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SOLUTION OF A DIFFERENTIAL-DIFFERENCE EQUATION VIA AN ANSATZ METHOD
Keywords:
ansatz method, differential-difference equation, pantograph, exact solution.DOI:
https://doi.org/10.17654/0974165823004Abstract
The differential-difference equation $\phi^{\prime}(t)=\alpha \phi(t)+\beta \phi(-t)$ is a special case of the pantograph equation $\phi^{\prime}(t)=\alpha \phi(t)+\beta \phi(\gamma t)$ at $\gamma=-1$. The pantograph is the device responsible for gathering the current in electric trains. The present paper develops an ansatz method to solve the current model. The developed ansatz method is capable to obtain the solution in exact wave form which has not been reported in the literature. At special cases of the involved parameters, the obtained exact solutions are in full agreement with the corresponding ones in the literature via a series approach. However, the present method is direct, simpler, and can be further extended to include other complex models of the same type.
Received: November 2, 2022;
Accepted: December 14, 2022;
References
H. I. Andrews, Third paper: calculating the behaviour of an overhead catenary system for railway electrification, Proc. Inst. Mech. Eng. 179 (1964), 809-846.
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.
G. Gilbert and H. E. H. Davtcs, Pantograph motion on a nearly uniform railway overhead line, Proc. Inst. Electr. Eng. 113 (1966), 485-492.
P. M. Caine and P. R. Scott, Single-wire railway overhead system, Proc. Inst. Electr. Eng. 116 (1969), 1217-1221.
J. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 322 (1971), 447 468.
L. Fox, D. Mayers, J. R. Ockendon and A. B. Tayler, On a functional differential equation, IMA J. Appl. Math. 8 (1971), 271-307.
T. Kato and J. B. McLeod, The functional-differential equation Bull. Am. Math. Soc. 77 (1971), 891-935.
A. Iserles, On the generalized pantograph functional-differential equation, Eur. J. Appl. Math. 4 (1993), 1-38.
V. A. Ambartsumian, On the fluctuation of the brightness of the milky way, Doklady Akad Nauk USSR 44 (1994), 223-226.
J. Patade and S. Bhalekar, On analytical solution of Ambartsumian equation, Natl. Acad. Sci. Lett. 40 (2017), 291-293. DOI 10.1007/s40009-017-0565-2.
F. M. Alharbi and A. Ebaid, New analytic solution for Ambartsumian equation, Journal of Mathematics and System Science 8 (2018), 182-186.
DOI: 10.17265/2159-5291/2018.07.002
H. O. Bakodah and A. Ebaid, Exact solution of Ambartsumian delay differential equation and comparison with Daftardar-Gejji and Jafari approximate method, Mathematics 6 (2018), 331.
N. O. Alatawi and A. Ebaid, Solving a delay differential equation by two direct approaches, Journal of Mathematics and System Science 9 (2019), 54-56. DOI: 10.17265/2159-5291/2019.02.003
A. Ebaid, A. Al-Enazi, B. Z. Albalawi and M. D. Aljoufi, Accurate approximate solution of Ambartsumian delay differential equation via decomposition method, Math. Comput. Appl. 24(1) (2019), 7.
A. A. Alatawi, M. Aljoufi, F. M. Alharbi and A. Ebaid, Investigation of the surface brightness model in the Milky Way via homotopy perturbation method, J. Appl. Math. Phys. 8(3) (2020), 434-442.
E. A. Algehyne, E. R. El-Zahar, F. M. Alharbi and A. Ebaid, Development of analytical solution for a generalized Ambartsumian equation, AIMS Mathematics 5(1) (2019), 249-258. DOI: 10.3934/math.2020016
S. M. Khaled, E. R. El-Zahar and A. Ebaid, Solution of Ambartsumian delay differential equation with conformable derivative, Mathematics 7 (2019), 425.
D. Kumar, J. Singh, D. Baleanu and S. Rathore, Analysis of a fractional model of the Ambartsumian equation, Eur. Phys. J. Plus 133 (2018), 133-259.
A. Ebaid, C. Cattani, A. S. Al Juhani and E. R. El-Zahar, A novel exact solution for the fractional Ambartsumian equation, Adv. Differ. Equ. 2021, Paper No. 88, 18 pp. https://doi.org/10.1186/s13662-021-03235-w.
W. Li and Y. Pang, Application of Adomian decomposition method to nonlinear systems, Adv. Differ. Equ. 2020, Paper No. 67, 17 pp.
https://doi.org/10.1186/s13662-020-2529-y.
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acad., Boston, MA, USA, 1994.
A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput. 166 (2005), 652-663.
A. Ebaid, Approximate analytical solution of a nonlinear boundary value problem and its application in fluid mechanics, Z. Naturforschung A 66 (2011), 423-426.
J. S. Duan and 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.
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.
E. H. Aly, A. Ebaid and 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.
C. Chun, A. Ebaid, M. Lee and E. H. Aly, An approach for solving singular two point boundary value problems: analytical and numerical treatment, ANZIAM J. 53 (2012), 21-43.
A. Ebaid, M. D. Aljoufi and A.-M. Wazwaz, An advanced study on the solution of nanofluid flow problems via Adomian’s method, Appl. Math. Lett. 46 (2015), 117-122.
A. Ebaid, Analytical solutions for the mathematical model describing the formation of liver zones via Adomian’s method, Computational and Mathematical Methods in Medicine, Volume 2013, Article ID 547954, 8 pages.
A. Ebaid and N. Al-Armani, A new approach for a class of the blasius problem via a transformation and Adomian’s method, Abstract and Applied Analysis, Volume 2013, Article ID 753049, 8 pages.
A. Alshaery and A. Ebaid, Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method, Acta Astronautica 140 (2017), 27-33.
H. O. Bakodah and 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), 115.
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(3) (2014), 77-85.
Z. Ayati and J. Biazar, On the convergence of homotopy perturbation method, J. Egypt. Math. Soc. 23 (2015), 424-428.
A. Ebaid, A. F. Aljohani, E. H. Aly, Homotopy perturbation method for peristaltic motion of gold-blood nanofluid with heat source, International Journal of Numerical Methods for Heat & Fluid Flow 30(6) (2020), 3121-3138. https://doi.org/10.1108/HFF-11-2018-0655
A. Ebaid and H. K. Al-Jeaid, On the exact solution of the functional differential equation $y^{prime}(t)=a y(t)+b y(-t)$, Advances in Differential Equations and Control Processes 26(2022), 39-49.
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