This book was conceived while I was revising my engineering mathematics textbook. I noticed that in many engineering and scientific problems the nature of the boundary condition changes, say from a Dirichlet to a Neumann condition, along a particular boundary. Although these mixed boundary value problems appear in such diverse fields as elasticity and biomechanics, there are only two books (by Sneddon and Fabrikant) that address this problem and they are restricted to the potential equation. The purpose of this book is to give an updated treatment of this subject.