Uniform early time asymptotic solutions for parabolic partial differential equations

Abstract

A uniform early-time representation is sought for the solution of the parabolic equation $$\frac{\partial }{{\partial t}}U(x,t) = \left( {\frac{{\partial ^2 }}{{\partial x^2 }} - \sum (x)} \right)U(x,t) - \int\limits_0^t {K(x,t - \tau ) U(x,\tau )d\tau + S(x,t)}$$ subject to appropriate initial and boundary conditions. The procedure is based on the use of the Laplace transformation\(u(x,\lambda ) = \int\limits_0^{ + \infty } {\exp ( - \lambda ^2 t) U(x,t) ds}\). A boundary value problem is obtained for an ordinary differential equation, in which the complex parameter λ appears. The asymptotic theory of ordinary differential equations involving a large parameter is employed. Specific attention is devoted to the question of the asymptotic matching of the boundary conditions. An asymptotic expansion for λ → ∞ is obtained for the solution of the transformed problem. Then a term-by-term Laplace inverse-transformation is applied, yielding a formal early-time representation of a solution of the original problem. Finally it is proved that the latter is indeed the early-time representation of the unique solution of the problem, and that it is validuniformly with respect to x. The early-time representation takes the form of an asymptotic series involving powers of t, as well as iterated error functions.