Developed analytical approach for a special kind of differential-difference equation: Exact solution

Volume 33, Issue 4, pp 381--389 https://dx.doi.org/10.22436/jmcs.033.04.05

Publication Date: January 28, 2024 Submission Date: October 25, 2023 Revision Date: November 18, 2023 Accteptance Date: December 19, 2023

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Authors

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. A. Ebaid - Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia. 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.

Abstract

This paper investigates a differential-difference equation with a variable coefficient of exponential order in the form \( \phi'(t)=\alpha \phi(t)+\beta e^{\sigma t} \phi(- t)\). In literature, periodic solution has been obtained at the special case \(\sigma=0\). In this paper, an effective approach is developed to determine the exact solution in terms of exponential and trigonometric functions. In addition, the exact solution is expressed in terms of exponential and hyperbolic functions under specific conditions of the involved parameters. Exact solutions of several special cases are derived and found in full agreement with the corresponding results in the relevant literature. Some theoretical results are presented and proved which can be generalized to include other complex models. The behavior of the obtained shows periodicity in the absence of \(\sigma\) while the damped oscillations are shown graphically when \(\sigma\) is assigned to negative values.

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Keywords

• Ansatz method
• differential-difference equation
• exact solution

MSC

•  34A99
•  34A30

References

• [1] M. R. Abbott, Numerical method for calculating the dynamic behaviour of a trolley wire overhead contact system for electric railways, Comput. J., 13 (1970), 363–368
  • View Article
  • Google Scholar


• [2] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers Group, Dordrecht (1994)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] A. A. Alatawi, M. Aljoufi, F. M. Alharbi, A. Ebaid, Investigation of the surface brightness model in the Milky Way via homotopy perturbation method, J. Appl. Math. Phys., 8 (2020), 434–442
  • View Article
  • Google Scholar


• [4] N. O. Alatawi, A. Ebaid, Solving a delay differential equation by two direct approaches, J. Math. Syst. Sci., 9 (2019), 54–56
  • View Article
  • Google Scholar


• [5] E. A. Algehyne, E. R. El-Zahar, F. M. Alharbi, A. Ebaid, Development of analytical solution for a generalized Ambartsumian equation, AIMS Math., 5 (2020), 249–258
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] F. M. Alharbi, A. Ebaid, New Analytic Solution for Ambartsumian Equation, J. Math. Syst. Sci., 8 (2018), 182–186
  • View Article
  • Google Scholar


• [7] A. Alshaery, A. Ebaid, Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method, Acta Astronaut., 140 (2017), 27–33
  • View Article
  • Google Scholar


• [8] E. H. Aly, A. Ebaid, R. Rach, Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions, Comput. Math. Appl., 63 (2012), 1056–1065
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] V. A. Ambarzumian, On the fluctuation of the brightness of the milky way, C. R. (Doklady) Acad. Sci. URSS (N.S.), 44 (1994), 223–226
  • View Article
  • Google Scholar


• [10] H. I. Andrews, Third paper: Calculating the behaviour of an overhead catenary system for railway electrification, Proc. Inst. Mech. Eng., 179 (1964), 809–846
  • View Article
  • Google Scholar


• [11] Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc., 23 (2015), 424–428
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] 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
  • View Article
  • Google Scholar
  • MATH


• [13] H. O. Bakodah, A. Ebaid, The Adomian decomposition method for the slip flow and heat transfer of nanofluids over a stretching/shrinking sheet, Rom. Rep. Phys., 70 (2018), 1–15
  • View Article
  • Google Scholar


• [14] P. M. Caine, P. R. Scott, Single-wire railway overhead system, Proc. Inst. Electr. Eng., 116 (1969), 1217–1221
  • View Article
  • Google Scholar


• [15] C. Chun, A. Ebaid, M. Y. Lee, E. Aly, An approach for solving singular two point boundary value problems: analytical and numerical treatment, ANZIAM J. Electron. Suppl., 53 (2012), E21–E43
  • View Article
  • Google Scholar
  • MATH


• [16] J.-S. Duan, R. Rach, A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl. Math. Comput., 218 (2011), 4090–4118
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] A. Ebaid, Approximate analytical solution of a nonlinear boundary value problem and its application in fluid mechanics, Z. Naturforschung A, 66 (2011), 423–426
  • View Article
  • Google Scholar


• [18] A. Ebaid, A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method, J. Comput. Appl. Math., 235 (2011), 1914–1924
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] A. Ebaid, Analytical solutions for the mathematical model describing the formation of liver zones via Adomian’s method, Comput. Math. Methods Med., 2013 (2013), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] A. Ebaid, Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel, Comput. Math. Appl., 68 (2014), 77–85
  • View Article
  • Google Scholar
  • MATH


• [21] A. Ebaid, N. Al-Armani, A new approach for a class of the Blasius problem via a transformation and Adomian’s method, Abstr. Appl. Anal., 2013 (2013), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] 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), 1–9
  • View Article
  • MathSciNet
  • Google Scholar


• [23] 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
  • View Article
  • Google Scholar
  • MATH


• [24] A. Ebaid, A. F. Aljohani, E. H. Aly, Homotopy perturbation method for peristaltic motion of gold-blood nanofluid with heat source, Int. J. Numer. Methods Heat Fluid Flow, 30 (2020), 3121–3138
  • View Article
  • Google Scholar


• [25] A. Ebaid, M. D. Aljoufi, A. M. Wazwaz, An advanced study on the solution of nanofluid flow problems via Adomian’s method, Appl. Math. Lett., 46 (2015), 117–122
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] A. Ebaid, C. Cattani, A. S. Al Juhani, E. R. El-Zahar, A novel exact solution for the fractional Ambartsumian equation, Adv. Difference Equ., 2021 (2021), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] L. Fox, D. F. Mayers, J. R. Ockendon, A. B. Tayler, On a functional differential equation, J. Inst. Math. Appl., 8 (1971), 271–307
  • View Article
  • Google Scholar


• [28] G. Gilbert, H. E. H. Davies, Pantograph motion on a nearly uniform railway overhead line, Proc. Inst. Electr. Eng., 113 (1966), 485–492
  • View Article
  • Google Scholar


• [29] A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1–38
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] T. Kato, J. B. McLeod, The functional-differential equation y0(x) = ay(x) + by(x), Bull. Amer. Math. Soc., 77 (1971), 891–935
  • View Article
  • MathSciNet
  • Google Scholar


• [31] S. M. Khaled, E. R. El-Zahar, A. Ebaid, Solution of Ambartsumian delay differential equation with conformable derivative, Mathematics, 7 (2019), 1–10
  • View Article
  • Google Scholar


• [32] D. Kumar, J. Singh, D. Baleanu, S. Rathore, Analysis of a fractional model of the Ambartsumian equation, Eur. Phys. J. Plus, 133 (2018), 133–259
  • View Article
  • Google Scholar


• [33] W. Li, Y. Pang, Application of Adomian decomposition method to nonlinear systems, Adv. Difference Equ., 2020 (2020), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] J. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. A, 322 (1971), 447–468
  • View Article
  • Google Scholar


• [35] J. Patade, S. Bhalekar, On analytical solution of Ambartsumian equation, Nat. Acad. Sci. Lett., 40 (2017), 291–293
  • View Article
  • MathSciNet
  • Google Scholar


• [36] A.-M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166 (2005), 652–663
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH