The determination of eigen-functions of a certain Sturm-Liouville equation and its application to problems of heat-transfer

Summary

In this paper a generalized Sturm-Liouville differential system has been considered. Any arbitrary functionf(x) can be expanded in terms of the eigen-functions of this differential equation. These eigen-functions are expanded as an infinite series of the eigen-functions of an auxiliary equation with identical boundary conditions. The eigen-functions are orthogonal over a finite interval of integration and the Sturm-Liouville system is reduced to an infinite set of linear simultaneous algebraic equations for the coefficients of the series. These equations contain the characteristic parameter λ and may be solved by standard iterative or relaxation methods. A special case is considered in detail, for which the auxiliary equation becomes Bessel’s equation. A more general case can be reduced to the above type by a suitable change of both dependent and independent variables. Three physical problems are posed and it is indicated that they can be solved exactly.