A new transform method for evolution partial differential equations

We introduce a new transform method for solving initial-boundary-value problems for linear evolution partial differential equations with spatial derivatives of arbitrary order. This method is illustrated by solving several such problems on the half-line {t > 0, 0 0,0 < xj < ∞, j = 1, 2}. For equations in one space dimension this method constructs q(x, t) as an integral in the complex k-plane involving an x-transform of the initial condition and a t-transform of the boundary conditions. For equations in two space dimensions it constructs q (x 1 , x 2 , t) as an integral in the complex (k 1 , k 2 )-planes involving an (x 1 , x 2 )-transform of the initial condition, an (x 2 , t)-transform of the boundary conditions at x 1 = 0, and an (x 1 , t)-transform of the boundary conditions at x 2 = 0. This method is simple to implement and yet it yields integral representations which are particularly convenient for computing the long time asymptotics of the solution.