A Novel Approach for Solving Nonlinear Time Fractional Fisher Partial Differential Equations

Rana T.  Shwayyea

doi:10.24297/jam.v22i.9558

Authors

Rana T. Shwayyea Department of Physics, College of Education / University of AL-Qadisiyah

DOI:

https://doi.org/10.24297/jam.v22i.9558

Keywords:

Inverse of Laplace transform, LRPS method, fractional Fisher equations

Abstract

This study focuses on solving non-linear time fractional Fisher partial differential equations using analytical series solutions. The authors consider the Caputo fractional derivative in their formulas, which adds accuracy to the results. They introduce a novel method called LRPS which proves to be a powerful tool for obtaining precise analytical and numerical solutions for these equations. The LRPS method emphasizes precision, effectiveness, and practical application, making it suitable for various fields such as physics, engineering, and finance. Due to the importance of accuracy, effectiveness and method of application in this approach, it is highlighted that accurate solutions can be obtained when there is a pattern in the parts of the series, while approximate estimates are provided otherwise. The LRPS method is presented as a powerful technique specifically designed for solving nonlinear fractional Fisher partial differential equations.

Downloads

Download data is not yet available.

References

Abu Arqub, O., Application of residual power series method for the solution of time-fractional Schrödinger equations in one-dimensional space. Fundamenta Informaticae, 166(2), 87-110,2019. DOI:10.3233/FI-2019-1795.

Akdemir, A. O., Dutta, H., & Atangana, A. (Eds.)., Fractional order analysis: theory, methods and applications. John Wiley & Sons,2020, ISBN: 9781119654162, 1119654165

Alabedalhadi, M., Al-Smadi, M., Al-Omari, S., Baleanu, D., & Momani, S., Structure of optical soliton solution for nonlinear resonant space-time Schrödinger equation in conformable sense with full nonlinearity term. Physica Scripta, 95(10), 105215,2020,DOI:10.1088/1402-4896/abb739.

Al-Smadi, M., & Arqub, O. A., Computational algorithm for solving Fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates. Applied Mathematics and Computation, 342, 280-294,2019.

Bira, B., Raja Sekhar, T., & Zeidan, D., Exact solutions for some time‐ fractional evolution equations using Lie group theory. Mathematical Methods in the Applied Sciences, 41(16), 6717-6725,2018.

AL-Smadi, M. H., & Gumah, G. N., On the homotopy analysis method for fractional SEIR epidemic model. Research Journal of Applied Sciences, Engineering and Technology, 7(18), 3809-3820,2014, DOI: 10.19026/rjaset.7.738

Al-Smadi, M., Freihat, A., Arqub, O. A., & Shawagfeh, N., A Novel Multistep Generalized Differential Transform Method for Solving Fractional-order Lü Chaotic and Hyperchaotic Systems. Journal of Computational Analysis & Applications, 19(1),2015.

Baleanu, D., Golmankhaneh, A. K., Golmankhaneh, A. K., & Baleanu, M. C., Fractional electromagnetic equations using fractional forms. International Journal of theoretical Physics, 48(11), 3114-3123,2009, Doi.org/10.1007/s10773-009-0109-8

Kumar, S., A new fractional modeling arising in engineering sciences and its analytical approximate solution. Alexandria Engineering Journal, 52(4), 813-819,2013, Doi.org/10.1016/j.aej.2013.09.005

Al-Smadi, M., Freihat, A., Khalil, H., Momani, S., & Ali Khan, R., Numerical multistep approach for solving fractional partial differential equations. International Journal of Computational Methods, 14(03), 1750029,2017. Doi.org/10.1142/S0219876217500293

Al-Smadi, M., Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation. Ain Shams Engineering Journal, 9(4), 2517-2525,2018, Doi.org/10.1016/j.asej.2017.04.006 .

Jleli, M., Kumar, S., Kumar, R., & Samet, B., Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform method. Alexandria Engineering Journal, 59(5), 2859-2863,2020, Doi.org/10.1016/j.aej.2019.12.022

Momani, S., Freihat, A., & Al-Smadi, M., Analytical study of fractional-order multiple chaotic FitzHugh-Nagumo neurons model using multistep generalized differential transform method. In Abstract and Applied Analysis (Vol. 2014). Hindawi, 2014, January, Doi.org/10.1155/2014/276279

Al‐Smadi, M., Abu Arqub, O., & Gaith, M., Numerical simulation of telegraph and Cattaneo fractional‐type models using adaptive reproducing kernel framework.Mathematical Methods in the Applied Sciences, 44(10), 8472-8489,2021, Doi.org/10.1002/mma.6998.

