Rational function approximations in the numerical solution of Cauchy-type singular integral equations

Abstract

Cauchy-type singular integral equations of the second kind with constant coefficients are solved via rational and polynomial approximations. Rational functions, similar to that of polynomials, have the property that for r( t) rational and for many of the weight functions w( t) encountered in practice, R(x) ≡ ∫ -1 1 w(t)r(t) t − x dt is also rational. Hence, approximations by rational functions is feasible. Rational function approximations in the solution of the dominant equation results in a linear algebraic system which possesses block-diagonal structure. It is further shown that the determinant of the coefficient matrix is bounded below away from zero and stability is ensured under fairly non-restrictive conditions. For the complete Cauchy-type singular integral equation, i.e. the equation with both the principal and regular parts, gaussian quadrature in conjunction with the rational function method is synthesized in the construction of a “hybrid” scheme. Error estimates and convergence are established. A variety of problems from Aerodynamics and Fracture Mechanics are solved and presented as a basis of comparison to polynomial-based schemes.