Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integrodifferential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal method in solving these problems.


Introduction:
Fractional integro-differential equations (FIDE's) occur in many applications in the sciences (physics, engineering, finance, biology) (1). In most of the problems the analytical solution cannot be found, and hence finding a good approximate solution using numerical methods will be very helpful (2).
Many researchers studied and discussed the numerical solution of FVFIDE's. Mittal and Nigam (1) in 2014 used Adomian decomposition approach to find numerical solution to FIDE's of Volterra type with Caputo fractional derivative. Huang et al (3) in 2011 used Taylor expansion series for solving (approximately) a class of linear fractional integrodifferential equations including two types Fredholm and Volterra. Mohammed (2) in 2014 investigated numerical solution of LFIDE's by the least squares method with the aid of shifted Chebyshev polynomial. Maleknejad et al (4) in 2013 presented a numerical scheme, based on the cubic B-spline wavelets for solving fractional integrodifferential equations. Mohamed et al (5) in 2016 introduced an analytical method, called homotopy analysis transform method (HATM) which is a combination of HAM and Laplace decomposition method, this scheme is applied to linear and nonlinear fractional integro-differential equations.
Shwayyea and Mahdy (6) in 2016 investigated the numerical solution of linear fractional integrodifferential equations by the least squares method with the aid of shifted Laguerre polynomial. Oyedepo et al (7) in 2016 proposed two numerical methods for solving FIDE's the proposed methods are the least squares method with the aid of Bernstein polynomials function as the basis. Senol and Kasmaei (8) in 2017 developed with perturbation-iteration algorithm to obtain approximate solutions of some FIDEs. Alkan and Hatipoglu (9) in 2017 study sinc-collocation method for solving Volterra-Fredholm integrodifferential equations of fractional order. Syam (10) in 2017 modified the version of the fractional power series method to extract an approximate solution of the model. The method is a combination of the generalized fractional Taylor series and the residual functions. Hamoud and Ghadle (11) in 2018 applied the Adomian decomposition and the modified Laplace Adomian decomposition methods to find the approximate solution of a nonlinear FVFIDE. Hamoud et al (12) in 2018 studied the existence and uniqueness theorems for FVFIDE's. different types of problems. Wang and Wang (13) in 2013 used Lagrange collocation method to solve Volterra-Fredholm integral equations, this method transforms the system of the linear integral equations into matrix form via Lagrange collocation points. Mustafa and Muhammad (14) in 2014 introduced a numerical method for solving linear Volterra-Fredholm integro-differential equations of the 1st order using three types of Lagrange polynomial including OLP, MLP and BLP. Mustafa and Ghanim (15) in 2014 used Lagrange polynomials for solving linear Volterra-Fredholm integral equations by three types including OLP, MLP and BLP. Liu et al (16) in 2017 solved the two-dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Pan and Huang (17) in 2017 presented a modified barycentric rational interpolation method for solving two-dimensional integral equations. Tian and He (18) in 2018 used barycentric rational interpolation collocation method to solve higherorder boundary value problems. Wu et al (19) in 2018 find numerical solution of a class of nonlinear partial differential equations using Barycentric interpolation collocation method.
This study aims to find numerical solutions of LFVFIDE of the following form: ( ) = ( ) ( ) + ( ) ( ) denote the 'Caputo fractional derivative' of ( ); ( ), ( ), 1 ( , ) and 2 ( , ) are continuous functions, are real variables in [a,b] and ( ) is the indefinite function to be determined using OLP, MLP and BLP. Section (2), introduces some necessary definitions and mathematical preliminaries which are required for establishing our results. While section (3) presents the derivation of the proposed methods. Section (4) proposes the general algorithm for the method. Test examples are given in section (5) including general and special cases of LFVFIDE to improve the capability of the proposed method to solve various type of equation in addition with LFVFIDE, in all the test examples ( ) is chosen in such a way that we know the exact solution. The exact solution is used only to show that the numerical solution obtained with our method is true.

Preliminaries
There are different definitions of 'fractional integral' sometimes defined in (0, ∞) and sometimes called the left-sided integrals are given below:

General Algorithm for Methods
To evaluate numerical solutions of LFVFIDE using OLP, MLP and BLP, the following steps are introduced: Step 1: assume ℎ = − , ℕ , ( ) = 0 (the initial condition is given).
Step 2: put = + ℎ , with 0 = and = , = 0,1, ⋯ , . Step 4: Solve the system ( U ⃑ ⃑ = ) using step3 and Gauss elimination method with partial pivoting. Note that, we can use another method to solve the system in step 4 like LU decomposition method, but the computational cost of computing a solution via Gaussian elimination or LU is the same.

Numerical Applications
In this section, four numerical examples are considered to confirm the efficiency of the above methods for solving LFVFIDE's. MATLAB\R2018a are used to apply the algorithms. with the true solution ( ) = ( ) . where ℎ represent the generalized hyper geometric function in MATLAB. Table 1 shows the absolute error by using OLP, MLP, and BLP with n=5. Table 2 contains the maximum error by using OLP, MLP, and BLP with n=4, 5,8,10. Such that ‖ ‖ ∞ represents the maximum absolute error and R.T. represents running time.  With the true solution ( ) = − 3 .
where represent the gamma function. Table 3 represents the absolute error by using OLP, MLP, and BLP with n=5. Table 4 contains the maximum error by using OLP, MLP, and BLP. with n=4,5,8,10, with the best results obtain in (23) using Laguerre polynomials.     Table 5 represents the absolute error by using OLP, MLP, and BLP with n=5. Table 6 contains the maximum error by using OLP, MLP, and BLP with n=4,5,8,10 .  (Note that in this case k 1 (x,t)=0 and k 2 (x,t)=0 ). Table 7 represents the absolute error by using OLP, MLP, and BLP with n=5. Table 8 contains the maximum error by using OLP, MLP, and BLP with n=4,5,8,10, with the best results obtain in (24) using fractional Euler's method.

Conclusion:
In this study, Lagrange polynomials including: (OLP), (MLP), (BLP) are applied for solving the LFVFIDE's. According to results from examples, we conclude that:  The three methods offer the force and capability of the introduced algorithms.  The faster method is MLP due to the point that computing the integral part appearing in Eq. (18) is easier than that in Eq. (11) and Eq. (21), and this result agrees with references (14) and (15).  As n (the degree of polynomials) increases, the error term is decreased in all methods except when the exact solution is a polynomial with low degree (we can satisfy with low degree).  Also from Tables (2, 4 and 6) one can note that the BLP method has no results for n=8 and n=10 because of difficulty for finding the fractional derivatives which are so hard to compute exactly by hand or by using MATLAB within reason of shape of BLP appear in Eq. (18) and this result doses not agree with the other papers since the fractional derivative appears here only not in the other .  Due to the same reason, we suggest using numerical integration instead of exact value to avoid the difficulty of finding the integration in Eq. (21).  Methods can be extended and applied to nonlinear FVFIDE, in this case the problem transforms to nonlinear system of equation which can be solved by using Newton's method.