Power series solutions of partial differential equations



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The results of a preliminary investigation into the feasibility of obtaining approximate numerical solutions to problems governed by partial differential equations by means of a generalized method of Frobenius are presented. The method of solution utilizes products of power series in the independent variables of a given problem. All problems considered are governed by linear second-order equations. Example applications are made to an initial-value problem characterized by the one-dimensional wave equation and to a boundary-value problem characterized by Laplace's equation. The results for these two example applications are recognized to be equivalent to the exact analytical solutions of the problems. A study of the application of the method to the solution of a hypothetical set of boundary-value problems governed by partial differential equations with variable coefficients is also presented. The results of the investigation appear promising in certain areas. Recommendations for areas of future study are included.