COLLOCATION TECHNIQUES FOR SOLVING MULTI-ORDER FRACTIONAL DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS.
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Date
2018-06
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UNIVERSITY OF ILORIN
Abstract
Multi-order fractional differential and integro-differential equations are increasingly being used to model many physical phenomena which occur in fluid mechanics, engineering and other areas of science. Unfortunately, exact solutions do not exist for most of this class of problems in closed form. The necessity to nd approximate solutions to such problems thus arises. Therefore, an approximate solutions through numerical treatment is desired. The Multi-order fractional integro-differential equations considered in this work are generally in the form:
n
Xi
D ((y(x)) + piy(i)(x) + J (y(x)) = f(x)
=0
subject to the conditions; yk(0) = k; k = 0; 1; 2:::n 1
where D and J denote the fractional order derivative and integral of the function y(x) respectively. The objectives of the study were to: (i) construct new orthogonal polynomials for the general class of Multi-order Fractional Differential Equations (MFDEs) and Multi-order Fractional Integro-differential Equations (MFIDEs); (ii) use Standard and Perturbed Collocation methods to solve the class of problems in (i); (iii) examine the absolute estimated errors; and (iv) examine the accuracy of the proposed methods by comparing the results obtained with exact solutions where such exist. The use of Standard Collocation Method (SCM) and Perturbed Collocation Method (PCM) to nd the numerical solution of multi-order fractional differential and integro-differential equations using the constructed orthogonal polynomials as basis function in the approximation used were tested. Some numerical examples were considered to illustrate the accuracy of the methods. From the computational view, the proposed methods are reliable and in total agreement with some known results in the literature. The results obtained include:
(i) new orthogonal polynomials basis functions were constructed;
(ii) Standard Collocation Method (SCM) successfully solved linear and nonlinear multi-order fractional differential and integro-differential equations;
(iii) Perturbed Collocation Method (PCM) successfully solved linear and nonlinear multi-order fractional differential and integro-differential equations;
(iv) the absolute estimated errors closely approximate the exact solution. The study con-cluded that Standard and Perturbed Collocation Methods serve as alternative and additional ways of nding numerical solutions of multi-order fractional differential and integro-differential
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equations. The use of newly constructed orthogonal polynomial as basis function for solving multi-order fractional differential and integro-differential equations is recommended, and also, the proposed methods can be extended to solve integral and partial fractional order differential equations.
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Keywords
COLLOCATION TECHNIQUES, MULTI-ORDER FRACTIONAL DIFFERENTIAL, INTEGRO-DIFFERENTIAL EQUATIONS