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An adaptation of Adomian decomposition for numeric-analytic integration of strongly nonlinear and chaotic oscillators. (English) Zbl 1120.70303
Summary: A novel form of an explicit numeric-analytic technique is developed for solving strongly nonlinear oscillators of engineering interest. The analytic part of this technique makes use of Adomian Decomposition Method (ADM), but unlike other analytical solutions it does not rely on the functional form of the solution over the whole domain of the independent variable. Instead it discretizes the domain and solves multiple IVPs recursively. ADM uses a rearranged Taylor series expansion about a function and finds a series of functions which add up to generate the required solution. The present method discretizes the axis of the independent variable and only collects lower powers of the chosen step size in series solution. Each function constituting the series solution is found analytically. It is next shown that the modified ADM can be used to obtain the analytical solution,in a piecewise form. For nonlinear oscillators such a piecewise solution is valid only within a chosen time step. An attempt has been made to address few issues like the order of local error and convergence of the method. Emphasis has been on the application of the present method to a number of well known oscillators. The method has the advantage of giving a functional form of the solution within each time interval thus one has access to finer details of the solution over the interval. This is not possible in purely numerical techniques like the Runge-Kutta method, which provides solution only at the two ends of a given time interval, provided that the interval is chosen small enough for convergence. It is shown that the present technique successfully overcomes many limitations of the conventional form of ADM. The present method has the versatility and advantages of numerical methods for being applied directly to highly nonlinear problems and also have the elegance and other benefits of analytical techniques.
MSC:
70-08Computational methods (mechanics of particles and systems)
70K55Transition to stochasticity (chaotic behavior)