An Efficient Method For Solving The Conformable Cauchy Reaction-diffusion Partial Differential Equations

Lingxia Liu

doi:10.6180/jase.202307_26(7).0005

An Efficient Method For Solving The Conformable Cauchy Reaction-diffusion Partial Differential Equations

Mathematics / Statistics

Lingxia LiuThis email address is being protected from spambots. You need JavaScript enabled to view it.¹

¹Department of Mathematics, Weifang University, Weifang 261061, Shandong, China

Received: May 1, 2022
Accepted: July 14, 2022
Publication Date: October 4, 2022

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

Download Citation: ||https://doi.org/10.6180/jase.202307_26(7).0005

ABSTRACT

In this paper, we propose a new technique for solving conformable Cauchy reaction-diffusion equations (CRFEs). These equations are widely used as models for spatial effects in engineering, biology and ecology sciences. The conformable derivatives are considered in Khalil sense. This method is based on perturbation theory and the conformable Laplace transformation (CLHPM). The solutions presented in this work can be used to obtain the closed form of the solutions if they are needed. The outcomes display that new technique is of high validity, more convenient and effective to use. The results presented in this paper show that the CLHPM is a powerful mathematical tool for solving other nonlinear conformable equations.

Keywords: CLT; Homotopy perturbation method; CRFEs; Conformable derivative.

