The Numerical Solution of Two-parameter Eigenvalue Problems in Ordinary Differential Equations with an Application to the Problem of Diffraction by a Plane Angular Sector

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Abstract
In this paper we are concerned with the numerical solution of Sturm–Liouville eigenvalue problems associated with a system of two second order linear ordinary differential equations containing two spectral parameters. Such problems are of importance in applied mathematics and frequently arise in solving boundary value problems for the Helmholtz equation or Laplace's equation by the method of separation of variables. The numerical method proposed here is a departure from the usual techniques of solving eigenvalue problems associated with ordinary differential equations and appears capable of considerable generalization. Briefly, the idea is to replace the given problem by a related initial boundary value problem and then to use the powerful numerical techniques currently available for such problems. The technique developed is illustrated in application to the important problem of diffraction by a plane angular sector. It appears that, in theory, the method described here is capable of generalization to systems of ordinary differential equations containing more than two spectral parameters.

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