Numerical Approach for Solving the Fractional Pantograph Delay Differential Equations

Hajishafieiha, Jalal; Abbasbandy, Saeid

doi:https://doi.org/10.1155/2022/4134892

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Research Article | Open Access

Volume 2022 | Article ID 4134892 | https://doi.org/10.1155/2022/4134892

Numerical Approach for Solving the Fractional Pantograph Delay Differential Equations

Jalal Hajishafieiha¹and Saeid Abbasbandy¹

Academic Editor: Abdellatif Ben Makhlouf

Received19 Jun 2022

Accepted29 Aug 2022

Published13 Sept 2022

Abstract

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter , which is obtained using the collocation and least-squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in this study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, -polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed.

1. Introduction

Since the 17th century, many problems in physics, mechanics, economics, biology, etc., have been investigated and solved by ordinary differential equations. Ordinary differential equations are suitable models for describing processes that occur at the moment, meaning that the process rate of change depends only on its current state, not in its past. However, time delays in many processes cannot be ignored. For example, a signal needs time to reach the desired point and a driver to react to a possible accident. Delays are also used to describe some aspects of infectious diseases: the initial contagion, drug treatment, and immunity to the disease. In modeling the control of blood carbon dioxide levels, the time interval of blood oxygen recovery in the lungs and the stimulation of the chemical receptor in the brainstem are considered as delays in the model [1]. Delayed differential equations have been studied for more than 200 years [2]. In the 19th century, Euler, Lagrange, and Laplace studied delayed differential equations. In the late 1930s and early 1940s, Voltaire introduced a number of delayed differential equations during his studies on the hunting-hunter model and was the first to study these equations in an organized manner.

In recent decades, fractional delay differential equations have had an important role in engineering and natural sciences. Applications of these equations include hydrology, signal processing, control theory, medical sciences, networks, cell biology, climate models, infectious diseases, navigation prediction, circulating blood, population dynamics, oncolytic virotherapy, delayed plant disease model, the body reaction to carbon dioxide, and many others [3–6].

Pantograph differential equations are one of the types of delay differential equations that were first created by studying on an electric locomotive [7]. It is difficult to find an accurate and analytical solution to this problem, so approximate and numerical methods should be used to solve these problems. Different numerical methods have been used to solve fractional pantograph differential equations such as Hermit wavelets method [8], spectral method [9, 10], fractional-order Bernoulli wavelet method [11], fractional-order Boubaker polynomials [12], Taylor collocation method [13, 14], Laguerre–Gauss collocation method [15], Legendre multiwavelet collocation method [16], two-step Runge–Kutta of order one method [17], one–Leg method [18], variational method [19], collocation method [20], generalized differential transform scheme [21], modified the predictor-corrector scheme [22], and Bernoulli wavelet method [23].

Various polynomials have been used to solve the pantograph differential equations. The authors in [24] have used a numerical method based on Hermit polynomials to obtain the approximate solution of the generalized pantograph differential equations with variable coefficients. In this study, the solution of the pantograph equation is approximated using the collocation method and creating matrix operators. Rabiei and Ordokhani [12] have used fractional-order Boubaker polynomials to approximate fractional pantograph differential equations in any arbitrary interval. The properties of these polynomials have been used to construct pantograph operational matrices and fractional integrals. Then, by the least-squares method, a nonlinear system is obtained, which is solved using Newton’s iterative method. Sedaghat et al. [25] have used the shifted Chebyshev polynomials and Chebyshev pantograph operator matrix and transformed the pantograph differential equation into a system of equations. Also, using error analysis, the number of polynomials used in the approximation is obtained. Isah et al. [26] have used Genocchi polynomials to approximate the generalized fractional pantograph equations under boundary and initial conditions. With the help of the properties of these polynomials, the Genocchi delay operator matrix is produced.

