THE SOLUTION OF A HYPERSINGULAR INTEGRAL EQUATION OCCURRING IN POTENTIAL PROBLEMS OVER AN ELLIPTIC DISC BY A SPECTRAL METHOD

Abstract

Often in scattering theory, two-dimensional singular integrals arise over a circular or elliptic region. So far, formulas either for evaluating Cauchy principal value integrals over circular and elliptic regions or for computing hypersingular integrals over circular region are available. Thus, in the present paper, we derive a formula that evaluates the dominant part of a hypersingular integral equation that appears in a variety of physical problems involving an elliptic disc in terms of trigonometric and associated Legendre functions $P_{\nu }^{\mu }$. Expanding the integrand function in terms of Fourier series in the azimuthal direction, orthogonal polynomials in the radial variable, and the Bessel function $J_{\nu }$ in terms of $P_{\nu }^{\mu }$, utilizing recurrence relations for $P_{\nu }^{\mu }$ and $J_{\nu }$, and incorporating the value of the Weber–Schafheitlin integral, we derive an expression for the hypersingular integral in terms of hypergeometric functions. Using relationships of hypergeometric functions and their connections with $P_{\nu }^{\mu }$, finally the dominant integral is expressed in terms of $P_{\nu }^{\mu }$. This result generalizes previously known integral identities which are applicable to weakly singular kernels on elliptic disc or to hypersingular integrals on circular discs. The formula obtained is verified by recovering known results for the integral over a circular disc. In particular, the added mass due to heaving circular and elliptic discs in an unbounded fluid are recovered. A spectral method for the solution of a class of hypersingular integral equations over an elliptic disc is proposed and numerical results for a simple case are presented, showing the numerical convergence of the method.