DOUBLE PERTURBATION COLLOCATION METHOD FOR SOLVING FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

Abstract

In this work, we proposed a computational technique called the Double Perturbation Collocation Method (DPCM) for the numerical solution of fractional Riccati differential equation. The DPCM requires the addition of a perturbation term to the approximate solution in terms of the shifted Chebyshev polynomials basis function. This function is substituted into a slightly perturbed fractional Riccati equation. The fractional derivative is in the Caputo sense. The resulting equation is simplified and then collocated at some equally spaced points. Thus resulted into system of equations which are then solved by implementing Gaussian elimination method for linear to obtain the unknown constants and for the case of nonlinear, Newton linearization scheme of appropriate orders are used to linearize. The values of the constants obtained are then substituted back into the perturbed approximate solution. Results obtained with DPCM compared favourably well with existing results in literature and the exact solutions where such existed in closed form. Some numerical examples are included to illustrate the accuracy, simplicity and computational cost of the method.

Keywords: Riccati., collocation, double - perturbation, fractional, linearization


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