An Integral Transform Associated with Boundary Conditions Containing an Eigenvalue Parameter

Abstract

It has long been known that certain integral transforms and Fourier-type series can be used to solve many classical boundary and initial value problems in separable coordinate systems. More recently, it has been shown that these classical transforms and series are spectral representations associated with an ordinary differential system which results on applying separation of variables to the given boundary value problem. This has been the basis for recent work concerned with systematically generating the proper spectral representation needed to solve a given problem. See [1]-[4] for a list of references. We shall consider the problem of finding the associated spectral representation when the resulting ordinary differential system has the eigenvalue parameter occurring in both the equation and one boundary condition. Moreover, the differential equation is to be satisfied on a semi-infinite interval, thus leading to a singular problem which does not seem to have been studied before. In §2 by using a transformation due to B. Friedman (which we modify appropriately for our singular case), we first give a formal derivation of the spectral representation, and then we rigorously prove the result. In §3 our representation is applied to solve an initial-boundary value problem arising in the theory of diffusion and heat flow in one dimension. We should bear in mind that even in cases where solutions are already known, our method systematically yields alternative representations which are often more rapidly convergent and from which asymptotic expansions of solutions with respect to parameters can often be found. In the problem to be considered a representation of the solution can also be found easily by a straight-forward application of the Laplace transform in time t, while the new transform derived in §2 yields the solution when applied in space x. The new transform, however, can be applied when the coefficients in the boundary value problem are time dependent, a situation which, in general, precludes the use of the Laplace transform.. As a general rule [1] there should be a spectral representation associated with each ordinary differential system resulting from applying separation of variables to the original boundary value problem, and each spectral representation should lead to a different representation of the solution of the original problem.