Graduate Thesis Or Dissertation
 

Reduction of Fredholm integral equations with Green's function kernels to Volterra equations

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  • G. F. Drukarev has given a method for solving the Fredholm equations which arise in the study of collisions between electrons and atoms. He transforms the Fredholm equations into Volterra equations plus finite algebraic systems. H. Brysk observes that Drukarev's method applies generally to a Fredholm integral equation (I-λ G)u = h with a Green's function kernel. In this thesis connections between the Drukarev transformation and boundary value problems for ordinary differential equations are investigated. In particular, it is shown that the induced Volterra operator is independent of the boundary conditions. The resolvent operator can be expressed in terms of the Volterra operator for regular λ. The characteristic values of G satisfy a certain transcendental equation. The Neumann expansion provides a means for approximating this resolvent and the characteristic values. To illustrate the theory several classical boundary value problems are solved by this method. Also included is an appendix which relates the resolvent operator mentioned above and the Fredholm resolvent operator.
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