An Operational Matrix Technique Based on Chebyshev Polynomials for Solving Mixed Volterra-Fredholm Delay Integro-Differential Equations of Variable-Order

Raslan, Kamal R.; Ali, Khalid K.; Mohamed, Emad M. H.; Younis, Jihad A.; Abd El salam, Mohamed A.

doi:https://doi.org/10.1155/2022/6203440

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Advances in Fractional Functional Analysis

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Research Article | Open Access

Volume 2022 | Article ID 6203440 | https://doi.org/10.1155/2022/6203440

An Operational Matrix Technique Based on Chebyshev Polynomials for Solving Mixed Volterra-Fredholm Delay Integro-Differential Equations of Variable-Order

Kamal R. Raslan,¹Khalid K. Ali,¹Emad M. H. Mohamed,¹Jihad A. Younis,²and Mohamed A. Abd El salam¹

Academic Editor: Emanuel Guariglia

Received21 Aug 2021

Accepted15 Mar 2022

Published09 Apr 2022

Abstract

In this work, an algorithm for finding numerical solutions of linear fractional delay-integro-differential equations (LFDIDEs) of variable-order (VO) is introduced. The operational matrices are used as discretization technique based on shifted Chebyshev polynomials (SCPs) of the first kind with the spectral collocation method. The proposed VO-LFDIDEs have multiterms of integer, fractional-order derivatives for delayed or nondelayed and mixed Volterra-Fredholm integral terms. The introduced model is a more general form of linear fractional VO pantograph, neutral, and mixed Fredholm–Volterra integro-differential equations with delay arguments. Caputo’s VO fractional derivative operator is used to generate the matrices of the derivative. Operational matrices are presented for all terms. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, some examples are included to improve the validity and applicability of the techniques. Finally, comparisons between the proposed method and other methods were used to solve this kind of equation.

1. Introduction

In recent few decades, fractional calculus has shown to be a useful implement to formulate many problems in science and engineering where the fractional derivatives and integrals can be used for the description of the properties of various real materials in different branches of science [1–7]. In 1993, Samko and Ross [8] have introduced the variable order derivative (VOD) operator just as a generalization of the fractional-order derivative, and they provided some of its main properties. In this introduced operator, the order of the derivative is a function of the independent variables such as time and space variables. With this generalization, many applications have been established in mechanics, physics, signal processing, and control [9–12]. One of the most important tools in studying the prime numbers is the Riemann zeta-function which embodies both additive and multiplicative structures in a single function, the fractional generalization of it is also found in [13–17].

Consequently, a generalized kind of differential equation appears named VO fractional differential equations (VOFDEs) [18–24]. In VOFDEs, the differential operator is VO, and the derivative changes in general concerning the independent variable or concerning an external functional behavior [8, 25]. Analytical solutions for the VOFDEs are difficult to obtain because the kernel of the VO’s operator has a variable exponent, hence, there will be increasingly rapid developments in numerical approaches to VOFDEs [26–29]. The delay, neutral delay FDEs, and fractional integro-differential equations (FIDEs) are considered as a generalization and development of FDEs, and dealing with them analytically in most of the cases is difficult [30–34]. The VO fractional delay differential equations (VOFDDEs) are a kind of generalization for the fractional delay differential equations (FDDEs) [35–38]. VOFDDEs did not receive the attention that the FDDEs accomplished; however, the potential to describe complicated behavior by the VO of differentiation or integration is clear, all this was a motivation to start studying this type of equation. Soon after, a variety of definitions have been offered for variable order integral and derivative operators such as Riemann–Liouville (RL) [39], Grünwald–Letnikov [40], Caputo, and Fabrizio [41] derivatives.

In this work, we use operational matrices discretization technique based on shifted Chebyshev polynomials (SCPs) of the first kind with the spectral collocation method to solve the following type of linear fractional delay-integro-differential equations of variable-order (LFDIDEs): under the conditions where , , and are positive real delay arguments, and the known functions , , , , and are well defined, additionally, is an unknown function to be determined. The symbole denotes the variable-order derivative (VOD) operator in the Caputo sense.

Corollary 1. The independent variable of (1) belongs to , which is the intersection of the intervals of the different delayed arguments and , i.e., and

The introduced model (1) is a more general form of linear fractional VO pantograph, neutral, and mixed Fredholm–Volterra integro-differential equations with delay arguments. Several methods have been presented for solving VO differential equations [42–46]. For example, Ganji et al. applied the first-kind CPs to obtain the solution of variable order differential equations [47]. Doha et al. solved variable-order fractional Volterra integro-differential equations by shifted Legendre–Gauss–Lobatto collocation method [39]. A numerical method based on the Jacobi polynomials for solving variable-order differential equations has been proposed in [48]. Furthermore, Ganji et al. have introduced a numerical scheme based on the Bernstein polynomials to solve variable order diffusion-wave equations [49]. As a notation, the proposed model (1) is a generalization of our previous reports [50–54] and other work [55–57] as well.

