A Note on the Disadvantages Associated with the Treatment of an Eigenvalue as an Additional Dependent Variable in Differential Eigenvalue Problems

Abstract

A novel technique of dealing with differential eigenvalue problems has recently been introduced by Wadsworth and Wilde . The differential equation is expressed as a set of simultaneous first-order differential equations, the eigenvalue λ being regarded as an additional variable by adding the equation to the basic set. The differential eigenvalue problem is thus reduced to a set of non-linear first-order differential equations with two-point boundary conditions. This treatment of the problem, although novel, suffers from two serious disadvantages. First, it introduces non-linearity into an otherwise linear set of equations. Thus, the solution can no longer be obtained by linear combinations of independent particular solutions. One method of solving the non-linear systems is by assigning arbitrary starting values at one boundary and performing a step-by-step integration to the other boundary where in general the boundary conditions are not satisfied. The problem can be solved by adjustment of the initial assigned arbitrary values until the given conditions at the other boundary are satisfied. A second method and the one used by Wadsworth and Wilde is to estimate the unknown boundary values at both boundaries and integrate inwards to a meeting point. Changes can then be made to the unknown boundary values to make the two branches of the curve fit together.