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 omega(k), in the domain l(t)<x<infinity, 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-iomega(k)t]rho(k) along a time-dependent contour, or an integral of exp[ikx-iomega(k)t]rho(k, &kmacr;) over a fixed two-dimensional domain. The functions rho(k) and rho(k,&kmacr;) 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.
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