution of linear integral equations by means of the Sinc collocation method based on the double exponential transformation, abbreviated the DE transformation, is considered. We first apply the method to the Volterra integral equation of the second kind and then to the Volterra equation of the first kind. This method is also applied to the Fredholm integral equation of the second kind. For the Volterra equations we employed a formula for numerical indefinite integration developed by Muhammad and Mori obtained by applying the DE transformation incorporated into the Sinc expansion of the integrand, while for the Fredholm equation we employed the conventional DE transformation for definite integrals. An error analysis of the method is given and in every case a convergence rate of O ( exp ( - cN / log N ) ) for the error is established where N is a parameter representing the number of terms of the Sinc expansion. Also, the condition of the matrix of the main system of linear equations is watched through an estimate of the condition number returned by the program. Numerical examples show the convergence rate mentioned above and confirm the high efficiency of the present method.