NUMERICAL METHODS FOR SOLVING SINGULAR AND MULTI-ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

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Date

2021-01

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UNIVERSITY OF ILORIN

Abstract

This thesis considers the construction of canonical polynomials with the aid of Lanczos and Ortiz method. The research work is focussed on the approximate solution of a special class of Fractional Inte- gro Differential Equations (FIDEs) of the singular and multi-order Volterra types. Fractional calculus as a branch of Mathematics deals with the integrals and derivatives of arbitrary orders, appearing in the fields of Engineering and Sciences. Many researchers that have worked in this area remarked that rigorous works were involved in obtaining the approximate solutions to the general class of problem considered (see (1) and (2)) and yet the results obtained were less accurate. Thus, this study was geared towards overcoming the draw- backs of the existing methods.The aim of this study, therefore was to propose three numerical methods and to construct canonical poly- nomials that provide less rigorous works in terms of computational costs with improved accuracy. The objectives of the study were to: (i) construct canonical polynomials to be used as basis functions for the general class of singular and multi-order fractional integro dif- ferential equations; (ii) demonstrate the formulation of the proposed methods on the general class of (1) and (2) using the constructed canonical polynomials as basis functions; (iii) carry out the formu- xii n ∫ lated three proposed methods on some numerical examples; and (iv) compare the results obtained with the avialable exact solutions in the literature. The general form of the class of problems considered in the study is given as: Dαy(x) + Σi=0 piyi(x) + λ x y(t) √x − tdt = f (x) (1) subject to the conditions yk(0) = αk, k = 0, 1, 2, · · · , n − 1, 0 ≤ x, t ≤ 1, (2) where Dαy(x) is the αth Caputo derivative of y(x); f (x) is a given smooth function; pJis are constants; x and t are given real variables in the interval [0, 1]; y(x) is the unknown function to be determined and (1) together with conditions (2) is referred to as singular and multi-order fractional integro differential equations. The methods considered for solving the problems were the Standard Collocation Method (SCM), Perturbation Collocation Method (PCM) and Ex- ponentially Fitted Collocation Method (EFCM). In each case, the proposed methods (SCM, PCM, EFCM) assumed approximate solu- tion using the constructed canonical polynomials as basis functions which is then substituted into the problems considered. For the case of the PCM, the right hand side of the problem considered is per- 0 xiii Σ turbed by adding a perturbation term denoted by HN (x), where N is the degree of the approximant used. Similarly, for the case of the EFCM, the problem considered is slightly perturbed by adding n−1 τiTN −i+1(x) i=1 where n is the order of the problem considered, τi(i ≥ 0) are the free tau parameters to be determined and the conditions are fitted with ex- ponential of the form τnea which consists of one tau-parameter to be determined along with the unknown constants. Hence, all the three proposed methods resulted into algebraic linear system of equations after collocating at some equally spaced interior points by which the solutions of the unknown constants were obtained using the Guas- sian elimination method or Maple 18 software. The findings of the study were that: (i) canonical polynomials used as basis functions for the general class of singular and multi-order fractional integro differential equations were successfully constructed; (ii) three proposed numerical methods demonstrated on the general class of singular and multi-order fractional integro differential equations were successfully carried out; (iii) numerical examples were successfully solved by the three pro- xiv posed numerical methods; and (iv) results obtained using the three proposed numerical methods were comparatively better than those available in the literature in terms of accuracy. The study concluded that SCM, PCM and EFCM are alternative methods of finding numerical solutions to singular and multi-order fractional integro differential equations. It is therefore recommended that the methods be adopted in solving singular and multi-order frac- tional integro differential equations for better accuracy.

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Keywords

NUMERICAL METHODS, SINGULAR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS, MULTI-ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

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