Analytical study to solving the inhomogeneous pantograph delay equation: the exact solution
Authors
E. R. El-Zahar
- Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia.
- Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt.
A. Ebaid
- Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia.
L. F. Seddek
- Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia.
- Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, 44519, Egypt.
Abstract
This paper solves the pantograph delay equation which includes inhomogeneous term. The inhomogeneous term is in the form of a class of exponential functions. An efficient transformation is introduced to reduce the inhomogeneous pantograph delay differential equation (IPDDE) to the homogeneous pantograph delay differential equation (HPDDE), which is a homogeneous model. It is found that the solution of the IPDDE depends mainly on the solution of HPDDE. It is well-known in the literature that the analytical solution of the HPDDE is already obtained in a closed series form. Such ready solution of the HPDDE is invested in this paper and accordingly, the solution of the IPDDE under consideration is established. Also, several exact solutions of the present model are determined at specific conditions and values of the parameters. The solutions of some examples in the literature are obtained in exact forms as special cases of the current results. Moreover, the properties of the obtained solutions are theoretically and graphically addressed.
Share and Cite
ISRP Style
E. R. El-Zahar, A. Ebaid, L. F. Seddek, Analytical study to solving the inhomogeneous pantograph delay equation: the exact solution, Journal of Mathematics and Computer Science, 34 (2024), no. 2, 152--161
AMA Style
El-Zahar E. R., Ebaid A., Seddek L. F., Analytical study to solving the inhomogeneous pantograph delay equation: the exact solution. J Math Comput SCI-JM. (2024); 34(2):152--161
Chicago/Turabian Style
El-Zahar, E. R., Ebaid, A., Seddek, L. F.. "Analytical study to solving the inhomogeneous pantograph delay equation: the exact solution." Journal of Mathematics and Computer Science, 34, no. 2 (2024): 152--161
Keywords
- Pantograph
- delay
- inhomogeneous
- exact solution
- series solution
MSC
References
-
[1]
N. O. Alatawi, A. Ebaid, Solving a delay differential equation by two direct approaches, J. Math. Syst. Sci., 9 (2019), 54–56
-
[2]
A. B. Albidah, N. E. Kanaan, A. Ebaid, H. K. Al-Jeaid, Exact and numerical analysis of the pantograph delay differential equation via the homotopy perturbation method, Mathematics, 11 (2023), 1–14
-
[3]
A. H. S. Al-Enazy, A. Ebaid, E. A. Algehyne, H. K. Al-Jeaid, Advanced Study on the Delay Differential Equation y0(t) = ay(t) + by(ct), Mathematics, 9 (2022), 1–13
-
[4]
H. S. Ali, E. Alali, A. Ebaid, F. M. Alharbi, Analytic solution of a class of singular second–order boundary value problems with applications, Mathematics, 7 (2019), 1–10
-
[5]
A. F. Aljohani, A. Ebaid, E. A. Algehyne, Y. M. Mahrous, P. Agarwal, M. Areshi, H. K. Al–Jeaid, On solving the chlorine transport model via Laplace transform, Sci. Rep., 12 (2022), 1–11
-
[6]
R. Alrebdi, H. K. Al-Jeaid, Accurate Solution for the Pantograph Delay Differential Equation via Laplace Transform, Mathematics, 11 (2023), 1–15
-
[7]
A. Atangana, B. S. T. Alkaltani, A novel double integral transform and its applications, J. Nonlinear Sci. Appl., 9 (2016), 424–434
-
[8]
H. O. Bakodah, A. Ebaid, Exact solution of Ambartsumian delay differential equation and comparison with Daftardar-Gejji and Jafari approximate method, Mathematics, 6 (2018), 1–10
