Spectral Parameter Power Series Method Research Articles

Conformal mapping of right circular quadrilaterals

We study conformal mappings from the unit disc to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm–Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the coefficient functions contain the accessory parameters t, λ of the mapping problem. The parameter λ is designed in such a way that for fixed t, it plays the role of an eigenvalue of the Sturm–Liouville problem. Further, for each t a particular solution (an elliptic integral) is known a priori, as well as its corresponding spectral parameter λ. This leads to insight into the dependence of the image quadrilateral on the parameters, and permits application of a recently developed spectral parameter power series method for numerical solution. Rate of convergence, accuracy and computational complexity are presented for the resulting numerical procedure, which in simplicity and efficiency compares favourably with previously known methods for this type of problem.

Spectral parameter power series for Sturm–Liouville problems

AbstractWe consider a recently discovered representation for the general solution of the Sturm–Liouville equation as a spectral parameter power series (SPPS). The coefficients of the power series are given in terms of a particular solution of the Sturm–Liouville equation with the zero spectral parameter. We show that, among other possible applications, this provides a new and efficient numerical method for solving initial value and boundary value problems. Moreover, due to its convenient form the representation lends itself to numerical solution of spectral Sturm–Liouville problems, effectively by calculation of the roots of a polynomial. We discuss the examples of the numerical implementation of the SPPS method and show it to be equally applicable to a wide class of singular Sturm–Liouville problems as well as to problems with spectral parameter‐dependent boundary conditions. Copyright © 2009 John Wiley & Sons, Ltd.