Solution of A 2-D weak singular integral equation with constraint

Abstract

In this paper, the solution of a 2-D weak singular integral equation of the first kind

$$\int_0^\pi {\int_{ - \infty }^\infty {p\left( {s,\psi } \right)} } dsk\left( \psi \right)d\psi = F\left( {r,\theta } \right)$$

subjected to constraint

$$p\left( {s,\psi } \right) = 0, for \left( {s,\psi } \right) = \left( {r,\theta } \right) \notin Q = \left\{ {\left( {r,\theta } \right)|F\left( {r,\theta } \right) > C_* } \right\}$$

is found and listed

$$p = p\left( {r,\theta } \right) = \left\{ {2/\left[ {\pi ^2 k\left( {\psi _0 } \right)} \right]} \right\}\sqrt {F\left( {r,\theta } \right) - c_* } \left( {0 \leqslant r \leqslant r_* } \right)$$

where (s, ψ) is a local polar coordinating with origin at M (r, θ), (r, θ) is the global polar coordinating with origin at O(0, 0): k and F are given continuous functions; ψ₀ is a constant; F(r_*, θ)=c_* (const.) is the boundary contour of considering range Q.

The method used can be extended to 3-D cases.