A method for fractional Volterra integro-differential equations by Laguerre polynomials

The main purpose of this study is to present an approximation method based on the Laguerre polynomials for fractional linear Volterra integro-differential equations. This method transforms the integro-differential equation to a system of linear algebraic equations by using the collocation points. In addition, the matrix relation for Caputo fractional derivatives of Laguerre polynomials is also obtained. Besides, some examples are presented to illustrate the accuracy of the method and the results are discussed.

Here, n ∈ Z + , λ∈ R, K(x, t), p(x), and g(x) are given functions, y(x) is the unknown function to be determined, D α y(x) indicates the Caputo fractional derivative of y(x). Now, we give the definition and the basic properties of the Caputo fractional derivative as follows.
Besides, the Caputo fractional derivative of a constant function is zero and the Caputo fractional differentiation operator is linear [53].
The aim of this study is to give an approximate solution of problem (1)-(2) in the form where a i are unknown coefficients, N is chosen any positive integer such that N ≥ n, and L i (x) are the Laguerre polynomials of order i defined in Ref. [54] as Besides, the main purpose of the solution method presented in this paper is to obtain the Caputo fractional derivative of the Laguerre polynomials in terms of the Laguerre polynomials and to give a matrix representation for this relation. The Caputo fractional derivative of the Laguerre polynomials is mentioned in Ref. [51,[55][56][57]. While these matrix relations have been given depending on approximate matrices, the relation proposed in this paper is new, exact, and simpler than the former ones. This paper is organized as follows: In Sect. 2, the main matrix relations of the terms in Eq. (1) are established. In Sect. 3, the collocation method which is used to find the solution is introduced. In Sect. 4, some numerical examples are solved and their comparison with the existing results in the literature are presented to verify the accuracy and efficiency of the proposed method. The conclusion is given in Sect. 5.

Main matrix relations
In this section, we construct the matrix forms of each term of Eq. (1). Firstly, we can write the approximate solution (3) in the matrix form where Now, we will state a theorem that gives the Caputo fractional derivative of Laguerre polynomials in terms of Laguerre polynomials.
Theorem Let L i (x) be Laguerre polynomial of order i, then the Caputo fractional derivative of L i (x) in terms of Laguerre polynomials is found as follows: where α denotes the ceiling function which is the smallest integer greater than or equal to α.
Proof Let us begin deriving the Laguerre polynomials with the definition of them: By the linearity of Caputo fractional derivative, we get Using the Caputo fractional derivative of x k , k = 0, 1, 2, . . . , At this step, by taking x 1-α out of the series and using the Laguerre series of the function x k given by Lebedev [58] x we have relation (5) and the proof is completed.

Matrix relation for the differential part
Now, we will write the matrix form of the differential part of Eq. (1). The fractional part is obviously seen as The right-hand side of this equation can be expressed as where S α is an (N + 1) dimensional square matrix denoted by Here, the S k,i terms in the entries of S α are defined as follows: Then, by using relations (4) and (7), the fractional differential part of Eq. (1) can be expressed as

Matrix relation for conditions
The relation between L(x) and its derivatives of integer order is given by Yüzbaşı [44] as where the matrix M is defined by By using relation (9), the corresponding matrix forms of the conditions defined in (2) can be written as Here, the matrix L(0)M j is named U j where it is an 1 × (N + 1) dimensional matrix. Hence, Eq. (10) becomes U j A = c j , j = 0, 1, . . . , n -1.

Method of solution
To obtain the approximate solution of Eq. (1), we compute the unknown coefficients by using the following collocation method. Firstly, let us substitute the matrix forms (4) and (8) into Eq. (1), and thus we obtain the matrix equation By substituting the collocation points x s > 0 (s = 0, 1, . . . , N ) into Eq. (11), we have a system of matrix equations where v(x s ) = This system can be written in the compact form: where Denoting the expression in parenthesis of Eq. (13) by W, the fundamental matrix equation for Eq. (1) is reduced to WA = G, which corresponds to a system of (N + 1) linear algebraic equations with unknown Laguerre coefficients a 0 , a 1 , . . . , a N . Finally, to obtain the solution of Eq. (1) under conditions (2), we replace or stack the n rows of the augmented matrix [W; G] with the rows of the augmented matrix [U j ; c j ]. In this way, the Laguerre coefficients are determined by solving the new linear algebraic system.

Numerical examples
In this section, we apply the proposed method to four examples existing in the literature and to a test example constructed for this method. We have performed all of the numerical computations using Mathcad 15. We also use the collocation points by using the formula x s = [1 -cos( (s+1)π N+1 )]/2, s = 0, 1, . . . , N .
Example 1 Consider the following fractional integro-differential equation: By solving this system, we get a 0 = 2, a 1 = -4, a 2 = 2. When we substitute the determined coefficients into Eq. (3), we get the exact solution.
Using the homotopy analysis method, this problem was also solved by Awawdeh et al. [19]. They found the approximate solution for N = 5, but they did not state the numerical results of the errors of their method. Besides, Sahu et al. [32] found the approximate solution with the maximum absolute error 4.2 × 10 -15 by the Legendre wavelet Petrov-Galerkin method for N = 6. If the results are compared, it can be said that the proposed method is better than the other methods since the exact solution is found for N = 2.  This problem was also solved by Awawdeh et al. [19] with the homotopy analysis method. They found the approximate solution for N = 5, but they did not state the numerical results of the errors of their method. Besides, Sahu et al. [32] found the approximate solution with the maximum absolute error 1.1 × 10 -16 by the Legendre wavelet Petrov-Galerkin method for N = 6. If the results are compared, it can be said that the proposed method is better than the other methods since the exact solution is found for N = 1.
Example 3 Consider the following fractional integro-differential equation: Let N = 2, the collocation points become x 0 = 0.25, x 1 = 0.75, x 2 = 1. Here, the matrices in the main matrix relation of this problem are given as follows: This problem was also solved by Awawdeh et al. [19] and they found the approximate solution by the homotopy analysis method for N = 5. By the proposed method, we have found the exact solution of the problem for N = 2. Apparently, our method is better than the other method.
The main matrix equation of this problem and the conditions are given as The absolute errors of our method are compared with three methods: linear scheme, quadratic scheme, and linear-quadratic scheme for the fractional integro-differential equations of Kumar et al. [28] for N = 5 in Table 1. It is seen that our method gives better results than the other methods.

Conclusion
In this study, a collocation method based on Laguerre polynomials has been developed for solving the fractional linear Volterra integro-differential equations. For this purpose, the matrix relation for the Caputo fractional derivative of the Laguerre polynomials has been obtained for the first time in the literature. Using these relations and suitable collocation points, the integro-differential equation has been transformed into a system of algebraic equations. The method is faster and simpler than the other methods in the literature, and better than the homotopy analysis and Legendre wavelet method.