The Method of Fokas for Solving Linear Partial Differential Equations

The classical methods for solving initial-boundary-value problems for linear partial differential equations with constant coefficients rely on separation of variables, and specific integral transforms. As such, they are limited to specific equations, with special boundary conditions. Here we review a method introduced by Fokas, which contains the classical methods as special cases. However, this method allows also for the equally explicit solution of problems for which no classical approach exists. In addition, it is easy to elucidate which boundary-value problems are well posed and which are not. We provide examples of problems posed on the positive half-line and on the finite interval. Some of these examples have solutions obtainable using classical methods, others do not. For the former examples, it is illustrated how the classical methods may be recovered from the more general approach of Fokas.