zbMATH — the first resource for mathematics

Numerical verification and error analysis of asymptotic solutions of strongly nonlinear oscillation. (Chinese. English summary) Zbl 1117.65342
Summary: The multiple scales method of G. E. Kuzmak [Prikl. Math. Mekh. 23, 515–526 (1959; Zbl 0089.29803)] and of J. C. Luke [Proc. R. Soc. Lond., Ser. A 292, 403–412 (1966; Zbl 0143.13603)] can be efficiently applied to obtain the solutions of strongly nonlinear oscillators with slowly varying parameters. A technique of numerical order verification is applied to verify that the asymptotic solutions are uniformly valid for small parameter. A numerical comparison of error shows that the asymptotic solutions obtained by the multiple scales method of Kuzmak-Luke [loc. cit.] are uniformly valid, and the errors are about one-tenth of the small parameter $${\varepsilon}$$.
MSC:
 65L05 Numerical methods for initial value problems 65L70 Error bounds for numerical methods for ordinary differential equations 65G20 Algorithms with automatic result verification 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems, general theory