Theoretical and Numerical Study for Volterra − Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

In this paper


Introduction
Te concept of fractional calculus has attracted interest of mathematicians and physicists because of its efectiveness in describing various phenomena in biology [1][2][3], medicine [4][5][6][7], physics [8,9], and fnance [10].A signifcant number of real-world problems are represented by diferential equations involving fractional derivative.Tis notion of fractional derivation generalises the classical derivative of integer order to a derivation of a noninteger order.Te two most used defnitions of fractional integration and derivation are the Riemann− Liouville (R-L) sense and the Caputo sense.Tey are mostly used for their hight explication of memory efect which is needed in diferent real problems especially for medicine, physics, and fnance.Te issue is that most diferential equations lack explicit analytical solutions.
For this, many researchers head for numerical methods to obtain approached solutions in efective way.Now, the challenge for mathematicians is to develop suitable and accurate algorithms to get the required solutions.
Te existence and uniqueness of the solution of such equations have been widely studied in many papers, as can be seen in [11][12][13][14][15].
In order to expand the feld of study of this equations, numerical methods are introduced.It is well known that there are numerous numerical methods for solving diferent equations.Spectral methods (Galerkin, collocation, Tau, Petrov− Galerkin) take the great interest of researchers because of the efciency of its numerical schemes.Tese methods consist of expressing the solution as a fnite expansion of some known functions defned all over the domain which imports it a global character.In virtue of the special properties of the chosen basis functions, spectral methods gain fast convergence rate and hight accuracy compared to other numerical methods.Orthogonal polynomials, also known as eigenfunctions of the Sturm− Liouville problem are the most commonly used basis functions [16][17][18][19][20].
Collocation methods are becoming increasingly popular for solving a wide variety of diferential equations.For instance, spectral collocation methods are used for integral and integro-diferential equations (21), for multiorder FDEs [22], for multiorder FEs with multiple delays [23], for twodimensional FIDs with weakly singular kernel [24], and also for systems of FDEs [25].
Tere has been a lot of interest in solving FI-DEs [26][27][28][29], specially using collocation methods.In [30], the authors used a wavelet collocation method to solve Fredholm FI-DEs.Doha et al. in [31] used a Shifted Jacobi polynomials to build a Gausscollocation method for FI-Des, namely, linear and nonlinear Volterra, Fredholm, mixed Volterra− Fredholm, and system of Volterra equations.In [32], the authors investigated frst kind of Bessel polynomials in a collocation technique to approximate the solution of a linear Fredholm− Volterra FI-DEs of multihight order.In [33], an implementation of the collocation method using shifted Legendre polynomials coupled with Gauss− Legendre quadrature is presented.Rahimkhani and Ordokhani [34] used alternative Legendre functions for nonlinear FI-DEs.We can see also other works like [35][36][37].
Herein, we propose an efcient spectral method to approximate the solution of (1)- (2).We are heading towards writing the solution as a fnite expansion of S-Cheb-3, and we describe the steps of the shifted Chebyshev− Gauss collocation method based on the zeros of Chebyshev polynomials of the third kind (Cheb-3).After substituting the approximation in the studied problem and considering the collocation, we evaluate integrals using Chebyshev− Gauss quadrature.Depending on the linearity of G 1 and G 2 , we study two cases.In the linear case where , we obtain a simple algebraic system that we solve by a Gauss elimination algorithm.For the second case , we obtain a nonlinear system for which we calculate the Jacobian matrix and solve it using a Newton algorithm.Tis scheme is considered a suitable method because of its simplicity and accurate results obtained for diferent examples.One of its advantages is that it treats diferent types of integro-diferential equation, namely, Volterra, Fredholm, and mixed Volterra− Fredholm in both linear and nonlinear cases.Moreover, the use of Cheb-3 is of great importance; in virtue of the several properties, these polynomials simplify the calculations; it is a fact that they constitute an orthogonal basis that can be written as a compact combination of Cheb-2 hands to generate an accurate algorithm of approximation to gain the spectral accuracy, which is the well-known advantage of spectral methods.Furthermore, an error analysis is established and numerical results are exposed to validate the efectiveness of the proposed method.
Te paper is organized as follows: In Section 2, we introduce some important mathematical concepts useful for the elaboration of the method.Section 3 is devoted to develop the numerical method; the linear and nonlinear cases are studied separately in Sections 3.1 and 3.2.Section 4 presents error analysis based on some defnitions and useful lemmas.Numerical examples are studied in Section 5, and results are presented and discussed in a clear way.Finally, we summarize all details in conclusion.

