The non-characteristic Cauchy problem for the diffusion equation in plane geometry is re-examined and an analytic solution is derived. There is a well known series solution to this classic problem which involves derivatives of all orders of the Cauchy data. In this work, we present a new series solution to the problem which does not require higher derivatives of the Cauchy data. The solution is expressed in a series of Laguerre polynomials in the time variable with coefficients involving new specialized functions of the space variable. A complete analysis of these specialized functions is presented and their properties exploited to great effect in a number of numerical examples. All computations are performed using an efficient numerical algorithm based on a truncated form of the series solution.0266-5611/94/051165+19%19.50 0 1994 1OP Publishing LtdOnce again, substituting series (5.1) into the above equation and collecting like powers
Elastic scattering of positronium by hydrogen atoms is investigated. A method is devised wherein the effects of the long-range van der Waals attraction between the two atoms are accurately incorporated into a trial wave function in the form of a pseudo-state. The properly adjusted trial function yields a van der Waals coefficient that is within 0.03% of the known value. Improved phase shifts in the absence of exchange are reported.
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