Transforming Boundary Conditions to Initial Conditions for Ordinary Differential Equations

Abstract

One of the interesting and important classes of transformations in the solution of boundary value problems is the class of transformations from a boundary value to an initial value problem. A classical example is given in Blasius' solution of the steady, two-dimensional, incompressible boundary layer equations with uniform mainstream velocity [1]. This technique, when it applies, simplifies the process of obtaining numerical solutions. Otherwise a trial-and-error technique generally has to be used in order to match the boundary condition at the other point. A recent contribution by Klamkin [2] has greatly extended the range of application of the technique, and as a result, a much wider class of problems, including some simultaneous differential equations, can be solved in this manner. In the present paper, the technique is extended to cases where the boundary conditions are specified over finite intervals, i.e., from zero to a finite value of the independent variable, with one or more boundary conditions required to be specified at the initial point. It also shows that the transformation treated in previous works is not the only type of transformation possible. Similar to the class of problems treated by Blasius [1] and Klamkin [2], the boundary conditions at the initial point have to be homogeneous for the method to apply. It can be homogeneous or nonhomogeneous at the other point. The method given in this paper is developed in terms of a oneor two-parameter group of transformations, which is seen to be as simple here as in its application to the similarity analysis [3]. The concept can best be illustrated by an example. Consider the problem of steady heat conduction with linear heat generation and power-law thermal conductivity. The energy equation can be written as