The application of approximate product-integration to the numerical solution of integral equations

Abstract This paper presents numerical methods of solving integral equations in which formulae for numerical integration are derived from the kernel functions of the equations by means of the technique of ‘approximate product-integration’. Provided certain product-moments of the kernel functions exist, the methods are applicable to singular equations. By adopting matrix methods, the problem of determining the necessary weights for the integration formulae is made to depend on the inversion of certain alternant matrices; tables of the necessary inverse matrices already exist. The methods prove to be essentially the same for the solution of equations of both Volterra and Fredholm types. The initial solutions may be improved in certain cases, by using expressions for the errors arising from the use of the approximate numerical integration formulae.