Then, by converting this equation into a system of equations, the approximate solution is obtained. The authors in [27] chose Bernstein polynomials to solve the pantograph differential equations numerically and also studied the stability of the method. Using the spectral Galerkin method and shifted Legendre polynomial properties, Al-Suyuti et al. [28] transformed the multiorder fractional pantograph equations into a system of linear equations and solved this system with appropriate methods. Cakmak and Alkan [29] have used Fibonacci polynomials and collocation method to solve the system of nonlinear pantograph equations under initial conditions. In this method, solving the generated system of equations approximates the given differential equation. Yüzbaşı et al. [30] use the residual correction of Hermit polynomial solutions to solve the generalized pantograph differential equations. Hermit polynomials solutions are obtained by collocation method and transform the problem into a system of equations and obtain unknown coefficients. Javadi et al. [31] have first produced shifted orthonormal Bernstein polynomials and integral and delayed operator matrices of these polynomials. Using the properties of these polynomials, they have converted the generalized pantograph equations into a system of linear equations and proved the stability of this method. Wang et al. [32] has used shifted Chebyshev polynomials to solve generalized fractional pantograph equations with variable coefficients. By producing the generalized pantograph operational matrix using the properties of these polynomials and creating a system of equations and solving this system, the numerical solution of the pantograph equation is obtained. Akkaya et al. [33] have used first Boubaker polynomials to approximate the solution of the pantograph equations with variable coefficients. In this study, the authors have obtained the numerical solution of differential equations by using matrix operators and converting pantograph equations into a system of equations. Ezz–Eldien et al. [34] have approximated the numerical solution of multiorder fractional neutral pantograph equations with the help of Chebyshev polynomials and spectral tau and collocation methods. In this study, the linear and nonlinear pantograph equations and the system of fractional multipantograph equations have been investigated.

We consider a new method for the numerical solution of the following fractional pantograph delay differential equation problem:where , is the unknown function, and the functions and are the known functions defined in . Indeed, is the Caputo fractional derivative of order .

2. Basic Concepts

There are many types of fractional derivatives and integrals which are suggested by Riemann, Liouville, Riesz, Letnikov, Grünwald, Weyl, Marchaud, and Caputo. In this study, we will consider the fractional pantograph delay differential equation problem in the Caputo sense.

Definition 1. The Caputo fractional derivative of order th, for a function and is integrable on , is written as follows:where is Euler’s Gamma function (or Euler’s integral of the second kind) and .

Definition 2. The Riemann–Liouville fractional integral of order th is defined as follows:For , we know thatAs we know, using the polynomials is very useful for finding the solutions to the differential equation problems, especially in engineering applications. It is due to the simple application of them. Recently, a new class of polynomials equipped with an auxiliary parameter has been introduced by the first author in [35] and some applications of it have been shown in [36–39]. This class is introduced as follows.

Definition 3. (see [35]). The -polynomial functions is defined as follows:where is the second kind of Chebyshev polynomial and is an auxiliary real parameter.
The following equations are also established as follows:See [35–37], for more properties.

3. The Implementation and Error Analysis

In this method, the solution of (1) is approximated by the following truncated series form:where are the unknown coefficients. The collocation points of the present method on the interval are defined as such that . Therefore, the unknown coefficients and the unknown auxiliary parameter, , are obtained by using the collocation method on (1) and hence solving the following nonlinear system of equations:for . Above residual and initial conditions produce a nonlinear system of equations. To solve this nonlinear system of equations, we used the Mathematica software version 12.0 and the FindMinimum command.

Now, we want to specify a bound for norm solution using Gronwall’s inequality. Consider the fractional pantograph delay differential (1); by integration by of the two sides of (1) and using properties (4), (5), and (6), we will havewhere

Now, we calculate the absolute value (10):where

With a simple change of variables, we have

According to , we have

So, from the above inequality, we haveand hence, Gronwall’s inequality [40] concludes that

3.1. The Rate of Convergence Theorem

In this section, we want to analyze the convergence rate of the present method. For this purpose, first, we express the rate of convergence of Chebyshev polynomials of the second kind.

