Method for Solving Moving Boundary Value Problems for Linear Evolution Equations

We introduce a method of solving initial boundary value problems for linear evolution equations in a time-dependent domain, and we apply it to an equation with dispersion relation , in the domain [Math Processing Error], [Math Processing Error]. We show that the solution of this problem admits an integral representation in the complex [Math Processing Error] plane, involving either an integral of [Math Processing Error] along a time-dependent contour, or an integral of [Math Processing Error] over a fixed two-dimensional domain. The functions [Math Processing Error] and [Math Processing Error] can be computed through the solution of a system of Volterra linear integral equations. This method can be generalized to nonlinear integrable partial differential equations.