2. Basic Definitions

In this preliminary section, the definitions and properties of Caputo’s VO fractional derivative and the required mathematical tools will be introduced. Also, briefly, the SCPs of the first kind, we will present what we need from their properties.

2.1. Caputo Variable-Order’s Operator

Definition 2. The Caputo’s fractional derivative operator of order operates a function in [42, 43] as and .

Remark 3. In this work, we write the Variable-Order Caputo’s fractional symbol instead of for short.

Remark 4. The following useful property according to Definition 2: where

Definition 5. The Caputo’s VO fractional derivative operator of order operates where can be defined as [42, 58] and . Definition 5 can be reformulated if we assume that; the starting time in a perfect situation, then:

Remark 6. By (7), we can get the following formulas: (i)The Caputo’s VO operator is linearwhere and are any two constants. (ii)As the ordinary differentiation operator the Caputo’s VO operator as well, such that and is constant.

2.2. Shifted First Kind of Chebyshev Polynomials

Next, let us introduce some properties of the SCPs [59]. It is well known that the classical CPs are defined on by the three-term recurrence relation:

Let the SCPs be denoted by , which are defined on . Then, can be generated by using the following recurrence relation: where

The analytic form of the SCPs of degree is given by:

The orthogonality condition of the SCPs is where

A function , square integrable in , may be expressed in terms of shifted Chebyshev polynomials as where the coefficients are given by

In practice, only the first -terms shifted Chebyshev polynomials are considered. The special values will be of important use later.

In the approximation theory, the series in (15) can be approximated by taking the first terms as follows: where is a vector, and if we assume that .

From (12) and (17), can be written as the following form: where is square lower triangle matrix with size given by: and its entries elements are given by:

For example, if , then the square matrix is given by:

Now, from (19), we can obtain the derivative of the matrix as

3. Operational Matrices

In this section, we introduce the generalized operational matrices for , , , and according to fractional calculus.

Lemma 7. The order derivative of the row vector can be in the following relation form [50, 60]: where is the operational matrix of derivative for and can be obtained from

Proof (see [50, 60]). The row vector represents in terms of the vector in the following form [50, 60]: where is square upper triangle matrix with size given by:

Corollary 8. The order derivative of the delay row vector can be represented as [50, 60]

According to the previous lemmas with the fractional calculus by using the Caputo’s variable-order fractional derivative, we introduce the following theorem.

Theorem 9. The th variable-order fractional derivative of the vector can be written as where where diagonal matrix with size , its elements given by:

Proof.

Corollary 10. Relation (29) satisfies for according to Definition 2 by induction with and

Corollary 11. The th variable-order fractional derivative of the delay vector can be written in the following form: Form (18), we get and By putting in the relation (37), we obtain the matrix form Consequently, by substituting the matrix form (24) into (37), we have the matrix relation By substituting the matrix forms (29) into (38), we have the matrix relation By using (35) and (39), we get Similar form (28) and (40), we obtain

3.1. Matrix Representation for Fredholm Integral Terms

Now, we try to find the matrix form corresponding to the integral term. Assume that can be expanded to univariate Chebyshev series concerning , as follows

Then the matrix representation of the kernel function is given by where

Substituting the relations (29) and (46) in the present integral part, we obtained where or

So, the present integral term can be written as

3.2. Matrix Representation for Volterra Integral Terms

Now, we try to find the matrix form corresponding to the integral term. By the same way, can be expanded as (45)

Then, the matrix representation of the kernel function is given by where

Substituting the relations (29) and (53) in the present integral part, we obtained where or

So, the present integral term can be written as:

3.3. Conditions’ Matrix Relation

Finally, we can obtain the matrix form for condition (2) by using (36) on the form:

Via, Equations (41), (42), (43), (44), (51), and (58), then, Equation (1) converted to

We can write (60) in the form: where

The collocation points are defined in this form:

By substituting (63) in (60), then (60) can be turned into the following system:

where , and are block matrices, also and are block matrices defined in this form:

Here, the system in (64) in addition to the supplementary conditions can be solved numerically (the inverse matrix method is preferred when be invertible) to determine the unknown vector . Hence, the approximate analytical solution defined in series (18) can be calculated.

