On some dual integral equations involving Legendre and associated Legendre functions

Abstract

Dual integral equations involving Legendre and associated Legendre functions as kernels are considered in this paper. Except for one pair, these dual integral equations are first reduced to solving some appropriate ordinary differential equations. Invoking the inversion fonnulae for Abel integral equations, closed-form. solutions to these dual integral equations are obtained in most cases and in other cases they are reduced to some appropriate Fredholm integral equation of the second kind. For the exceptional dual integral equations pair which involves the associated Legendre function as kernel, a direct method of the use of Abel integral equation is applied to obtain the closed-form solution. As an example of application of these dual integral equations a problem arising from mathematical physics is considered.