Solution of a class of two-dimensional integral equations

The two-dimensional integral equation1π∫ ∫Dϕ(r,θ)Rα dS=f(r0,θ0)defined on a circular disk D : r0⩽a,0⩽θ0 ⩽2π, is considered in the present paper. Here R in the kernel denotes the distance between two points P(r,θ) and P0(r0,θ0) in D, and 0<α<2 or 2<α<4. Based on some known results of Bessel functions, integral representations of the kernel are established for 0<α<2 and 2<α<4, respectively, and employed to solve the corresponding two-dimensional integral equation. The solutions of the weakly singular integral equation for 0<α<2 and of the hypersingular integral equation for 2<α<4 are obtained, respectively.