On the relationship of some Fredholm integral equations of the first kind to a family of boundary value problems

It is shown that the solution of a class of forced convection heat transfer problems can be used as a vehicle for exhibiting a correspondence between certain boundary value problems and their associated integral equations. If the solution of the boundary value problem is known then the solution of the integral equation can be found by a simple calculation. This generalizes the relationship of certain solutions of Laplace's equation to integral equations having logarithmic or Cauchy kernels. When the boundary value problems have separable solutions, the solution of the integral equation can be reduced to the verification of a set of identities μn ∫E P(t) φn(t) k(x - t) dt = φn(x), x ε{lunate} E, where the {φn} form an orthonormal set on E. Once the method of solution has been derived from the physical approach it can be put on a firm mathematical basis