In recent research the effect of different type fractional derivative to nonlinear evolution equations plays a vital rule in the various branches of science and engineering. The nonlinear physical phenomena are expressed by nonlinear partial differential equations which are characteristics on the field of solid-state physics, plasma physics, fluid mechanics, chemical physics, mechanics, biology, chemistry, and so on. To visualize and identify their properties, it is essential to find the exact and multi solitons of the related nonlinear partial differential equation. In this work, we investigate more soliton solutions for novel truncated M-fractional Cahn–Allen (t-MfCA) models to secure different soliton solutions via the unified scheme. This model has significant in the area of mathematical physics and also known as reaction–diffusion. The obtained solutions are expressed in turns of trigonometric, hyperbolic and rational function solution under the condition on the free constraints. This work offers the kink, singular soliton, different type interaction of kink and lump wave for the numerical value of the free constraints. Form the obtained solution it is shown that the implement method is more informal, effective and reliable as compared to other methods. The calculation and all analytic solutions are verified by computational software MAPLE 18.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 6) |
| DOI | 10.11648/j.ijtam.20220806.11 |
| Page(s) | 112-120 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Cahn–Allen Models, Unified Scheme, Reaction–Diffusion
| [1] | M. Kapoor, Exact solution of coupled 1D non-linear Burgers' equation by using Homotopy Perturbation Method (HPM): A review, j. Phys. Commun. 4 (9) (2020) 095017. |
| [2] | D Baleanu, HK Jassim, Exact solution of two-dimensional fractional partial differential equations, Fractal Fract. 4 (2) (2020) 21. |
| [3] | M. M. Roshid, M. F. Karim, A. K. Azad, M. M. Rahman, T. Sultana, New solitonic and rogue wave solutions of a Klein–Gordon equation with quadratic nonlinearity, Partial Diff Equ Appl Math 3 (2021) 100036. |
| [4] | H. O. Roshid, M. F. Hoque, M. A. Akbar, New extended (G'/G)-expansion method for traveling wave solutions of non-linear partial differential equations (NPDEs) in mathematical physics, Ital. J. Pure Appl. Math. 33 (33) (2014) 175-190. |
| [5] | S. Kumar, A. Kumar, A. M. Wazwaz, New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur. Phys. J. Plus 135 (2020) 870. |
| [6] | H. O. Roshid, N. Rahman, M. A. Akbar, Traveling waves solutions of nonlinear Klein Gordon equation by extended (G′/G)-expansion method, Ann. Pure Appl. Math 3 (2013) 10-16. |
| [7] | W. Ma, Travelling wave solutions to a seventh order generalized KdV equation, Phys. Lett. A 180 (3) (1993) 221-224. |
| [8] | K. K. Ali, A. M. Wazwaz, M. S. Osman, Optical soliton solutions to the generalized nonautonomous nonlinear Schrödinger equations in optical fibers via the sine-Gordon expansion method, Optik 208 (2020) 164132. |
| [9] | M. Hossen, H. O. Roshid, M. Ali, Modified Double Sub-equation Method for Finding Complexiton Solutions to the (1 + 1, 1+1) Dimensional Nonlinear Evolution Equations, Int. J. Appl. Math. Comput. Sci. 3 (1) (2017) 679-697. |
| [10] | A. M. Wazwaz, Bright and dark optical solitons for (3+ 1)-dimensional Schrödinger equation with cubic–quintic-septic nonlinearities, Optik 255 (2021) 165752. |
| [11] | M. M. Roshid, H. O. Roshid, M. Z. Ali, H. Rezazadeh, Kinky periodic pulse and interaction of bell wave with kink pulse wave propagation in nerve fibers and wall motion in liquid crystals, Partial Diff Equ Appl Math 2 (2020) 100012. |
| [12] | R. Lin, Y. Zeng, W. X. Ma, Solving the KdV hierarchy with self-consistent sources by inverse scattering method, Phys. A: Stat. Mech. Appl 291 (1-4) (2001) 287-298. |
