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 < x < ∞}, and on the quarter‐plane {t > 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(x1, x2, t) as an integral in the complex (k1, k2)‐planes involving an (x1, x2)‐transform of the initial condition, an (x2, t)‐transform of the boundary conditions at x1 = 0, and an (x1, t)‐transform of the boundary conditions at x2 = 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.