Theorem 1 (see [41]). Suppose is a continuous function of bounded variation on , that are Chebyshev polynomials of the second kind, and is total variation of on that satisfies the following Lipschitz condition:

Then, we have, for and ,where .

Theorem 2. If satisfies the conditions of Theorem 1, then we have, for and ,for some real constant .

Proof. According to (8) and Theorem 1, we havewhere .

Theorem 3. If is the exact solution to the fractional pantograph delay differential equation (1) and satisfies the conditions of Theorem 2, then we have, for and ,for some real constant .

Proof. According to (10), for the numerical solution of equation (1), we haveSubtracting (12) from (10) givesUsing Theorem 2, we haveAs a result,Since the exact solution of (1) is often unknown for noninteger values , the following error can help us to get a more accurate numerical solution:

4. Numerical Examples

In this section, we show the advantage and high accuracy of the present method by presenting and analyzing five examples. Let be the number of collocation points, we use maximum absolute error to verify and validate the results where the following example codes are written by Mathematics software version 12.0.

4.1. Example 1

Assume the linear fractional pantograph delay differential equation problem (1) is as follows:where is the exact solution and the function is obtained by the exact solution at .

The error obtained from the present method is compared with the fractional-order Boubaker polynomials method [12] in Table 1. In continuation, the fractional-order Bernoulli wavelet method [11] and the present method are compared in Table 2. By observing these tables, the high accuracy of the present method is confirmed in comparison with the other two methods. In Figures 1 and 2, graphs of numerical and exact solutions are plotted for different values and at , , , and at , , , respectively. In these figures, it can be seen that as the value of to 2 approaches, the approximate solutions converge to the exact solution.

4.2. Example 2

Assume the neutral fractional pantograph delay differential equation problem (1) is as follows:where is the exact solution and the function is obtained as before at .

In Table 3, the results of this method are compared with the following methods. Two-step Runge–Kutta of order one method [17], one-leg method [18], variational iteration method (m = 6) [19], fractional-order Boubaker polynomials method [12], Bernoulli wavelets method [23], spectral method based on modification of the hat functions [10], Taylor wavelets method [14], and fractional-order Bernoulli wavelet method [11].

The results of this table show the tangible advantage of the proposed method in comparison with other methods. In Table 4, by increasing the number of collocation points, the maximum error decreases, which also confirms the convergence theorem. In Figure 3, numerical solutions converge to the exact solution of the problem as the value of to 1 approaches.

4.3. Example 3

Assume the nonlinear fractional pantograph delay differential equation problem (1) is as follows:where is the exact solution at . In Table 5, the results of the proposed method are compared to the following methods: fractional-order Boubaker polynomials method [12], Bernoulli wavelet method [11], spectral method [10], Bernoulli wavelets method [23], and Taylor wavelets method [14].

This table displays the high accuracy and efficiency of the present method compared to other methods. In Table 6, by increasing the number of collocation points, the maximum error decreases, which confirms the convergence theorem. In Figure 4, the graphs of numerical and exact solutions of the problem are plotted in different values of at , , and . It can be observed that, as the value of to 2 approaches, numerical solutions converge to the exact solution. Also, in this figure, the absolute error graph is plotted in .

(a)

(b)

4.4. Example 4

Assume the neutral fractional delay differential equation problem (1) is as follows:where , arbitrary , is the exact solution, and the function is obtained as before at . In Table 7, the results of the proposed method are compared to the following methods: fractional-order Boubaker polynomials method [12], spectral method [10], Bernoulli wavelets method [23], and Taylor wavelets method [14].

In Table 8, the maximum error results of the proposed method are compared with discontinuous Galerkin method [42], Taylor wavelets method [14], and Bernoulli wavelets method [23]. In this table, the results of maximum error decrease by increasing the number of collocation points. Observing these two tables shows the high accuracy of the present method compared to other methods. In Figure 5, the graphs of numerical and exact solutions of the problem are plotted in different values of at , , , , and . It can be observed that, as the value of to 1 approaches, numerical solutions converge to the exact solution. Also, in this figure, the absolute error graph is plotted in .

