New Approach of Non-Linear Fractional Differential Equations Analytical Solution by Akbari-Ganji’s Method

Document Type : Original Article

Authors

• Ziba Khalili ¹
• Ali Bararpour ¹
• Khashayar Hosseinzadeh ¹
• Bahram Jafari ²
• Davood Domiri Ganji ¹

¹ Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

² Faculty of Engineering Modern Technologies, Amol University of Special Modern Technologies (AUSMT), Amol, Iran

Abstract

Keywords

• Fractional Differential Equations
• Akbari-Ganji’s Method
• Mathematical Methods

• Sinan, M., Shah, K., Khan, Z. A., Al-Mdallal, Q., & Rihan, F. (2021). On semianalytical study of fractional-order kawahara partial differential equation with the homotopy perturbation method. Journal of Mathematics, 2021. doi:10.1155/2021/6045722.
• Das, P., Rana, S., & Ramos, H. (2019). Homotopy perturbation method for solving Caputo‐type fractional‐order Volterra‐Fredholm integro‐differential equations. Computational and Mathematical Methods, 1(5). doi:10.1002/cmm4.1047.
• Javeed, S., Baleanu, D., Waheed, A., Shaukat Khan, M., & Affan, H. (2019). Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations. Mathematics, 7(1), 40. doi:10.3390/math7010040.
• Gómez-Aguilar, J., Yépez-Martínez, H., Torres-Jiménez, J., Córdova-Fraga, T., Escobar-Jiménez, R., & Olivares-Peregrino, V. (2017). Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. Advances in Difference Equations, 2017(1). doi:10.1186/s13662-017-1120-7.
• Inas, A. A., & Eladdad, E. E. (2022). On Solving Partial Differential Equations by a Coupling of the Homotopy Perturbation Method and a New Integral Transform. Applied Mathematics and Information Sciences, 16(1), 51–57. doi:10.18576/amis/160106.
• Shabeeb, M. A., Aljaboori, M. A., & Jasim, S. H. (2018). Homotopy Perturbation Method For Solving Coupled Of Partial Differential Equation. مجلة میسان للدراسات الأکادیمیة, 326. doi:10.54633/2333-017-034-029.
• Ziane, D., & Cherif, M. H. (2018). Variational iteration transform method for fractional differential equations. Journal of Interdisciplinary Mathematics, 21(1), 185–199. doi:10.1080/09720502.2015.1103001.
• Sontakke, B. R., Shelke, A. S., & Shaikh, A. S. (2019). Solution of Non-Linear Fractional Differential Equations By Variational Iteration Method and Applications. Far East Journal of Mathematical Sciences (FJMS), 110(1), 113–129. doi:10.17654/ms110010113.
• Singh, B. K., & Kumar, P. (2017). Fractional Variational Iteration Method for Solving Fractional Partial Differential Equations with Proportional Delay. International Journal of Differential Equations, 2017. doi:10.1155/2017/5206380.
• Sakar, M. G., & Ergören, H. (2015). Alternative variational iteration method for solving the time-fractional Fornberg-Whitham equation. Applied Mathematical Modelling, 39(14), 3972–3979. doi:10.1016/j.apm.2014.11.048.
• Demir, D. D., & Zeybek, A. (2017). The Numerical Solution of Fractional Bratu-Type Differential Equations. ITM Web of Conferences, 13, 01008. doi:10.1051/itmconf/20171301008.
• Bansal, M. K., & Jain, R. (2016). Analytical solution of bagley Torvik equation by generalize differential transform. International Journal of Pure and Applied Mathematics, 110(2), 265–273. doi:10.12732/ijpam.v110i2.3.
• Ünal, E., & Gökdoğan, A. (2017). Solution of conformable fractional ordinary differential equations via differential transform method. Optik, 128, 264–273. doi:10.1016/j.ijleo.2016.10.031.
• Yang, X. J., Tenreiro Machado, J. A., & Srivastava, H. M. (2016). A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach. Applied Mathematics and Computation, 274, 143–151. doi:10.1016/j.amc.2015.10.072.
• Panda, A., Santra, S., & Mohapatra, J. (2022). Adomian decomposition and homotopy perturbation method for the solution of time fractional partial integro-differential equations. Journal of Applied Mathematics and Computing, 68(3), 2065–2082. doi:10.1007/s12190-021-01613-x.
• Sadeghinia, A., & Kumar, P. (2021). One Solution of Multi-term Fractional Differential Equations by Adomian Decomposition Method: Scientific Explanation. Current Topics on Mathematics and Computer Science Vol. 6, 120–130. doi:10.9734/bpi/ctmcs/v6/11542d.
• Haq, F., Shah, K., ur Rahman, G., & Shahzad, M. (2018). Numerical solution of fractional order smoking model via laplace Adomian decomposition method. Alexandria Engineering Journal, 57(2), 1061–1069. doi:10.1016/j.aej.2017.02.015.
• Verma, P., & Kumar, M. (2022). Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative. Engineering with Computers, 38(1), 661–678. doi:10.1007/s00366-020-01061-4.
• Khalouta, A., & Kadem, A. (2019). A New Method to Solve Fractional Differential Equations: Inverse Fractional Shehu Transform Method. An International Journal (AAM), 14(2), 926–941. http://pvamu.edu/aam
• Baskonus, H. M., & Bulut, H. (2015). On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method. Open Mathematics, 13(1), 547–556. doi:10.1515/math-2015-0052.
• Khalouta, A., & Kadem, A. (2020). An efficient method for solving nonlinear time-fractional wave-like equations with variable coefficients. Tbilisi Mathematical Journal, 12(4). doi:10.32513/tbilisi/1578020573.
• Jafari, H., Tajadodi, H., & Baleanu, D. (2015). A numerical approach for fractional order riccati differential equation using B-spline operational matrix. Fractional Calculus and Applied Analysis, 18(2), 387–399. doi:10.1515/fca-2015-0025.
• Keshavarz, E., Ordokhani, Y., & Razzaghi, M. (2016). A numerical solution for fractional optimal control problems via Bernoulli polynomials. JVC/Journal of Vibration and Control, 22(18), 3889–3903. doi:10.1177/1077546314567181.
• Garrappa, R. (2018). Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial. Mathematics, 6(2), 16. doi:10.3390/math6020016.
• Nagdy, A. S., & Hashem, K. M. (2020). Numerical solutions of nonlinear fractional differential equations by variational iteration method. Journal of Nonlinear Sciences and Applications, 14(02), 54–62. doi:10.22436/jnsa.014.02.01.
• Meng, Z., Yi, M., Huang, J., & Song, L. (2018). Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials. Applied Mathematics and Computation, 336, 454–464. doi:10.1016/j.amc.2018.04.072.
• Loverro, A. (2004). Fractional calculus: history, definitions and applications for the engineer. Rapport technique, Department of Aerospace and Mechanical Engineering, Univeristy of Notre Dame, Notre Dame, United States.
• Akbari, M. R., Ganji, D. D., Nimafar, M., & Ahmadi, A. R. (2014). Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach. Frontiers of Mechanical Engineering, 9(4), 390–401. doi:10.1007/s11465-014-0313-y