| [13] | W. X. Ma, R. Zhou, L. Gao, Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+ 1) dimensions, Mod. Phys. Lett. A 24 (21) (2008) 1677-1688. |
| [14] | M. B. Hossen, H. O. Roshid, M. Z. Ali, Multi-soliton, breathers, lumps and interaction solution to the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equation, Heliyon 5 (10) (2019) e02548. |
| [15] | M. M. Hossain, H. O. Roshid, M. A. N. Sheikh, Bilinear form of the regularized long wave equation and its multi-soliton solutions, Partial Diff Equ Appl Math 6 (2022) 100422. |
| [16] | M. M. Roshid, T. Bairagi, H. O. Roshid, M. M. Rahman, Lump, interaction of lump and kink and solitonic solution of nonlinear evolution equation which describe incompressible viscoelastic Kelvin–Voigt fluid, Partial Diff Equ Appl Math 5 (2022) 100354. |
| [17] | H. O. Roshid, M. M. Roshid, N. Rahman, M. R. Pervin, New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation, Propuls. Power Res. 6 (1) (2017) 49-57. |
| [18] | M. M. Roshid, H. O. Roshid, Exact and explicit travelling wave solutions to two nonlinear evolution equations which describe incompressible visco-elastic Kelvin-Voigt fluid, Heliyon 4 (8) (2018) e00756. |
| [19] | N. F. M. Noor, M. S. Ullah, H. O. Roshid, M. Z. Ali, Novel dynamics of wave solutions for Cahn–Allen and diffusive predator–prey models using MSE scheme, Partial Diff Equ Appl Math 3 (1) (2021) 100017. |
| [20] | W. X. Ma, Z. Zhu, Solving the (3+ 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput. 218 (24) (2012) 11871-11879. |
| [21] | M. M. Hossain, H. O. Roshid, M. A. N. Sheikh, Topological soliton, singular soliton and other exact travelling wave solution for Burger-Hexley Equation, Asian J. Math. & Comput. Res. 6 (4) (2015) 312-331. |
| [22] | Z. Islam, M. M. Hossain, M. A. N, Sheikh, Exact Travelling wave Solutions to Benney-Luke Equation, GANIT J. Bangladesh Math. Soc. 37 (2017) 1-14. |
| [23] | M. M. Hossain, H. O. Rashid, M. A. N. Sheikh, Abundant Exact Travelling Solutions of the (2+1) Dimensional Couple Broer-Kaup Equation, J. found. appl. phys. 3 (1) (2016) 2394-3688. |
| [24] | S.-W. Yao, R. Manzoor, A. Zafar, M. Inc, S. Abbagari, A. Houwe, Exact soliton solutions to the Cahn–Allen equation and Predator–Prey model with truncated M-fractional derivative, Results Phys. 37 (2022) 105455. |
| [25] | S. Javeed, S. Saif, D. Baleanu, New exact solutions of fractional Cahn–Allen equation and fractional DSW system, Adv Differ Equ 2018 (2018) 459. |
| [26] | F. Tascan, A. Bekir, Travelling wave solutions of the Cahn–Allen equation by using first integral method. J. Comput. Appl. Math. 207 (1) (2009) 279–282. |
| [27] | H. Bulut, S. S. Atas, H. M. Baskonus, Some novel exponential function structures to the Cahn–Allen equation, Cogent Phys 3 (2016) 1240886. |
| [28] | M. S. Ullah, A. Abdeljabbar, H. O. Roshid, M. Z. Ali, Application of the unified method to solve the Biswas-Arshed model, results phys. 42 (2022) 105946. |
| [29] | E. Ilhan, I. O. Kıymaz, A generalization of truncated M-fractional derivative and applications to fractional differential equations, Appl. math. nonlinear sci. 5 (1) (2020) 171–188. |
APA Style
Mohammad Mobarak Hossain, Md. Mamunur Roshid, Md. Abu Naim Sheikh, Mohammad Abu Taher, Harun-Or-Roshid. (2022). Novel Exact Soliton Solutions of Cahn–Allen Models with Truncated M-fractional Derivative. International Journal of Theoretical and Applied Mathematics, 8(6), 112-120. https://doi.org/10.11648/j.ijtam.20220806.11
ACS Style
Mohammad Mobarak Hossain; Md. Mamunur Roshid; Md. Abu Naim Sheikh; Mohammad Abu Taher; Harun-Or-Roshid. Novel Exact Soliton Solutions of Cahn–Allen Models with Truncated M-fractional Derivative. Int. J. Theor. Appl. Math. 2022, 8(6), 112-120. doi: 10.11648/j.ijtam.20220806.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ijtam.20220806.11,
