Least Squares Technique for Solving Volterra Fractional Integro-Differential Equations Based on Constructed Orthogonal Polynomials

A numerical method is presented in this paper to solve fractional integro-differential equations. In the sense of Caputo, the fractional derivative is considered. The method proposed is the Standard Least Squares Method (SLSM) with the aid of orthogonal polynomials constructed as basic functions. The suggested method reduces this type of problem to the solution of system of linear algebraic equations and then solved using Maple18. Some numerical examples are provided to show the accuracy and applicability of the presented method. Numerical results show that when applied to fractional integro- differential equations, the method is easy to implement and accurate.


INTRODUCTION
Fractional calculus is a field dealing with integral and derivatives of arbitrary orders, and their applications in science, engineering and other fields. In recent years, the assistance of fractional differentiation for mathematical modeling of realworld physical problems has increased dramatically (e.g., earthquake modeling, reducing the spread of virus, control the memory behavior of electric socket, etc.). There are many fascinating or exciting books about fractional calculus and fractional differential equations (Caputo, 1967;Munkhammar, 2005;Podlubny, 1999).
Many fractional integro-differential equations (FIDEs) are often difficult to solve and hence may not have analytical or exact solutions in the interval of consideration, so approximate and numerical methods must be used. Several numerical methods to solve the FIDEs have been given such as Adomian Decomposition Method (Mittal & Nigam, 2008) (Rawashdeh, 2006). Rawashdeh (2006) proposed a numerical solution of integro-differential fractional equations using the method of collocation in which polynomial spline functions was used to find the approximate solution. Momani and Qaralleh (2006) suggested an efficient method for the solution of the systems of fractional integrodifferential equations solution using Adomian decomposition method (ADM). Also, Mittal and Nigam (2008), employed Adomian Decomposition Method for the solution of fractional integro-differential equations. ADM requires the construction of Adomian polynomials which was reported demanding to construct. Mohammed (2014) applied least squares method and shifted Chebyshev polynomial for the solution of fractional integro-differential equations. In the work, the author employed shifted Cheyshev polynomial of the first kind as basis function and the result was presented graphically. Taiwo and Fesojaye, (2015) applied perturbation least-Squares Chebyshev method for solving fractional order integro-differential equations. In their work, an approximate solution taken together with the Least -Squares method is utilized to reduce the fractional integro-differential equations to system of algebraic equations, which are solved for the unknown constants associated with the approximate solution.
Mohamed et al., (2016) applied homotopy analysis transform method for the solving fractional integro-differential equation in the work, Laplace transforms was used to reduce a differential equation to an algebraic equation. Oyedepo, et al., (2016) suggested a numerical method called Numerical Studies for the resolution of fractional Integro differential equations using the least square method and polynomials of Berntein. Also, Oyedepo and Taiwo (2019), applied standard least squares method for solving fractional integro-differential equations using constructed orthogonal as basic functions.
The main objective of this work is to find the numerical solution of the Volttera type fractionalintegro differential equation using the standard less square method based on the orthogonals constructed as basic functions. The general form of the problem class considered in this work is as follows: With the following supplementary conditions: are given smooth functions, are real constant, and are real variables varying [0, 1] and is the unknown function to be determined.

Definition 1:
Fraction Calculus involves differentiation and integration of arbitrary order (all real numbers and complex values). Example etc.

Definition 2:
Gamma function is defined as: This integral converges when real part of is positive .
Where is a positive integer.

Definition 3:
Beta function is defined as:

Definition 4:
Riemann -Liouville fractional integral is defined as: denotes the fractional integral of order

Definition 5:
Riemann -Liouville fractional derivative denoted is defined as:

Definition 6:
Riemann-Liouville fractional derivative defined as: is positive integer with the property that

Definition 7:
The Caputor Factional Derivative is defined as: Where is a positive integer with the property that For example, if the caputo fractional derivative is: Hence, we have the following properties: (1) We defined absolute error as: where is the exact solution and is the approximate solution.

MATERIALS AND METHODS
In this section, we constructed our orthogonal polynomials using the general weight function of the form: = .
This corresponds to quartic functions for and , respectively, satisfying the orthogonality conditions in the interval under consideration. According to Gram-Schmidt orthogonalization process, the orthogonal polynomial Valid in the interval [a, b] with the leading term , is given as: In this work the method assumed an approximate solution with the orthogonal polynomial as basis function as: Where denotes the orthogonal polynomial of degree where , are constants.

Demonstration of the Proposed Method
In this section, we demonstrated the two proposed methods mentioned above.

Standard Least Squares Method (SLSM):
The standard least square method with the orthogonal polynomials constructed as the basis is applied in order to find the numerical solution of the fractional integro-differential equation of the type and this method is based on approximating the unknown function by assuming an approximation solution from the defined in (28).

Consider Equation operating with on both sides as follows:
Substituting into : Hence, the residual equation is obtained as: Where is the positive weight function defined in the interval, . In this work, we take for simplicity. Thus: (34)

Numerical Examples
In this section, we have shown on some examples the method discussed above on the general integratio n -differential equations. The problems are solved using the constructed orthogonal polynomials as basic function. The examples are solved to illustrate the computational cost accuracy and efficiency of the proposed methods using Maple 18.

CONCLUSION
The study showed that the method with the constructed orthogonal polynomials is successfully used for solving FIDEs in a wide range with three examples. The method gives more realistic series solutions that converge very rapidly in fractional equations. The results obtained showed that the method is powerful when compared with the exact solutions and also show shown that there is a similarity between the exact and the approximate solution. Calculation showed that SLSM is a powerful and efficient technique to find a very good solution for this type of equation as well as analytical solutions to numerous physical problems in science and engineering. Also, the results were presented in graphical forms to further demonstrate the method.