Numerical integration via power series expansions



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A method for obtaining numerical solutions to initial value problems by implementing a generalized method of Frobenius for digital computers is presented. Power series which are equivalent to the Taylor's series expansion and which represent the solution of differential equations over finite intervals of the independent variable are constructed with recurrence formulas. A programming approach for overcoming difficulties in obtaining the necessary power series coefficients for nonlinear differential equations is developed. Example applications of the method are made to several second-order nonlinear differential equations containing trigonometric and other transcendental functions. Flow diagram solutions for several examples are included. Evaluation and computer-implemented convergence tests for power series which allow one to obtain numerical solutions of specified accuracy are discussed. The accuracy of the power series expansion method is demonstrated with numerical results, and the method is shown to have some outstanding characteristics when compared to standard integration schemes. Conclusions and recommendations for further study are included.