author = {Mohammad Mobarak Hossain and Md. Mamunur Roshid and Md. Abu Naim Sheikh and Mohammad Abu Taher and Harun-Or-Roshid},
title = {Novel Exact Soliton Solutions of Cahn–Allen Models with Truncated M-fractional Derivative},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {8},
number = {6},
pages = {112-120},
doi = {10.11648/j.ijtam.20220806.11},
url = {https://doi.org/10.11648/j.ijtam.20220806.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220806.11},
abstract = {In recent research the effect of different type fractional derivative to nonlinear evolution equations plays a vital rule in the various branches of science and engineering. The nonlinear physical phenomena are expressed by nonlinear partial differential equations which are characteristics on the field of solid-state physics, plasma physics, fluid mechanics, chemical physics, mechanics, biology, chemistry, and so on. To visualize and identify their properties, it is essential to find the exact and multi solitons of the related nonlinear partial differential equation. In this work, we investigate more soliton solutions for novel truncated M-fractional Cahn–Allen (t-MfCA) models to secure different soliton solutions via the unified scheme. This model has significant in the area of mathematical physics and also known as reaction–diffusion. The obtained solutions are expressed in turns of trigonometric, hyperbolic and rational function solution under the condition on the free constraints. This work offers the kink, singular soliton, different type interaction of kink and lump wave for the numerical value of the free constraints. Form the obtained solution it is shown that the implement method is more informal, effective and reliable as compared to other methods. The calculation and all analytic solutions are verified by computational software MAPLE 18.},
year = {2022}
}
TY - JOUR T1 - Novel Exact Soliton Solutions of Cahn–Allen Models with Truncated M-fractional Derivative AU - Mohammad Mobarak Hossain AU - Md. Mamunur Roshid AU - Md. Abu Naim Sheikh AU - Mohammad Abu Taher AU - Harun-Or-Roshid Y1 - 2022/12/29 PY - 2022 N1 - https://doi.org/10.11648/j.ijtam.20220806.11 DO - 10.11648/j.ijtam.20220806.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 112 EP - 120 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20220806.11 AB - In recent research the effect of different type fractional derivative to nonlinear evolution equations plays a vital rule in the various branches of science and engineering. The nonlinear physical phenomena are expressed by nonlinear partial differential equations which are characteristics on the field of solid-state physics, plasma physics, fluid mechanics, chemical physics, mechanics, biology, chemistry, and so on. To visualize and identify their properties, it is essential to find the exact and multi solitons of the related nonlinear partial differential equation. In this work, we investigate more soliton solutions for novel truncated M-fractional Cahn–Allen (t-MfCA) models to secure different soliton solutions via the unified scheme. This model has significant in the area of mathematical physics and also known as reaction–diffusion. The obtained solutions are expressed in turns of trigonometric, hyperbolic and rational function solution under the condition on the free constraints. This work offers the kink, singular soliton, different type interaction of kink and lump wave for the numerical value of the free constraints. Form the obtained solution it is shown that the implement method is more informal, effective and reliable as compared to other methods. The calculation and all analytic solutions are verified by computational software MAPLE 18. VL - 8 IS - 6 ER -
Information
Email: