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 ω(k), in the domain l(t)<x<∞, 0<t<T. We show that the solution of this problem admits an integral representation in the complex k plane, involving either an integral of exp[ikx-iω(k)t]ρ(k) along a time-dependent contour, or an integral of exp[ikx-iω(k)t]ρ(k,k̅ ) over a fixed two-dimensional domain. The functions ρ(k) and ρ(k,k̅ ) 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.