Mathematical Preliminaries
First, we present necessary defnitions and mathematical tools required for the development of our study.

Some Essentials of Fractional Calculus.
Let α > 0 and Γ(α) be the gamma Euler function.
(ii) V n (ϰ) verify the recurrence relations

Essential Properties of Shifted Chebyshev Polynomials of the Tird Kind.
We defne the S-Cheb-3 using the change of variable (ii)  V n (ϰ) may be produced by employing recurrence relations (iii) Te analytical form of S-Cheb-3 of degree n in ϰ is (iv) Te zeros of the polynomial  V n (ϰ) are of the form (v) A function ϑ(ϰ) which is square integrable in [0, 1] can be expressed in terms of S-Cheb-3 as where for ı � 1, 2, . . .
Proof.Using 2ϰ − 1 � cos θ, the relation (18) becomes and after two integrations by parts, we get where Ten, From Teorem 1, Hence the uniform convergence.
Using the Gauss− Chebyshev integration formula, we get where τ q , ω q correspond to the m + 1 zeros of V m+1 and their respective weights such that From boundary condition (2) and approximation (19), we have Now, for numerical resolution, we use a Gauss elimination method to solve (32)-(34).

Nonlinear Case. We assume in this case
where r, s ≠ 1.By substituting the relation (19) in the equation (1) and using Teorem 3, we get Following the same steps described in the linear case, we obtain the scheme joined to the initial (34).Now in order to solve (34)- (36), we use a Newton method.For this, we defne F a vectorial function as such that for all j � 1, . . ., m and from boundary condition for j � m + 1

Complexity
Te Jacobian matrix of F is where for p � 1, . . ., m.

Error Analysis
In this section, we formulate an error analysis for the proposed method.To begin, we introduce some fundamental defnitions and lemmas that will be useful in the following (for more details see [39]).Here, we use C to denote a generic positive constant independent of m.We defne Π m : L 2 w (I) ⟶ P m as the orthogonal projection operator such that Defnition 5. We defne with where s is a non-negative integer.
Lemma 1 (see [39]).Assume that ϑ ∈ H s (I), we note the interpolation of ϑ at the Chebyshev Gauss points by I m ϑ which satisfes Lemma 2 (see [39]).Assume that ϑ ∈ H s w (I) and I m ϑ denotes the interpolation of ϑ at (m + 1) Chebyshev Gauss points corresponding to the weight function w(ϰ), then (51)

Error Analysis.
Here, we provide error analysis of the method described in (29) by using previous defnitions, lemmas, and Sobolev inequality.

