Advances in mathematical analysis for solving inhomogeneous scalar differential equation

Abdulrahman B. Albidah; Ibraheem M. Alsulami; Essam R. El-Zahar; Abdelhalim Ebaid; Abdulrahman B. Albidah; Ibraheem M. Alsulami; Essam R. El-Zahar; Abdelhalim Ebaid

doi:10.3934/math.20241134

AIMS Mathematics

2024, Volume 9, Issue 9: 23331-23343. doi: 10.3934/math.20241134

Previous Article Next Article

Research article

Advances in mathematical analysis for solving inhomogeneous scalar differential equation

1.
Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Majmaah, 11952, Saudi Arabia
2.
Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia
3.
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
4.
Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt
5.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia

Received: 13 July 2024 Revised: 22 July 2024 Accepted: 22 July 2024 Published: 02 August 2024
MSC : 34K06, 65L03

This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model's parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term.
- ordinary differential equation,
- delay,
- initial value problem
Citation: Abdulrahman B. Albidah, Ibraheem M. Alsulami, Essam R. El-Zahar, Abdelhalim Ebaid. Advances in mathematical analysis for solving inhomogeneous scalar differential equation[J]. AIMS Mathematics, 2024, 9(9): 23331-23343. doi: 10.3934/math.20241134

Related Papers:

Abstract

This paper considered a functional model which splits to two types of equations, mainly, advance equation and delay equation. The advance equation was solved using an analytical approach. Different types of solutions were obtained for the advance equation under specific conditions of the model's parameters. These solutions included the polynomial solutions of first and second degrees, the periodic solution and the hyperbolic solution. The periodic solution was invested to establish the analytical solution of the delay equation. The characteristics of the solution of the present model were discussed in detail. The results showed that the solution was continuous in the domain of the problem, under a restriction on the given initial condition, while the first derivative was discontinuous at a certain point and lied within the domain of the delay equation. In addition, some existing results in the literature were recovered as special cases of the current ones. The present successful analysis can be further generalized to include complex functional equations with an arbitrary function as an inhomogeneous term.

