A computer algebra approach to partial difference equations

Solutions to the Dirichlet problem for (elliptic) partial difference equations are needed both in the solutions of boundary value problems of elliptic partial differential equations and in random walk problems. Buchberger's Grobner basis algorithm is used to develop scMAPLE programs to find closed form expressions for the generating functions of such finite difference equations.;From the programs, data for the expected durations of the two-dimensional gambler's ruin problem are generated. From this data, relationships that may exist for these expected durations are conjectured.