Approximate Solution of Multi-Term Fractional Order Delay Differential Equations Using Homotopy Perturbation Method
Keywords:
Homotopy perturbation method, Fractional derivatives, Fractional delay differential equationsAbstract
In this paper the approximate solution of the non-linear equations of multi-term fractional order delay differential equations by using the homotopy perturbation method is considered. The fractional order derivative is communicated in the Caputo sense. In this methodology, the solutions are found in the form of a convergent power series with easily computed components. Finally, some examples are given to illustrate the obtained results, and then a comparison between the exact and the approximate results were given and they are presented in order to show the reliability and the accuracy of the proposed method.
References
[1] Mishra, H. K.; Tripathi, R.; “Homotopy Perturbation Method of Delay Differential Equation Using He’s Polynomial with Laplace Transform”, Proc. Natl. Acad. Sci. India Sect. A Phys. Sci., 2019.
[2] He, J. H.; “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng. 178(3–4), 257–262, 1999.
[3] Tarif, E. A.; “On the Coupling of the Homotopy Perturbation Method and New Integral Transform for Solving Systems of Partial Differential Equations” 2019(1), 2019.
[4] Jhinga, A.; Daftardar-Gejji, V.; “A new numerical method for solving fractional delay differential equations”, Comput. Appl. Math. 38(4), 2019.
[5] Shaikh, A.; Asif, J. M.; Hanif F.; Sadiq Ali Khan, M.; Inayatullah, S.; “Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA),” PLoS One 14(10), 1–22, 2019.
[6] Gepreel, K. A.; “The homotopy perturbation method applied to the nonlinear fractional Kolmogorov Petrovskii Piskunov equations”, Appl. Math. Lett. 24(8), 1428–1434, 2011.
[7] Fadhel, S.; Batool, I.; “Approximate Solutions of the Generalized Two-Dimensional Fractional Partial Integro-Differential Equations”, Al-Nahrain J. Sci. 22(2), 59–67, 2019.
[8] Daftardar-gejji, V.; Jafari, H.; “Solving a multi-order fractional differential equation using adomian decomposition”, 189, 541–548, 2007.
[9] Ji-Huan, H.; “Application of homotopy perturbation method to nonlinear wave equations,” Chaos Solitons and Fractals 26(3), 695–700, 2005.
[10] El-Dib, Y. O.; “On the coupling of the homotopy perturbation and frobenius method for exact solutions of singular nonlinear differential equations,” Nonlin. Sci. Lett. A9(3), 220–230, 2018.
[11] El-Shahed, M.; “Application of He’s Homotopy Perturbation Method to Volterra’s Integro-differential Equation,” Int. J. Nonlin. Sci. Numer. Simul. 6(2), 163–168, 2005.
[12] Ji-Huan, H.; “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Appl. Math. Comp. 151(1), 287–292, 2004.
[13] Ji-Huan, H.; “Homotopy perturbation method for solving boundary-value problems”, Phys. Lett. A-350(1–2), 87–88, 2006.
[14] Wazwaz, A. M.; “A new method for solving singular initial value problems in the second-order ordinary differential equations”, Appl. Math. Comp. 128(1), 45–57, 2002.
[15] Alshikhand, A.; Abdelrahim, A.; Mahgoub, M. M.; “Solving ordinary differential equations with variable coefficients”, J. Progr. Res. Math. 10(1), 15–22, 2016.
[16] Ji-Huan, H.; “Periodic solutions and bifurcations of delay differential equations,” Phys. Lett. A-347(4–6), 228–230, 2005.
[2] He, J. H.; “Homotopy perturbation technique,” Comput. Methods Appl. Mech. Eng. 178(3–4), 257–262, 1999.
[3] Tarif, E. A.; “On the Coupling of the Homotopy Perturbation Method and New Integral Transform for Solving Systems of Partial Differential Equations” 2019(1), 2019.
[4] Jhinga, A.; Daftardar-Gejji, V.; “A new numerical method for solving fractional delay differential equations”, Comput. Appl. Math. 38(4), 2019.
[5] Shaikh, A.; Asif, J. M.; Hanif F.; Sadiq Ali Khan, M.; Inayatullah, S.; “Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA),” PLoS One 14(10), 1–22, 2019.
[6] Gepreel, K. A.; “The homotopy perturbation method applied to the nonlinear fractional Kolmogorov Petrovskii Piskunov equations”, Appl. Math. Lett. 24(8), 1428–1434, 2011.
[7] Fadhel, S.; Batool, I.; “Approximate Solutions of the Generalized Two-Dimensional Fractional Partial Integro-Differential Equations”, Al-Nahrain J. Sci. 22(2), 59–67, 2019.
[8] Daftardar-gejji, V.; Jafari, H.; “Solving a multi-order fractional differential equation using adomian decomposition”, 189, 541–548, 2007.
[9] Ji-Huan, H.; “Application of homotopy perturbation method to nonlinear wave equations,” Chaos Solitons and Fractals 26(3), 695–700, 2005.
[10] El-Dib, Y. O.; “On the coupling of the homotopy perturbation and frobenius method for exact solutions of singular nonlinear differential equations,” Nonlin. Sci. Lett. A9(3), 220–230, 2018.
[11] El-Shahed, M.; “Application of He’s Homotopy Perturbation Method to Volterra’s Integro-differential Equation,” Int. J. Nonlin. Sci. Numer. Simul. 6(2), 163–168, 2005.
[12] Ji-Huan, H.; “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Appl. Math. Comp. 151(1), 287–292, 2004.
[13] Ji-Huan, H.; “Homotopy perturbation method for solving boundary-value problems”, Phys. Lett. A-350(1–2), 87–88, 2006.
[14] Wazwaz, A. M.; “A new method for solving singular initial value problems in the second-order ordinary differential equations”, Appl. Math. Comp. 128(1), 45–57, 2002.
[15] Alshikhand, A.; Abdelrahim, A.; Mahgoub, M. M.; “Solving ordinary differential equations with variable coefficients”, J. Progr. Res. Math. 10(1), 15–22, 2016.
[16] Ji-Huan, H.; “Periodic solutions and bifurcations of delay differential equations,” Phys. Lett. A-347(4–6), 228–230, 2005.
Downloads
Published
2020-06-04
Issue
Section
Articles
How to Cite
(1)
Approximate Solution of Multi-Term Fractional Order Delay Differential Equations Using Homotopy Perturbation Method. ANJS 2020, 23 (2), 60-66.
.jpg)