References

[1]	H. I. Andrews, Third paper: Calculating the behaviour of an overhead catenary system for railway electrification, Proc. Inst. Mech. Eng., 179 (1964), 809–846. https://doi.org/10.1243/PIME_PROC_1964_179_050_02 doi: 10.1243/PIME_PROC_1964_179_050_02
[2]	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. https://doi.org/10.1093/comjnl/13.4.363 doi: 10.1093/comjnl/13.4.363
[3]	G. Gilbert, H. E. H. Davtcs, Pantograph motion on a nearly uniform railway overhead line, Proc. Inst. Electr. Eng., 113 (1966), 485–492. https://doi.org/10.1049/piee.1966.0078 doi: 10.1049/piee.1966.0078
[4]	P. M. Caine, P. R. Scott, Single-wire railway overhead system, Proc. Inst. Electr. Eng., 116 (1969), 1217–1221. https://doi.org/10.1049/piee.1969.0226 doi: 10.1049/piee.1969.0226
[5]	J. R. Ockendon, 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. https://doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
[6]	L. Fox, D. F. Mayers, J. R. Ockendon, A. B. Tayler, On a functional differential equation, IMA J. Appl. Math., 8 (1971), 271–307. https://doi.org/10.1093/imamat/8.3.271 doi: 10.1093/imamat/8.3.271
[7]	T. Kato, J. B. McLeod, The functional-differential equation $y'(x) = ay(\lambda x)+by(x)$, Bull. Am. Math. Soc., 77 (1971), 891–935.
[8]	A. Iserles, On the generalized pantograph functional-differential equation, Eur. J. Appl. Math., 4 (1993), 1–38. https://doi.org/10.1017/S0956792500000966 doi: 10.1017/S0956792500000966
[9]	V. A. Ambartsumian, On the fluctuation of the brightness of the milky way, Doklady Akad Nauk USSR, 44 (1994), 244–247.
[10]	J. Patade, S. Bhalekar, On analytical solution of Ambartsumian equation, Natl. Acad. Sci. Lett., 40 (2017), 291–293. https://doi.org/10.1007/s40009-017-0565-2 doi: 10.1007/s40009-017-0565-2
[11]	H. O. Bakodah, A. Ebaid, Exact solution of Ambartsumian delay differential equation and comparison with Daftardar-Gejji and Jafari approximate method, Mathematics, 6 (2018), 331. https://doi.org/10.3390/math6120331 doi: 10.3390/math6120331
[12]	S. M. Khaled, E. R. El-Zahar, A. Ebaid, Solution of Ambartsumian delay differential equation with conformable derivative, Mathematics, 7 (2019), 425. https://doi.org/10.3390/math7050425 doi: 10.3390/math7050425
[13]	D. Kumar, J. Singh, D. Baleanu, S. Rathore, Analysis of a fractional model of the Ambartsumian equation, Eur. Phys. J. Plus, 133 (2018), 1330–259. https://doi.org/10.1140/epjp/i2018-12081-3 doi: 10.1140/epjp/i2018-12081-3
[14]	A. Ebaid, H. K. Al-Jeaid, On the exact solution of the functional differential equation $y'(t) = ay(t)+by(-t)$, Adv. Differ. Equ. and Contr., 26 (2022), 39–49.
[15]	N. A. M. Alshomrani, A. Ebaid, F. Aldosari, M. D. Aljoufi, On the exact solution of a scalar differential equation via a simple analytical approach, Axioms, 13 (2024), 129. https://doi.org/10.3390/axioms13020129 doi: 10.3390/axioms13020129
[16]	G. Adomian, Solving frontier problems of physics: The decomposition method, Boston: Springer Science & Business Media, 2013.
[17]	A. M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166 (2005), 652–663. https://doi.org/10.1016/j.amc.2004.06.059 doi: 10.1016/j.amc.2004.06.059
[18]	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. https://doi.org/10.1016/j.amc.2011.09.037 doi: 10.1016/j.amc.2011.09.037
[19]	J. Diblík, M. Kúdelcíková, Two classes of positive solutions of first order functional differential equations of delayed type, Nonlinear Anal., 75 (2012), 4807–4820. https://doi.org/10.1016/j.na.2012.03.030 doi: 10.1016/j.na.2012.03.030
[20]	A. Alshaery, A. Ebaid, Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method, Acta Astronaut., 140 (2017), 27–33. https://doi.org/10.1016/j.actaastro.2017.07.034 doi: 10.1016/j.actaastro.2017.07.034
[21]	W. J. Li, Y. N. Pang, Application of Adomian decomposition method to nonlinear systems, Adv. Differ. Equ., 67 (2020). https://doi.org/10.1186/s13662-020-2529-y
[22]	A. Ebaid, H. K. Al-Jeaid, H. Al-Aly, Notes on the perturbation solutions of the boundary layer flow of nanofluids past a stretching sheet, Appl. Math. Sci., 122 (2013), 6077–6085. http://dx.doi.org/10.12988/ams.2013.36277 doi: 10.12988/ams.2013.36277
[23]	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. https://doi.org/10.1016/j.camwa.2014.05.008 doi: 10.1016/j.camwa.2014.05.008
[24]	Z. Ayati, J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc., 23 (2015), 424–428. http://dx.doi.org/10.1016/j.joems.2014.06.015 doi: 10.1016/j.joems.2014.06.015
[25]	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. https://doi.org/10.2298/TSCI151005050K doi: 10.2298/TSCI151005050K
[26]	O. Nave, Modification of semi-analytical method applied system of ODE, Mod. Appl. Sci., 14 (2020), 75–81. https://doi.org/10.5539/mas.v14n6p75 doi: 10.5539/mas.v14n6p75
[27]	A. F. Yeniçerioğlu, C. Yazıcı, S. Pinelas, On the stability and behavior of solutions in mixed differential equations with delays and advances, Math. Method. Appl. Sci., 45 (2022), 4468–4496. https://doi.org/10.1002/mma.8049 doi: 10.1002/mma.8049
[28]	W. M. Abd-Elhameed, Y. H. Youssri, A. G. Atta, Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials, J. Math. Model., 12 (2024), 277–299. https://doi.org/10.22124/jmm.2024.25844.2295 doi: 10.22124/jmm.2024.25844.2295

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)