Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform

Mounirah Areshi; Adnan Khan; Rasool Shah; Kamsing Nonlaopon; Mounirah Areshi; Adnan Khan; Rasool Shah; Kamsing Nonlaopon

doi:10.3934/math.2022385

AIMS Mathematics

2022, Volume 7, Issue 4: 6936-6958. doi: 10.3934/math.2022385

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Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform

1.
Department of Mathematics, Faculty of science, University of Tabuk, Tabuk 71491, Saudi Arabia
2.
Department of Mathematics, Abdul Wali khan university Mardan 23200, Pakistan
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 09 December 2021 Revised: 17 January 2022 Accepted: 20 January 2022 Published: 28 January 2022
MSC : 34A34, 35A20, 35A22, 44A10, 33B15

In this paper, we find the solution of the time-fractional Newell-Whitehead-Segel equation with the help of two different methods. The newell-Whitehead-Segel equation plays an efficient role in nonlinear systems, describing the stripe patterns' appearance in two-dimensional systems. Four case study problems of Newell-Whitehead-Segel are solved by the proposed methods with the aid of the Antagana-Baleanu fractional derivative operator and the Laplace transform. The numerical results obtained by suggested techniques are compared with an exact solution. To show the effectiveness of the proposed methods, we show exact and analytical results compared with the help of graphs and tables, which are in strong agreement with each other. Also, the results obtained by implementing the suggested methods at various fractional orders are compared, which confirms that the solution gets closer to the exact solution as the value tends from fractional-order towards integer order. Moreover, proposed methods are interesting, easy and highly accurate in solving various nonlinear fractional-order partial differential equations.
Citation: Mounirah Areshi, Adnan Khan, Rasool Shah, Kamsing Nonlaopon. Analytical investigation of fractional-order Newell-Whitehead-Segel equations via a novel transform[J]. AIMS Mathematics, 2022, 7(4): 6936-6958. doi: 10.3934/math.2022385

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Abstract

In this paper, we find the solution of the time-fractional Newell-Whitehead-Segel equation with the help of two different methods. The newell-Whitehead-Segel equation plays an efficient role in nonlinear systems, describing the stripe patterns' appearance in two-dimensional systems. Four case study problems of Newell-Whitehead-Segel are solved by the proposed methods with the aid of the Antagana-Baleanu fractional derivative operator and the Laplace transform. The numerical results obtained by suggested techniques are compared with an exact solution. To show the effectiveness of the proposed methods, we show exact and analytical results compared with the help of graphs and tables, which are in strong agreement with each other. Also, the results obtained by implementing the suggested methods at various fractional orders are compared, which confirms that the solution gets closer to the exact solution as the value tends from fractional-order towards integer order. Moreover, proposed methods are interesting, easy and highly accurate in solving various nonlinear fractional-order partial differential equations.

