ON THE LARGE-TIME ASYMPTOTICS OF THE DIFFUSION EQUATION ON INFINITE DOMAINS

It is shown that expansions in similarity solutions provide a quick and economical method for assessing the large-time asymptotics of the diffusion equation on infinite and certain semi-infinite domains if Dirichlet or Neumann conditions are imposed. The similarity solutions are shown to form a basis for the Hilbert space {Mathematical expression}. This implies that initial conditions for the diffusion equation that are square integrable with respect to the exponentially-growing weight function {Mathematical expression} can be expanded in a discrete, infinite sum of mutually orthogonal similarity solutions, each having a different rate of amplitude decay. This leads to a rapid, almost effortless recognition of the large-time asymptotic behaviour of the solution. © 1990 Kluwer Academic Publishers.