Abstract
The block hybrid collocation method with three off-step points is proposed for the direct solution of fourth order ordinary differential equations. The interpolation and collocation techniques are applied on basic polynomial to generate the main and additional methods. These methods are implemented in block form to obtain the approximation at seven points simultaneously. Numerical experiments are conducted to illustrate the efficiency of the method. The method is also applied to solve the fourth order problem from ship dynamics.
Highlights
Fourth order ordinary differential equations (ODEs) arise in several fields such as fluid dynamics, beam theory, electric circuits, ship dynamics, and neural networks
The hybrid collocation method that generates the approximations to the general fourth order ODEs (1) is defined as follows: k j=0 j=1
[6] and Cortell [7] considered the conventional approach of reduction to system of first order ODEs
Summary
Fourth order ordinary differential equations (ODEs) arise in several fields such as fluid dynamics (see [1]), beam theory (see [2, 3]), electric circuits (see [4]), ship dynamics (see [5–7]), and neural networks (see [8]). Kayode [11, 12] developed collocation methods for the approximation of y at tn+5 with the predictor of orders five and six, respectively. These schemes [9, 11, 12] are implemented in predictor-corrector mode with the employment of Taylor series expansions for the computation of starting values. We are going to derive a block hybrid collocation method for the direct solution of general fourth order ODEs (1). We apply the interpolation and collocation technique on basic polynomials to derive the main and additional methods which are combined and used as block hybrid collocation method This method generates the approximation of y at four main points and three off-step points concurrently
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