REFERENCES

[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory And Applications of Fractional Differential Equations.204. Elsevier Science Limited, 2006.
[2] I. Podlubny, (1999) “An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications" Math. Sci. Eng 198: 340.
[3] J. Biazar and M. Eslami, (2011) “Differential transform method for nonlinear fractional gas dynamics equation" International Journal of Physical Sciences 6(5): 1203–1206.
[4] A. Zulfiqar and J. Ahmad, (2021) “Comparative study of two techniques on some nonlinear problems based using conformable derivative" Nonlinear Engineering 9(1): 470–482. DOI: 10.1515/nleng-2020-0030.
[5] M. Senol, L. Akinyemi, A. Ata, and O. S. Iyiola, (2021) “Approximate and generalized solutions of conformable type Coudrey–Dodd–Gibbon–Sawada–Kotera equation" International Journal of Modern Physics B 35(02): 2150021.
[6] S. Ganji, D. Ganji, and S. Karimpour, (2008) “Determination of the frequency-amplitude relation for nonlinear oscillators with fractional potential using he’s energy balance method" Progress In Electromagnetics Research C 5: 21–33.
[7] K. Srinivasa and H. Rezazadeh, (2021) “Numerical solution for the fractional-order one-dimensional telegraph equation via wavelet technique" International Journal of Nonlinear Sciences and Numerical Simulation 22(6): 767–780. DOI: 10.1515/ijnsns-2019-0300.
[8] M. Nadeem, J.-H. He, and A. Islam, (2021) “The homotopy perturbation method for fractional differential equations: part 1 Mohand transform" International Journal of Numerical Methods for Heat and Fluid Flow 31(11): 3490–3504. DOI: 10.1108/HFF-11-2020-0703.
[9] M.-X. Zhou, A. Ravi Kanth, K. Aruna, K. Raghavendar, H. Rezazadeh, M. Inc, and A. A. Aly, (2021) “Numerical solutions of time fractional zakharov-kuznetsov equation via natural transform decomposition method with nonsingular kernel derivatives" Journal of Function Spaces 2021: DOI: 10.1155/2021/9884027.
[10] M. S. Hashemi, M. Inc, and A. Yusuf, (2020) “On three-dimensional variable order time fractional chaotic system with nonsingular kernel" Chaos, Solitons & Fractals 133: 109628.
[11] H. Jafari and V. Daftardar-Gejji, (2006) “Solving a system of nonlinear fractional differential equations using Adomian decomposition" Journal of Computational and Applied Mathematics 196(2): 644–651.
[12] C. Cattani, T. A. Sulaiman, H. M. Baskonus, and H. Bulut, (2018) “On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems" Optical and Quantum Electronics 50(3): DOI: 10.1007/s11082-018-1406-3.
[13] M. Hashemi and D. Baleanu, (2016) “Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line" Journal of Computational Physics 316: 10–20. DOI: 10.1016/j.jcp.2016.04.009.
[14] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, (2015) “Exact solutions for the fractional differential equations by using the first integral method" Nonlinear engineering 4(1): 15–22.
[15] H. Aminikhah, A. R. Sheikhani, and H. Rezazadeh, (2016) “Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method" Boletim da Sociedade Paranaense de Matematica 34(2): 213–229. DOI: 10.5269/bspm.v34i2.25501.
[16] H. Rezazadeh, D. Kumar, T. A. Sulaiman, and H. Bulut, (2019) “New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation" Modern Physics Letters B 33(17): DOI: 10.1142/S0217984919501963.
[17] M. S. M. Shehata, H. Rezazadeh, E. H. M. Zahran, E. Tala-Tebue, and A. Bekir, (2019) “New Optical Soliton Solutions of the Perturbed Fokas-Lenells Equation" Communications in Theoretical Physics 71(11): 1275–1280. DOI: 10.1088/0253-6102/71/11/1275.
[18] H. Ahmad, M. N. Alam, and M. Omri, (2021) “New computational results for a prototype of an excitable system" Results in Physics 28: DOI: 10.1016/j.rinp.2021.104666.
[19] A. Yokus, H. Durur, D. Kaya, H. Ahmad, and T. A. Nofal, (2021) “Numerical comparison of Caputo and Conformable derivatives of time fractional Burgers-Fisher equation" Results in Physics 25: DOI: 10.1016/j.rinp.2021.104247.
[20] G. Wang and A.-M. Wazwaz, (2022) “On the modified Gardner type equation and its time fractional form" Chaos, Solitons and Fractals 155: DOI: 10.1016/j.chaos.2021.111694.
[21] G. Wang and A.-M. Wazwaz, (2022) “A NEW (3 + 1) -DIMENSIONAL KDV EQUATION AND MKDV EQUATION WITH THEIR CORRESPONDING FRACTIONAL FORMS" Fractals 30(4): DOI: 10.1142/S0218348X22500815.
[22] G. Wang, (2021) “A new (3 + 1)-dimensional Schrödinger equation: derivation, soliton solutions and conservation laws" Nonlinear Dynamics 104(2): 1595–1602. DOI: 10.1007/s11071-021-06359-6.
[23] G. Wang, (2021) “SYMMETRY ANALYSIS, ANALYTICAL SOLUTIONS and CONSERVATION LAWS of A GENERALIZED KdV-BURGERS-KURAMOTO EQUATION and ITS FRACTIONAL VERSION" Fractals 29(4): DOI: 10.1142/S0218348X21501012.
[24] G. Wang, (2021) “A novel (3+1)-dimensional sine-Gorden and a sinh-Gorden equation: Derivation, symmetries and conservation laws" Applied Mathematics Letters 113: DOI: 10.1016/j.aml.2020.106768.
[25] G.Wang, K. Yang, H. Gu, F. Guan, and A. Kara, (2020) “A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions" Nuclear Physics B 953: DOI: 10.1016/j.nuclphysb.2020.114956.
[26] M. T. Darvishi, M. Najafi, and A.-M. Wazwaz, (2021) “Some optical soliton solutions of space-time conformable fractional Schrödinger-type models" Physica Scripta 96(6): DOI: 10.1088/1402-4896/abf269.
[27] M. Darvishi, M. Najafi, and A.-M. Wazwaz, (2021) “Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions" Chaos, Solitons and Fractals 150: DOI: 10.1016/j.chaos.2021.111187.
[28] R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, (2014) “A new definition of fractional derivative" Journal of Computational and Applied Mathematics 264: 65–70. DOI: 10.1016/j.cam.2014.01.002.
[29] T. Abdeljawad, (2015) “On conformable fractional calculus" Journal of Computational and Applied Mathematics 279: 57–66. DOI: 10.1016/j.cam.2014.10.016.
[30] K. Hosseini, A. Bekir, and R. Ansari, (2017) “New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method" Optik 132: 203–209. DOI: 10.1016/j.ijleo.2016.12.032.
[31] A. Zheng, Y. Feng, andW.Wang, (2015) “The Hyers-Ulam stability of the conformable fractional differential equation" Mathematica Aeterna 5(3): 485–492.
[32] O. S. Iyiola and E. R. Nwaeze, (2016) “Some new results on the new conformable fractional calculus with application using D’Alambert approach" Progr. Fract.Differ. Appl 2(2): 115–122.
[33] K. Hosseini and R. Ansari, (2017) “New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method"Waves in Random and Complex Media 27(4): 628–636. DOI: 10.1080/17455030.2017.1296983.
[34] M. Eslami and H. Rezazadeh, (2016) “The first integral method for Wu–Zhang system with conformable timefractional derivative" Calcolo 53(3): 475–485. DOI: 10.1007/s10092-015-0158-8.
[35] A. Akbulut and M. Kaplan, (2018) “Auxiliary equation method for time-fractional differential equations with conformable derivative" Computers and Mathematics with Applications 75(3): 876–882. DOI: 10.1016/j.camwa.2017.10.016.
[36] N. F. Britton et al. Reaction-diffusion equations and their applications to biology. Academic Press, 1986.
[37] R. S. Cantrell and C. Cosner. Spatial ecology via reaction-diffusion equations. John Wiley & Sons, 2004.
[38] P. Grindrod. The theory and applications of reactiondiffusion equations: patterns and waves. Clarendon Press, 1996.
[39] J. Smoller. Shock waves and reaction—diffusion equations. 258. Springer Science & Business Media, 2012.
[40] K. Wang and S. Liu, (2016) “A new Sumudu transform iterative method for time-fractional Cauchy reaction–diffusion equation" SpringerPlus 5(1): DOI: 10.1186/s40064-016-2426-8.
[41] S. Kumar, (2013) “A new fractional modeling arising in engineering sciences and its analytical approximate solution" Alexandria Engineering Journal 52(4): 813–819. DOI: 10.1016/j.aej.2013.09.005.