(a)

(b)

4.5. Example 5

Assume the fractional pantograph delay differential equation problem (1) is as follows:where is the exact solution and the function is obtained by the exact solution at . In Table 9, the results of this method are reported in comparison with the following methods: collocation method [20], fractional-order Boubaker polynomials method [12], Bernoulli wavelets method [23], and spectral method [10]. This table displays the high accuracy and efficiency of the present method compared to other methods. In Table 10, the maximum error decreases by increasing the number of collocation points. In Figure 6, the graphs of numerical and exact solutions of the problem are plotted in different values of at , , and . It can be observed that, as the value of to 1 approaches, numerical solutions converge to the exact solution. Also, in this figure, the absolute error graph is plotted in .

(a)

(b)

5. Conclusion

As you can see in this study, the present method has more accurate results than the other method. A much smaller number of collocation points are used in the present method. The maximum absolute error in this method is much less than the other method. As shown in this study, the convergence of the present method is guaranteed. This method is easily implemented to solve the linear and nonlinear fractional pantograph delay differential equations. The simplicity of using -polynomials in fractional derivatives can be one of the advantage points of the present method, which creates less complexity to solve. In future studies, the use of these -polynomials to numerically solve the fractional nonlinear multipantograph and partial delay differential equations will be analyzed. According to the accurate results obtained from these polynomials in this study, we hope to observe the same accuracy in our future studies [43].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, New York, 2011.
E. Schmidt, “Über eine Klasse linearer funktionaler Differentialgleichungen,” Mathematische Annalen, vol. 70, no. 4, pp. 499–524, 1911.
View at: Publisher Site | Google Scholar
S. Irandoust-pakchin, H. Kheiri, and S. Abdi-mazraeh, “Efficient computational algorithms for solving one class of fractional boundary value problems,” Computational Mathematics and Mathematical Physics, vol. 53, no. 7, pp. 920–932, 2013.
View at: Publisher Site | Google Scholar
P. Kumar, D. Baleanu, V. S. Erturk, M. Inc, and V. Govindaraj, “A delayed plant disease model with Caputo fractional derivatives,” Advances in Continuous and Discrete Models, vol. 2022, no. 1, pp. 11–22, 2022.
View at: Publisher Site | Google Scholar
P. Kumar, V. S. Erturk, A. Yusuf, and S. Kumar, “Fractional time-delay mathematical modeling of Oncolytic Virotherapy,” Chaos, Solitons & Fractals, vol. 150, Article ID 111123, 2021.
View at: Publisher Site | Google Scholar
S. Masood, M. Naeem, R. Ullah, S. Mustafa, and A. Bariq, “Analysis of the fractional-order delay differential equations by the numerical method,” Complexity, Article ID 3218213, pp. 1–14, 2022.
View at: Publisher Site | Google Scholar
J. R. Ockendon and A. B. Tayler, “The dynamics of a current collection system for an electric locomotive,” Proceedings of the Royal Society of London, A. Mathematical and Physical Sciences, vol. 322, no. 1551, pp. 447–468, 1971.
View at: Publisher Site | Google Scholar
U. Saeed, “Hermite wavelet method for fractional delay differential equations,” Journal of Difference Equations, vol. 2014, Article ID 359093, pp. 1–8, 2014.
View at: Publisher Site | Google Scholar
Y. Yang and Y. Huang, “Spectral–collocation methods for fractional pantograph delay–integrodifferential equations,” Advances in Mathematical Physics, pp. 1–14, 2013.
View at: Publisher Site | Google Scholar
S. Nemati, P. Lima, and S. Sedaghat, “An effective numerical method for solving fractional pantograph differential equations using modification of hat functions,” Applied Numerical Mathematics, vol. 131, pp. 174–189, 2018.
View at: Publisher Site | Google Scholar
P. Rahimkhani, Y. Ordokhani, and E. Babolian, “Numerical solution of fractional pantograph differential equations by using generalized fractional–order Bernoulli wavelet,” Journal of Computational and Applied Mathematics, vol. 309, pp. 493–510, 2017.
View at: Publisher Site | Google Scholar
K. Rabiei and Y. Ordokhani, “Solving fractional pantograph delay differential equations via fractional–order Boubaker polynomials,” Engineering with Computers, vol. 35, no. 4, pp. 1431–1441, 2019.
View at: Publisher Site | Google Scholar
A. Anapali, Y. Öztürk, and M. Gulsu, “Numerical approach for solving fractional pantograph equation,” International Journal of Computer Application, vol. 113, no. 9, pp. 45–52, 2015.
View at: Publisher Site | Google Scholar
P. Vichitkunakorn, T. N. Vo, and M. Razzaghi, “A numerical method for fractional pantograph differential equations based on Taylor wavelets,” Transactions of the Institute of Measurement and Control, vol. 42, no. 7, pp. 1334–1344, 2020.
View at: Publisher Site | Google Scholar
A. H. Bhrawy, A. A. Al–Zahrani, Y. A. Alhamed, and D. Baleanu, “A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional pantograph equations,” Romanian Journal of Physics, vol. 59, no. 7-8, pp. 646–657, 2014.
View at: Google Scholar
S. A. Yousefi and A. Lotfi, “Legendre multiwavelet collocation method for solving the linear fractional time delay systems,” Open Physics, vol. 11, no. 10, pp. 1463–1469, 2013.
View at: Publisher Site | Google Scholar
A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, 2013.
W. S. Wang and S. F. Li, “On the one-leg -methods for solving nonlinear neutral functional differential equations,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 285–301, 2007.
View at: Google Scholar
X. Chen and L. Wang, “The variational iteration method for solving a neutral functional-differential equation with proportional delays,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2696–2702, 2010.
View at: Publisher Site | Google Scholar
Y. Muroya, E. Ishiwata, and H. Brunner, “On the attainable order of collocation methods for pantograph integro-differential equations,” Journal of Computational and Applied Mathematics, vol. 152, no. 1-2, pp. 347–366, 2003.
View at: Publisher Site | Google Scholar
Z. Odibat, V. S. Erturk, P. Kumar, A. Ben Makhlouf, and V. Govindaraj, “An implementation of the generalized differential transform scheme for simulating impulsive fractional differential equations,” Mathematical Problems in Engineering, vol. 2022, pp. 1–11, 2022.
View at: Publisher Site | Google Scholar
Z. Odibat, P. Kumar, and V. Govindaraj, “Dynamics of generalized Caputo type delay fractional differential equations using a modified Predictor-Corrector scheme,” Physica Scripta, vol. 96, no. 12, Article ID 125213, 2021.
View at: Publisher Site | Google Scholar
P. Rahimkhani, Y. Ordokhani, and E. Babolian, “A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations,” Numerical Algorithms, vol. 74, no. 1, pp. 223–245, 2017.
View at: Publisher Site | Google Scholar
S. Yalçinbaş, M. Aynigül, and M. Sezer, “A collocation method using Hermite polynomials for approximate solution of pantograph equations,” Journal of the Franklin Institute, vol. 348, no. 6, pp. 1128–1139, 2011.
View at: Publisher Site | Google Scholar
S. Sedaghat, Y. Ordokhani, and M. Dehghan, “Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4815–4830, 2012.
View at: Publisher Site | Google Scholar
A. Isah, C. Phang, and P. Phang, “Collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations,” International Journal of Differential Equations, pp. 1–10, 2017.
View at: Publisher Site | Google Scholar