References

[1]	J. Machado, V. Kiryakora, F. Mainardi, Reccent history of fractional calculus, Commun. Nonlinear Sci. , 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
[2]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent, Geophys. J. R. Astr. Soc. , 13 (1967), 529–539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x doi: 10.1111/j.1365-246X.1967.tb02303.x
[3]	S. Kemple, H. Beyer, Global and causal solutions of fractional differential equations, transform methods and special functions: Varna 96, Proceedings of 2nd international workshop (SCTP), Singapore, 96 (1997), 210–216.
[4]	S. Abbasbandy, An approximation solution of a nonlinear equation with Riemann-Liouville's fractional derivatives by He's variational iteration method, J. Comput. Appl. Math. , 207 (2007), 53–58. https://doi.org/10.1016/j.cam.2006.07.011 doi: 10.1016/j.cam.2006.07.011
[5]	H. Jafari, S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun. Nonlinear Sci. , 14 (2009), 2006–2012. https://doi.org/10.1016/j.cnsns.2008.05.008 doi: 10.1016/j.cnsns.2008.05.008
[6]	H. Jafari, S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun. Nonlinear Sci. , 14 (2009), 1962–1969. https://doi.org/10.1016/j.cnsns.2008.06.019 doi: 10.1016/j.cnsns.2008.06.019
[7]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[8]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, New York: Academic Press, 1999.
[9]	A. A. Kilbas, J. J. Trujillo, Differential equations of fractional order: Methods, results problems, Appl. Anal. , 78 (2001), 153–192. https://doi.org/10.1080/00036810108840931 doi: 10.1080/00036810108840931
[10]	K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equation, Nonlinear Dyn. , 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341 doi: 10.1023/A:1016592219341
[11]	T. Hayat, S. Nadeem, S. Asghar, Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model, Appl. Math. Comput. , 151 (2004), 153–161. https://doi.org/10.1016/S0096-3003(03)00329-1 doi: 10.1016/S0096-3003(03)00329-1
[12]	L. Debanth, Recents applications of fractional calculus to science and engineering, Int. J. Math. Sci. , 54 (2003), 3413–3442. https://doi.org/10.1155/S0161171203301486 doi: 10.1155/S0161171203301486
[13]	S. Momani, N. T. Shawagfeh, Decomposition method for solving fractional Riccati differential equations, Appl. Math. Comput. , 182 (2006), 1083–1092. https://doi.org/10.1016/j.amc.2006.05.008 doi: 10.1016/j.amc.2006.05.008
[14]	K. Joseph, Fractional dynamics: Recent advances, World Scientific, 2012.
[15]	D. Baleanu, J. A. T. Machado, A. C. Luo, Fractional dynamics and control, Springer Science & Business Media, 2011. https://doi.org/10.1007/978-1-4614-0457-6
[16]	S. A. El-Wakil, E. M. Abulwafa, E. K. El-Shewy, A. A. Mahmoud, Time-fractional KdV equation for plasma of two different temperature electrons and stationary ion, Phys. Plasmas. , 18 (1994), 092116. https://doi.org/10.1063/1.3640533 doi: 10.1063/1.3640533
[17]	S. A. El-Wakil, E. M. Abulwafa, E. K. El-Shewy, A. A. Mahmoud, Time-fractional study of electron acoustic solitary waves in plasma of cold electron and two isothermal ions, J. Plasma Phys. , 78 (2012), 641–649. https://doi.org/10.1017/S0022377812000530 doi: 10.1017/S0022377812000530
[18]	N. Laskin, G. Zaslavsky, Nonlinear fractional dynamics on a lattice with long range interactions, Physica A, 368 (2006), 38–54. https://doi.org/10.1016/j.physa.2006.02.027 doi: 10.1016/j.physa.2006.02.027
[19]	V. E. Tarasov, Fractional generalization of gradient and Hamiltonian systems, J. Phys. A-Math. Gen. , 38 (2005), 5929–5943. https://doi.org/10.1088/0305-4470/38/26/007 doi: 10.1088/0305-4470/38/26/007
[20]	I. Petras, Fractional-order nonlinear systems: Modeling, analysis and simulation, Springer Science & Business Media, 2011. https://doi.org/10.1007/978-3-642-18101-6
