Browsing by Author "Bryan, John Loyd"
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Item Numerical solutions of diffusion-type equations(1969) Bryan, John Loyd; Childs, S. Bart; Hedgcoxe, Pat G.; Bannerot, Richard B.; Hubbard, Martin G.A method of obtaining numerical solutions of a general class of boundary-value problems governed by the two-dimensional diffusion equation is investigated. The method employs a partial discretization of independent variables to reduce the problem of partial differential equations to a sequence of related boundary-value problems governed by a system of linear second-order ordinary differential equations. The generality of the method is demonstrated by applications to example problems involving both regular and irregular boundaries with boundary conditions of a general type. Application of separation of variables techniques to obtain closed-form solutions of a certain class of problems is presented and the results are used to indicate the accuracy of the method. An investigation into the stability characteristics of the resulting system of ordinary differential equations is also presented. It is concluded that the method appears to show promise as an easily implemented numerical method but that the full potential of the approach will not be realized until significant advances have been made in both computing hardware and software.Item Power series solutions of partial differential equations(1967) Bryan, John Loyd; Childs, S. Bart; Hubbard, Martin G.; Cox, Jimmy E.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.