O. R. Işik, Z. Güney, and M. Sezer, “Bernstein series solutions of pantograph equations using polynomial interpolation,” Journal of Difference Equations and Applications, vol. 18, no. 3, pp. 357–374, 2012.
View at: Publisher Site | Google Scholar
M. M. Alsuyuti, E. H. Doha, S. S. Ezz–Eldien, and I. K. Youssef, “Spectral Galerkin schemes for a class of multi-order fractional pantograph equations,” Journal of Computational and Applied Mathematics, vol. 384, Article ID 113157, 2021.
View at: Publisher Site | Google Scholar
M. Cakmak and S. Alkan, “A numerical method for solving a class of systems of nonlinear pantograph differential equations,” Alexandria Engineering Journal, vol. 61, no. 4, pp. 2651–2661, 2022.
View at: Publisher Site | Google Scholar
Ş. Yüzbaşı, E. Gök, and M. Sezer, “Residual correction of the Hermite polynomial solutions of the generalized pantograph equations,” New Trends in Mathematical Sciences, vol. 3, no. 2, pp. 118–125, 2015.
View at: Google Scholar
S. Javadi, E. Babolian, and Z. Taheri, “Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials,” Journal of Computational and Applied Mathematics, vol. 303, pp. 1–14, 2016.
View at: Publisher Site | Google Scholar
L. P. Wang, Y. M. Chen, D. Y. Liu, and D. Boutat, “Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials,” International Journal of Computer Mathematics, vol. 96, no. 12, pp. 2487–2510, 2019.
View at: Publisher Site | Google Scholar
T. Akkaya, S. Yalçinbaş, and M. Sezer, “Numeric solutions for the pantograph type delay differential equation using first Boubaker polynomials,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 9484–9492, 2013.
View at: Publisher Site | Google Scholar
S. S. Ezz–Eldien, Y. Wang, M. A. Abdelkawy, M. A. Zaky, A. A. Aldraiweesh, and J. T. Machado, “Chebyshev spectral methods for multi-order fractional neutral pantograph equations,” Nonlinear Dynamics, vol. 100, no. 4, pp. 3785–3797, 2020.
View at: Publisher Site | Google Scholar
S. Abbasbandy, “A new class of polynomial functions equipped with a parameter,” The Mathematical Scientist, vol. 11, no. 2, pp. 127–130, 2017.
View at: Publisher Site | Google Scholar
J. Hajishafieiha and S. Abbasbandy, “A new method based on polynomials equipped with a parameter to solve two parabolic inverse problems with a nonlocal boundary condition,” Inverse Problems in Science and Engineering, vol. 28, no. 5, pp. 739–753, 2019.
View at: Publisher Site | Google Scholar
J. Hajishafieiha and S. Abbasbandy, “A new class of polynomial functions for approximate solution of generalized Benjamin-Bona-Mahony-Burgers (gBBMB) equations,” Applied Mathematics and Computation, vol. 367, Article ID 124765, 2020.
View at: Publisher Site | Google Scholar
S. Abbasbandy and J. Hajishafieiha, “Numerical solution to the Falkner–Skan equation: a novel numerical approach through the new rational a–polynomials,” Applied Mathematics and Mechanics, vol. 42, no. 10, pp. 1449–1460, 2021.
View at: Publisher Site | Google Scholar
S. Abbasbandy and J. Hajishafieiha, “Numerical solution of the time–space fractional diffusion equation with Caputo derivative in time by a–polynomial method, Applications and Applied Mathematics,” International Journal, vol. 16, no. 2, pp. 881–893, 2021.
View at: Google Scholar
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer Science & Business Media, 2007.
K. Al, “–Khaled, on the Rate of convergence for the Chebyshev series,” Missouri Journal of Mathematical Sciences, vol. 14, no. 1, pp. 4–10, 2002.
View at: Google Scholar
H. Brunner, Q. Huang, and H. Xie, “Discontinuous Galerkin methods for delay differential equations of pantograph type,” SIAM Journal on Numerical Analysis, vol. 48, no. 5, pp. 1944–1967, 2010.
View at: Publisher Site | Google Scholar
G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics, Springer-Verlag, New York, 2002.

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