[21]	V. V. Uchaikin, Self-similar anomalous diffusion and Levystable laws, Phys-Usp. , 46 (2003), 821–849. https://doi.org/10.1070/PU2003v046n08ABEH001324 doi: 10.1070/PU2003v046n08ABEH001324
[22]	V. E. Tarasov, Fractional dynamics: Applications of fractional calculus to dynamics of particles, Fields and Media, Springer, 2011. https://doi.org/10.1007/978-3-642-14003-7_11
[23]	V. E. Tarasov, Gravitational field of fractal distribution of particles, Celest. Mech. Dyn. Astr. , 94 (2006), 1–15. https://doi.org/10.1007/s10569-005-1152-2 doi: 10.1007/s10569-005-1152-2
[24]	Z. M. Odibat, S. Kumar, N. Shawagfeh, A. Alsaedi, T. Hayat, A study on the convergence conditions of generalized differential transform method, Math. Meth. Appl. Sci. , 40 (2017), 40–48. https://doi.org/10.1002/mma.3961 doi: 10.1002/mma.3961
[25]	M. Ghoreishi, A. M. Ismail, N. H. M. Ali, Adomian decomposition method (ADM) for nonlinear wave-like equations with variable coefficient, Appl. Math. Sci. , 4 (2010), 2431–2444.
[26]	Y. Tan, S. Abbasbandy, Homotopy analysis method for quadratic Riccati differential equation, Commun. Nonlinear Sci., 13 (2008), 539–546. https://doi.org/10.1016/j.cnsns.2006.06.006 doi: 10.1016/j.cnsns.2006.06.006
[27]	B. Batiha, M. S. M. Noorani, I. Hashim, Numerical solution of sine-Gordon equation by variational iteration method, Phys. Lett. A, 370 (2007), 437–440. https://doi.org/10.1016/j.physleta.2007.05.087 doi: 10.1016/j.physleta.2007.05.087
[28]	Y. Qin, A. Khan, I. Ali, M. Al Qurashi, H. Khan, R. Shah, D. Baleanu, An efficient analytical approach for the solution of certain fractional-order, Energies, 13 (2020), 2725. https://doi.org/10.3390/en13112725 doi: 10.3390/en13112725
[29]	H. Khan, A. Khan, P. Kumam, D. Baleanu, M. Arif, An approximate analytical solution of the Navier-Stokes equations within Caputo operator and Elzaki transform decomposition method, Adv. Differ. Equ., 1 (2020), 1–23. https://doi.org/10.1186/s13662-020-03058-1 doi: 10.1186/s13662-020-03058-1
[30]	R. Shah, H. Khan, D. Baleanu, P. Kumam, M. Arif. The analytical investigation of time-fractional multi-dimensional Navier-Stokes equation, Alexandria Engineering Journal, 59 (2020). https://doi.org/10.1016/j.aej.2020.03.029
[31]	L. I. Yuanlu, Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun. Nonlinear Sci., 15 (2010), 2284–2292. https://doi.org/10.1016/j.cnsns.2009.09.020 doi: 10.1016/j.cnsns.2009.09.020
[32]	M. Ur-Rehman, R. A. Khan, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci., 16 (2011), 4163–4173. https://doi.org/10.1016/j.cnsns.2011.01.014 doi: 10.1016/j.cnsns.2011.01.014
[33]	M. Ali, I. Jaradat, M. Alquran, New computational method for solving fractional Riccati equation, J. Math. Comput. Sci., 17 (2017), 106–114. https://doi.org/10.22436/jmcs.017.01.10 doi: 10.22436/jmcs.017.01.10
[34]	M. Alquran, K. Al-Khaled, M. Ali, O. A. Arqub, Bifurcations of the time-fractional generalized coupled Hirota-Satsuma KdV system, Waves Wavelets Fract., 3 (2017), 31–39. https://doi.org/10.1515/wwfaa-2017-0003 doi: 10.1515/wwfaa-2017-0003
[35]	A. Newell, J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid. Mech., 38 (1968), 279–303. https://doi.org/10.1017/S0022112069000176 doi: 10.1017/S0022112069000176
[36]	M. Ali, M. Alquran, I. Jaradat, Asymptotic-sequentially solution style for the generalized Caputo time-fractional Newell Whitehead Segel system, Adv. Differ. Equ., 1 (2019), 1–9. https://doi.org/10.1186/s13662-019-2021-8 doi: 10.1186/s13662-019-2021-8
[37]	M. Alquran, M. Ali, M. Alsukhour, I. Jaradat, Promoted residual power series technique with Laplace transform to solve some time-fractional problems arising in physics, Results Phys., 19 (2020), 103667. https://doi.org/10.1016/j.rinp.2020.103667 doi: 10.1016/j.rinp.2020.103667

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