Hasan, S., Al-Smadi, M., El-Ajou, A., Momani, S., Hadid, S., & Al-Zhour, Z., Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system. Chaos, Solitons & Fractals, 143, 110506,2021, Doi.org/10.1016/j.chaos.2020.110506

Talafha, A. G., Alqaraleh, S. M., Al-Smadi, M., Hadid, S., & Momani, S., Analytic solutions for a modified fractional three wave interaction equations with conformable derivative by unified method. Alexandria Engineering Journal, 59(5),3731-3739,2020, Doi.org/10.1016/j.aej.2020.06.027

Al-Smadi, M., Freihat, A., Hammad, M. M. A., Momani, S., & Arqub, O. A., Analytical approximations of partial differential equations of fractional order with multistep approach. Journal of Computational and Theoretical Nanoscience, 13(11),7793-7801,2016, Doi.org/10.1166/jctn.2016.5780

Gambo, Y. Y., Ameen, R., Jarad, F., & Abdeljawad, T., Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives. Advances in Difference Equations, 2018(1), 1-13.

Hasan, S., El-Ajou, A., Hadid, S., Al-Smadi, M., & Momani, S., Atangana- Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system. Chaos, Solitons & Fractals, 133, 109624,2020, Doi.org/10.1016/j.chaos.2020.109624 .

He, J. H., Elagan, S. K., & Li, Z. B., Geometrical explanation of the fractionalcomplex transform and derivative chain rule for fractional calculus. Physics letters A,376(4), 257-259,2012, Doi.org/10.1016/j.physleta.2011.11.030

Khader, M. M., Saad, K. M., Hammouch, Z., & Baleanu, D., A spectral collocation method for solving fractional KdV and KdV-Burgers equations with non-singular kernel derivatives. Applied Numerical Mathematics, 161, 137-146,2021, Doi.org/10.1016/j.apnum.2020.10.024

Momani, S., Arqub, O. A., Freihat, A., & Al-Smadi, M., Analytical approximations for Fokker-Planck equations of fractional order in multistep schemes. Applied and computational mathematics, 15(3), 319-330,2016.

Arqub, O. A., El-Ajou, A., & Momani, S., Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations.Journal of Computational Physics, 293, 385-399,2015, Doi.org/10.1016/j.jcp.2014.09.034

Xu, F., Gao, Y., Yang, X., & Zhang, S., Constructing of fractional power series solutions to fractional Boussinesq equations using residual power series method.Mathematical Problems in Engineering, 2016.

Eriqat, T., El-Ajou, A., Moa'ath, N. O., Al-Zhour, Z., & Momani, S., A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations. Chaos, Solitons & Fractals, 138, 109957,2020, Doi.org/10.1016/j.chaos.2020.109957

Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J., Theory and applications of fractional differential equations (Vol. 204). Elsevier,2006.

El-Ajou, A., Arqub, O. A., & Momani, S., Approximate analytical solution of the nonlinear fractional KdV–Burgers equation: a new iterative algorithm. Journal of Computational Physics, 293, 81-95,2015, Doi.org/10.1016/j.jcp.2014.08.004

Al Qurashi, M. M., Korpinar, Z., Baleanu, D., & Inc, M. (2017). A new iterative algorithm on the time-fractional Fisher equation: Residual power series method. Advances in Mechanical Engineering, 9(9), Doi.org/10.1177/1687814017716009

Khan, N. A., Ayaz, M., Jin, L., & Yildirim, A. (2011). On approximate solutions for the time-fractional reaction-diffusion equation of Fisher type. Int. J. Phys. Sci, 6(10), 2483-2496, DOI: 10.5897/IJPS11.181.

Merdan, M. (2012). Solutions of time-fractional reaction-diffusion equation with modified Riemann-Liouville derivative. Int. J. Phys. Sci, 7(15), 2317-2326, DOI: 10.5897/IJPS12.027.