Numerical Examples
Tis section presents diferent examples to illustrate the efectiveness of the described method.Several examples of diferent cases are treated using MATLAB R2020a on a personal computer characterized with properties AMD Ryzen 5 5600X 6-Core Processor 3.70 GHz.To analyse the obtained results, diferent error values are calculated.We defne the absolute error the maximum absolute error and the root mean square error where N ϰ is a subdivision of [0, 1] and h the appropriate step.We defne also the relative error To study the numerical convergence of the method, the numerical rate of convergence according to relative error NRC is calculated as follows: where Example 1.We consider a linear Volterra equation with the condition where knowing that the exact solution is ϑ(ϰ) � ϰ 3/2 .
Tables 4 and 5  Table 6 corresponds to Example 5 of nonlinear case and shows values of approximate solutions of problems with different α on diferent points of [0, 1].For the last two columns, we can compare the approximate solution obtained for m � 2 and m � 8 and the exact solution values taken for ϰ � 0.25, 0.5, 0.75.
Figures 1 and 2 illustrate the behaviour of the RE and m avg (NRC) for Examples 1 and 4. We remark on each case that RE decreases the same way of m avg(NRC) which expresses the linear convergence of the approximate solution to the exact solution.
Figure 3 compares graphically the exact and the approximate solution obtained for m � 6 for Example 2. For the same example, Figure 4 depicts the values of AE throughout [0, 1] when m � 18.
In Figure 5, we plot RMSE and MAE in log scale versus m, proving the convergence of the method.
Figure 6 shows the obtained approximations for various values of α (0.5, 0.75, 0.9, 1) when m � 6 compared to the exact solution when α � 1. Te behaviour noticed here afrms the stability of the solution using the described method.
Finally, Figure 7 depicts the approximate solution's convergence to the exact solution when m takes the values 2, 6, 12 for Example 6.

Conclusion
In this article, the S-Cheb-3 was used in a Gauss-collocation method to approximate the solution of FI-DEs.Te fractional derivation is considered in the Caputo sense.Our investigation relies on using the S-Cheb-3 as spectral basis with the Cheb-3 zeros as collocation points to build an accurate algorithm applicable for linear and nonlinear equations.To this end, error analysis was established and several numerical examples of diferent types (Volterra, Fredholm, and mixed Volterra− Fredholm in both linear and nonlinear cases) were presented to show the simplicity and accuracy of the method.Te proposed method also guarantees the convergence of the approximate solution to the exact solution for each example and provides signifcant error values when compared to other methods.Furthermore, the experimental rate of convergence calculated for diferent examples exhibits the same behavior of the relative error obtained for diferent m, validating the convergence results.12 Complexity

4. 1 .
Defnitions and Lemmas Defnition 4. Let L 2 w (I) � ϑ, ϑ mesurable and ‖ϑ‖ w < ∞  , I � [0, 1] be the weighted space with 〈ϑ, θ〉 w �  1 0 ϑ(ϰ)θ(ϰ)w(ϰ)dϰ, ‖ϑ‖ w � 〈ϑ, ϑ〉 1/2 w .(45) present RMSE and RE with respect to m for examples of linear case (Examples 1-3).Using the technique described in Section 3.1, we observe that the values of RMSE and RE are decreasing when taking increasing values of m.Te NRC is also calculated for each example which takes the average value avg(NRC) � − 1, 6695 for Example 1, avg(NRC) � − 2.4633 for Example 2 and avg(NRC) � − 1, 7688 for Example 3. In Table depict RMSE and RE with respect to m for examples of nonlinear case (Examples 4 and 6).Te parameters used in the Newton's algorithm to solve the nonlinear systems is ε � 10 − 10 with 5 maximum number of iterations.We note that the error values are taking decreasing values while m increases.Te average value of NRC for Example 4 is avg(NRC) � − 1, 9912 with 3 iterations for Newton's algorithm and for Example 6avg(NRC) � − 1.8907.

Figure 1 :
Figure 1: Relative error and m NRC (in log scale) for diferent m for Example 1.

Figure 2 :Figure 3 :
Figure 2: Relative error and m NRC (in log scale) for diferent m for Example 4.

Figure 5 :Figure 6 :
Figure 5: RMSE and MAE (in log scale) for diferent m for Example 3.

Figure 7 :
Figure 7: Exact and approximate solutions for diferent m for Example 6.

Table 1 :
RMSE, RE, and NRC for diferent m for Example 1.

Table 2 :
RMSE, RE, and NRC for diferent m for Example 2.

Table 3 :
RMSE, RE, and NRC for diferent m for Example 3.

Table 4 :
RMSE, RE, and NRC for diferent m for Example 4.

Table 5 :
RMSE, RE, and NRC for diferent m for Example 6.

Table 6 :
Approximate solution values on diferent ϰ for diferent α comparing to the exact solution when α � 1 for m � 2, 8 for Example 5.