The iterative solution of non-linear ordinary differential equations in Chebyshev series. (English) Zbl 0133.08704

The paper is concerned with numerical solution of the first-order nonlinear differential equation (1) \(dy/dx = f(x, y)\) under the boundary condition of the form \(\alpha y(-1) +\beta y(+1) = \gamma\). To the equation (1), the author applies Newton’s method to compute a root of an equation and he derives an iterative formula \[ dy_i/dx = f(x, y_{i-1}) + (y_i - y_{i-1})f_y(x, y_{i-1}),\quad \alpha y(-1) +\beta y(+1) = \gamma, \quad (i=1, 2, 3, \ldots). \tag{2} \] To perform the iterative process (2) numerically, the author represents all functions appearing in (2) by their finite Chebyshev series and replaces the analytic operations involved by the arithmetic operations on the coefficients of these series using the formulas established by Clenshaw and the author [C. W. Clenshaw, Chebyshev series for mathematical functions. London: Her Majesty’s Stationary Office, Department of Scientific and Industrial Research (1962; Zbl 0114.07101), C. W. Clenshaw and H. J. Norton, Comput. J. 6, 88–92 (1963; Zbl 0113.11002)]. The formula (2) leads to a system of linear equations for the coefficients of the finite Chebyshev series representing the function \(y_i(x)\). Solving this system of linear equations, the author gets the finite Chebyshev series representing \(y_i(x)\) and repeats this process until the desired state of convergence is attained for the sequence of the finite Chebyshev series representing \(\{y_i(x)\}\). The extension of the method to the second-order nonlinear differential equations is also described. The paper contains various numerical examples and these illustrate the effectiveness of the author’s method though the mathematical foundation of the method is not given in the paper.
Reviewer: M. Urabe


65L10 Numerical solution of boundary value problems involving ordinary differential equations
65H10 Numerical computation of solutions to systems of equations
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