NEW WAVE SOLUTIONS OF TIME FRACTIONAL CHAFEE-INFANTE EQUATION WITH BETA DERIVATIVE

Abstract

In this article, we discuss the exact solutions forthe Chafee-Infante equation involving beta fractional derivative. Beta fractional derivative which is a local derivative, is a modification of conformable fractional derivative. Using the Modified Kudryashov Method, we obtain the general solution of the time fractional Chafee-Infante equation with the help of Wolfram Mathematica. We use chain rule and wave transform to convert the equation into integer order nonlinear ordinary differential equation. Hence, we don’t need any discretization, normalization, or reduction. Moreover, 3D graphical representations are given. With the help of these representations, we can have an idea on the physical and geometrical behavior of the solutions.

Keywords

• Beta fractional derivative
• Chafee-Infante equation
• analytical methods
• modified Kudryashov method
• fractional derivative

References

1. U. Younas, A. R. Seadawy, M. Younis, S. T. R. Rizvi, Construction of Analytical Wave Solutions to the Conformable Fractional Dynamical System of Ion Sound and Langmuir Waves, Waves in Random and Complex Media 32 (6) (2022) 2587-2605.
2. S. Khavale, K. Gaiwad, Two-Dimensional Generalized Magneto-Thermo-Viscoelasticity Problem for a Spherical Cavity with One Relaxation Time Using Fractional Derivative, International Journal of Thermodynamics 25 (2) (2022) 89-97.
3. W. Chen, H. Sun, X. Li, Fractional derivative modeling in mechanics and engineering, 1st Edtion, Springer, Singapore, 2022.
4. K. B. Oldham, Fractional Differential Equations in Electrochemistry, Advances in Engineering Software 41 (1) (2010) 9-12.
5. B. M. Vinagre, I. Podlubny, A. Hernandez, V. Feliu, Some Approximations of Fractional Order Operators Used in Control Theory and Applications, Fractional calculus and applied analysis 3 (3) (2000) 231-248.
6. M. Khazayinejad, S. S. Nourazar, On the Effect of Spatial Fractional Heat Conduction in MHD Boundary Layer Flow Using Gr-Fe3O4-H2O Hybrid Nanouid, International Journal of Thermal Sciences 172 (2022) 107265.
7. I. Dassios, T. Ker_ci, D. Baleanu, F. Milano, Fractional-order Dynamical Model for Electricity Markets, Mathematical Methods in the Applied Sciences 46 (7) (2023) 8349-8361.
8. I. Yalçınkaya, H. Ahmad, O. Tasbozan, A. Kurt, Soliton Solutions for Time Fractional Ocean Engineering Models With Beta Derivative, Journal of Ocean Engineering and Science 7 (5) (2022) 444-448.

9. M. F. Uddin, M. G. Hafez, S. A. Iqbal, Dynamical Plane Wave Solutions for the Heisenberg Model of Ferromagnetic Spin Chains With Beta Derivative Evolution and Obliqueness, Heliyon 8 (3) (2022) e09199 17 pages.
10. E. M. Ozkan, New Exact Solutions of Some Important Nonlinear Fractional Partial Differential Equations With Beta Derivative, Fractal and Fractional 6 (3) (2022) 173.
11. F. M. Al-Askar, C. Cesarano, W. W. Mohammed, The Influence of White Noise and the Beta Derivative on the Solutions of the BBM Equation, Axioms 12 (5) (2023) 447.
12. A. Atangana, Derivative With a New Parameter: Theory, Methods and Applications, 1st Edition, Academic Press, 2015.
13. M. A. Iqbal, M. A. Akbar, M. A. Islam, The nonlinear wave dynamics of fractional foam drainage and Boussinesq equations with Atangana's beta derivative through analytical solutions, Results in Physics 56 (2024) 107251.
14. L. Tang, Dynamical behavior and multiple optical solitons for the fractional Ginzburg-Landau equation with _-derivative in optical fibers, Optical and Quantum Electronics, 56 (2) (2024) 175.
15. D. Kumar, A. R. Seadawy, A. K. Joardar, Modified Kudryashov Method via New Exact Solutions for Some Conformable Fractional Di_erential Equations Arising in Mathematical Biology, Chinese Journal of Physics 56 (1) (2018) 75-85.
16. O. Tasbozan, Y. Cenesiz, A. Kurt, New Solutions for Conformable Fractional Boussinesq and Combined KdV-mKdV Equations Using Jacobi Elliptic Function Expansion Method, The European Physical Journal Plus 131 (7) (2016) 1-14.
17. M. Cina, A. Secer, M. Bayram, Solving Nonlinear Fractional PDEs Using Novel Wavelet Collocation Method, New Trends in Mathematical Sciences 10 (1) (2022).