Numerical integration of the boundary value problems

V. N. Maklakov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 2, pp. 354-365.

Résumé

An iterative procedure for numerical integration of boundary-value problems for nonlinear ordinary differential equations of the second order of arbitrary structure is suggested. The initial differential equation by algebraic transformation can be written as a linear inhomogeneous differential equation of the second order with constant coefficients; the right part of which is represented as a linear combination of the derivatives of the required function up to the second order and a differential equation of arbitrary structure under study. Taylor polynomials were used in the construction of the difference boundary value problem. This allowed to abandon the approximation of derivatives by finite differences. The degree of Taylor polynomials can be chosen as any natural number greater than or equal to two. Obtained inhomogeneous linear differential equation has three arbitrary coefficients. It is shown that the coefficient at the initial differential equations of any structure on the right side of the obtained non-homogeneous linear differential equation is associated with the convergence of the iterative procedure; and the coefficients at the derivatives of the required function affect the stability of difference boundary value problem at each iteration. The values of coefficients at the derivatives of the required function which ensure the stability of difference boundary value problem regardless of the type of the initial equation are theoretically set up. Numerical experiment showed that the coefficient providing the convergence of the iterative procedure depends on the type of the initial differential equation. Numerical experiments showed that the increase in the degree of the Taylor polynomial reduces the error between the exact and the obtained approximate solutions.

Détail

Keywords: nonlinear ordinary differential equations, boundary value problems, numerical methods, iterative procedures, Taylor series.

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