Piecewise homotopy methods for nonlinear ordinary differential equations. (English) Zbl 1137.65048
Summary: Piecewise homotopy perturbation methods are developed for the solution of nonlinear ordinary differential equations. These methods are based on the introduction of an artificial or book-keeping parameter and the expansion of the solution in a power series of this parameter and provide analytical solutions in open intervals which are smooth everywhere.
Three piecewise-adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size, a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size, are presented and applied to eight nonlinear ordinary differential equations. It is shown that piecewise-adaptive homotopy perturbation methods predict essentially the same solutions as MATLAB’s variable-step, variable-order solvers and variable-order transition matrix techniques provided that five-term approximations of the decomposition method are applied to both the displacement and the velocity.
It is also shown that piecewise homotopy perturbation techniques that use three-term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second-order accurate time-linearization technique when the same step is employed in both schemes.
Three piecewise-adaptive homotopy perturbation methods based on the use of either a fixed number of approximants and a variable step size, a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size, are presented and applied to eight nonlinear ordinary differential equations. It is shown that piecewise-adaptive homotopy perturbation methods predict essentially the same solutions as MATLAB’s variable-step, variable-order solvers and variable-order transition matrix techniques provided that five-term approximations of the decomposition method are applied to both the displacement and the velocity.
It is also shown that piecewise homotopy perturbation techniques that use three-term approximations to both the displacement and the velocity provide essentially the same results as those obtained with a second-order accurate time-linearization technique when the same step is employed in both schemes.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
decomposition method; homotopy perturbation technique; piecewise-adaptive methods; Picard’s theorem; nonlinear second-order ordinary differential equations; numerical examples; variable step sizeReferences:
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