-
[9]
N. Do˘gan, Solution of the system of ordinary differential equations by combined Laplace transform-Adomian decomposition method, Math. Comput. Appl., 17 (2012), 203–211
-
[10]
A. Ebaid, E. Alali, H. Saleh, The exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids, J. Assoc. Arab Univ. Basic Appl. Sci., 24 (2017), 156–159
-
[11]
A. Ebaid, A. Al-Enazi, B. Z. Albalawi, M. D. Aljoufi, Accurate approximate solution of Ambartsumian delay differential equation via decomposition method, Math. Comput. Appl., 24 (2019), 9 pages
-
[12]
A. Ebaid, W. Alharbi, M. D. Aljoufi, E. R. El–Zahar, The exact solution of the falling body problem in three–dimensions: Comparative study, Mathematics, 8 (2020), 1–15
-
[13]
A. Ebaid, H. K. Al-Jeaid, On the Exact Solution of the Functional Differential Equation y0(t) = ay(t) + by(-t), Adv. Differ. Equ. Control Processes, 26 (2022), 39–49
-
[14]
A. Ebaid, M. A. Al Sharif, Application of Laplace transform for the exact effect of a magnetic field on heat transfer of carbon–nanotubes suspended nanofluids, Z. Nature. A, 70 (2015), 471–475
-
[15]
A. Ebaid, A. M. Wazwaz, E. Alali, B. S. Masaedeh, Hypergeometric series solution to a class of second–order boundary value problems via Laplace transform with applications to nanofuids, Commun. Theor. Phys., 67 (2017),
-
[16]
E. R. El-Zahar, A. Ebaid, Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation, Axioms, 11 (2022), 1–11
-
[17]
S. S. Ezz-Eldien, On solving systems of multi-pantograph equations via spectral tau method, Appl. Math. Comput., 321 (2018), 63–73
-
[18]
O. R. Isik, T. Turkoglu, A rational approximate solution for generalized pantograph-delay differential equations, Math. Methods Appl. Sci., 39 (2016), 2011–2024
-
[19]
H. Jafari, M. Mahmoudi, M. H. Noori Skandari, A new numerical method to solve pantograph delay differential equations with convergence analysis, Adv. Difference Equ., 2021 (2021), 12 pages
-
[20]
S. Javadi, E. Babolian, Z. Taheri, Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials, J. Comput. Appl. Math., 303 (2016), 1–14
-
[21]
S. M. Khaled, The exact effects of radiation and joule heating on Magnetohydrodynamic Marangoni convection over a flat surface, Therm. Sci., 22 (2018), 63–72
-
[22]
S. M. Khaled, Applications of Standard Methods for Solving the Electric Train Mathematical Model With Proportional Delay, Int. J. Anal. Appl., 20 (2022), 1–12
-
[23]
S. M. Khaled, A. Ebaid, F. Al Mutairi, The exact endoscopic effect on the peristaltic flow of a nanofluid, J. Appl. Math., 2014 (2014), 11 pages
-
[24]
P. V. Pavani, U. L. Priya, B. A. Reddy, Solving differential equations by using Laplace transforms, Int. J. Res. Anal. Rev., 5 (2018), 1796–1799
-
[25]
H. Saleh, E. Alali, A. Ebaid, Medical applications for the flow of carbon-nanotubes suspended nanofluids in the presence of convective condition using Laplace transform, J. Assoc. Arab Univ. Basic Appl. Sci., 24 (2017), 206–212
-
[26]
S. Sedaghat, Y. Ordokhani, M. Dehghan, Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4815–4830
-
[27]
M. Sezera, A. Aky¨ uz-Das¸cıo ˇ glu, A Taylor method for numerical solution of generalized pantograph equations with linear functional argument, J. Comput. Appl. Math., 200 (2007), 217–225
-
[28]
J. Shen, T. Tang, L.-L. Wang, Spectral methods: algorithms, analysis and applications, Springer, Berlin, Heidelberg (2011)
-
[29]
E. Tohidi, A. H. Bhrawy, K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model., 37 (2013), 4283–4294