3.4. Convergence Analysis

Theorem 12. Assume that a function be times continuously differentiable and be the best square approximation of defined in (15). Then, we have where

Proof. Using Taylor expansion for as follows: where and . Assume then According to, that given in (18), we obtain Now, let , thus (71) rewritten as Since,
Then Hence, the proof is completed.

4. Numerical Examples and Results Analysis

In this section, we introduce some numerical examples for VO fractional delay-differential equation to illustrate the above results. All results are coded and obtained by using Mathematica package programming. Moreover, the absolute maximum error will be used in our computational results for the comparison between the exact and approximate solutions, where is defined as the below formula: where , , and are the space vectors, exact, and numerical solutions, respectively.

Example 1. Consider the following linear variable-order fractional delay integro-differential equation: The initial conditions are , , and the exact solution is where , , , ,, and The fundamental matrix equation of the problem (75) at is defined by: where

In Table 1, we list the absolute errors between the exact solutions and the approximate solutions using the proposed method with , and three choices of . In Figure 1 is the behavior of the absolute errors for the proposed method of different .

Example 2. Consider the following linear variable-order fractional delay integro-differential equation [44]: with the initial condition , . Problem (78) found in [44] with the exact solution given as at . Now, by applying the suggested technique introduced in the previous section with finite terms to (78), then, we have Hence, using the collocation method with the collection points that is given in the previous section. Then, the previous equation can be rewritten as follows:

The numerical solutions obtained by different values of together with the exact solution are displayed in Figure 2. From this figure, we see that the numerical solution converges to the exact one by increasing the number of basis functions. In Table 2, the comparisons between the absolute errors of the method are given in [44] and the presented method for different values of and and choices of and .

(a)

(b)

In Table 3 shows the absolute errors between the exact and approximate solutions for with and various choices of . In Figure 3 is the exact solution and numerical solutions (a), absolute error (b) at , , and

Example 3. Consider the following linear variable-order fractional delay integro-differential equation [45]: where , and the initial condition is equal to . The exact solution as given in [45] is given by . Now, by applying the suggested technique that introduced in the previous section with finite terms to (53) then we have hence, using the collocation method with the collection points that given in the previous section. Then, the previous equation can be rewritten as follows:

(a)

(b)

In Table 4, the comparison is the maximum absolute error of the method given in [45] and the presented method for different values of and in . The absolute error when for different values of is shown in Figure 4.

Example 4. Consider the following linear variable-order fractional delay integro-differential equation [45]: where , , and the initial condition is equal to . The exact solution of this equation is given by at (see [45]). Now, by applying the suggested technique that introduced in the previous section with finite terms to (8), then, we have Hence, using the collocation method with the collection points that given in the previous section. Then, the previous equation can be rewritten as follows:

By solving the algebraic system (86), we can obtain the vector . Subsequently, numerical solution according to (18) is obtained. Table 5 gives the comparison of the absolute error at some points obtained by the present method and method of [45] with different values of , and it is seen that our method gives more accurate results than the method of [45] at . In Figure 5, the exact solution and numerical solutions given (a), where the absolute error shown (b), are all with different values of and .

Example 5. Consider the following linear variable-order fractional integro-differential equation [39, 46]: where , , , and and the initial conditions is equal to ,. The exact solution of (87) according to [39, 46] is given by at . Now, by applying the suggested technique that introduced in the previous section with finite terms to (8), then, we have Hence, using the present collocation method with the collection points that given in the previous section. Then, the previous equation can be rewritten as follows:

(a)

(b)

By solving the algebraic system (89), we can obtain the vector . Subsequently, numerical solution (18) is obtained. In Table 6, a comparison between the AEs is obtained in [39, 46], and the results are obtained in this work for of for example (12). In Figure 6, we see the matching of the AEs values in this figure and that in Table 6. While Figure 7 displays the curves of exact and approximate solutions for example (12) with values of parameters listed in their caption.

5. Conclusion

In this work, the general form of mixed Fredholm–Volterra integro-differential equations with delay arguments of variable-order based on the operational matrix method with the Shifted Chebyshev polynomials (CPs) of the first kind is presented. The spectral collocation technique with the aid of CPs is used as an operational matrix method for solving the proposed model, which is reduced by the operational matrices to the matrix form. Caputo’s VO fractional derivative operator is used to generate the matrices of the derivative. The accuracy of the proposed technique is obtained by many numerical examples. Finally, we used codes written with the Mathematica package to calculate our numerical results and graphs.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors carried out the proofs and conceived the study. All authors read and approved the final manuscript.

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Copyright © 2022 Kamal R. Raslan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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