International Journal of Nonlinear Analysis and Applications
Analytical solutions for time-fractional Swift-Hohenberg equations via a modified integral transform technique
Document Type : Research Paper
Authors
1 Mathematics Department, Faculty of Computer Science and Mathematics, University of Thi-Qar, Thi-Qar, Iraq
2 Department of Mathematics, Delta State University, Abraka, PMB 1, Delta State, Nigeria
Abstract
In this work, the fractional natural transform decomposition method (FNTDM) is employed to obtain approximate analytical solutions for some time-fractional versions of the nonlinear Swift-Hohenberg (S-H) equation with the fractional derivatives taken in the sense of Caputo. The S-H equation models problems arising from fluid dynamics and describes temperature dynamics of thermal convection as well as complex pattern formation processes in liquid surfaces bounded along a horizontally well-conducting boundary. To explore the applicability, simplicity, and efficiency of the FNTDM, numerical simulations are provided for each of the considered problems to demonstrate the behavior of the obtained approximate solutions for different values of the fractional parameter index. The obtained simulations show a similar resemblance with those in existing related literature and further confirm the applicability of the considered method to even complex problems arising in diverse fields of applied mathematics and physics.
Keywords
[1] R.P. Agarwal, F. Mofarreh, R. Shah, W. Luangboon and K. Nonlaopon, An analytical technique based on natural transform to solve fractional-order parabolic equations, Entropy 23 (2021), 1086.
[2] H. Ahmad, A.R. Seadawy and T.A. Khan, Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm, Math. Comput. Simul. 177 (2020), 13–23.
[3] L. Akinyemi, M. S¸enol and S.N. Huseen, Modified homotopy methods for generalized fractional perturbed ZakharovKuznetsov equation in dusty plasma, Adv. Differ. Equ. 2021 (2021), no. 1, 1–27.
[4] F.T. Akyildiz, D.A. Siginer, K. Vajravelu and R.A.V. Gorder, Analytical and numerical results for the SwiftHohenberg equation, Appl. Math. Comput. 216 (2010), 221-226.
[5] H.A. Aljahdaly, R.P. Agarwal, R. Shah and T. Botmart, Analysis of the time fractional-order coupled Burgers equations with non-singular kernel operators, Math. 9 (2021), 2326.
[6] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci. 20 (2016), no. 2, 763-769.
[7] N. Attia, A. Akg¨ul and A. Nour, Numerical Solutions to the Time-Fractional Swift-Hohenberg Equation Using Reproducing Kernel Hilbert Space Method, Int. J. Appl. Comput. Math. 7 (2021), 194.
[8] O. Bazighifan, H. Ahmad and S-W. Yao, New oscillation criteria for advanced differential equations of fourth order, Math. 8 (2020), 728.
[9] F.B.M. Belgacem and R. Silambarasan, Theory of natural transform, Math. Eng. Sci. Aerosp. 3 (2012), no. 1, 105–135.
[10] A. Bokhari, D. Baleanu and R. Belgacem, Application of Shehu transform to Atangana-Baleanu derivatives, Int. J. Math. Comput. Sci. 20 (2019), 101–107.
[11] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. Int. 13 (1967), 529–539.
[12] M. Caputo and M. Fabricio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015), no. 2, 73–85.
[13] M. Cross and P. Hohenberg, Pattern formation outside of equilibrium, Rev. Modern Phys. 65 (1993), 851–862.
[14] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Berlin, 2010.
[15] U. Farooq, Khan, F. Tchier F, E. Hinca, D. Baleanu, H.B. Jebreen, New approximate analytical technique for the solution of time fractional fluid flow models, Adv. Differ. Equ. 2021 (2021), no. 1, 1–20.
[16] A. Fernandez and D. Baleanu, Classes of operators in fractional calculus: A case study, Math. Meth. Appl. Sci. 44 (2021), no. 11, 9143–9162.
[17] B. Ghanbari and S. Djilali, Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population, Chaos Solitons Fractals 138 (2020), 109960.
[18] A. Goswami, S. Rathore, J. Singh and D. Kumar, Analytical study of fractional nonlinear Schrodinger equation with harmonic oscillator, Discrete Continuous Dyn. Syst. Ser. S 14 (2021), 3589–3610.
[19] A. Goswami, Sushila, J. Singh and D. Kumar, Numerical computation of fractional Kersten-Krasil’shchik coupled KdV-mKdV system occurring in multi-component plasmas, AIMS Math. 5 (2020), 2346–2368.
[20] M.H. Heydari and M. Hosseininia, A new variable-order fractional derivative with non-singular Mittag–Leffler kernel: application to variable-order fractional version of the 2D Richard equation, Engin. Comput. 38 (2022), 1759—1770.
[21] P.C. Hohenberg and J.B. Swift, Effects of additive noise at the onset of Rayleigh-Benard convection, Phys. Rev. A 46 (1992), 4773–4785.
[22] S.N. Huseen, On analytical solution of time-fractional type model of the Fisher’s equation, Iraqi J. Sci. 61 (2020), 1419–1425.
[23] M. Inc, M.N. Kha, Ahmad, S.-W. Yao, H. Ahmad and P. Thounthong, Analysing time-fractional exotic options via efficient local meshless method, Res. Phys. 19 (2020), 103385.
[24] H. Jassim and M. Mohammed, Natural homotopy perturbation method for solving nonlinear fractional gas dynamics equations, Int. J. Nonlinear Anal. Appl. 12 (2021), no. 1, 812–820.
[25] B. Jin, R. Lazarov and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), 445–466.
[26] X. Ji and H. Tang, High-order accurate Runge-Kutta (local) discontinuous Galerkin methods for one-and twodimensional fractional diffusion equations, Numer. Math. Theory Meth. Appl. 5 (2012), no. 3, 333–358.
[27] N.A. Khan, N.U. Khan, M. Ayaz and A. Mahmood, Analytical methods for solving the time-fractional Swift–Hohenberg (S–H) equation, Comput. Math. Appl. 61 (2011), 2182-2185.
[28] A. Khalouta and A. Kadem, Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients, Acta Univ. Sapientiae Math. 11 (2019), 99–116.
[29] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[30] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1993.
[31] L. Lega, J.V. Moloney and A.C. Newell, Swift-hohenberg equation for lasers, Phys. Rev. Lett. 73 (1994), 2978–2981.
[32] S. Maitama, A hybrid natural transform homotopy perturbation method for solving fractional partial differential equations, Int. J. Differ. Equ. 2016 (2016) 9207869.
[33] M. Merdan, A numeric-analytic method for time-fractional Swift-Hohenberg (S-H) equation with modified Riemann-Liouville derivative, Appl. Math. Model. 37 (2013), 4224–4231.
[34] L. Noeiaghdam, S. Noeiaghdam and D. Sidorov, Dynamical control on the homotopy analysis method for solving nonlinear shallow water wave equation, J. Phys. Conf. Ser. 1847 (2021), 012010.
[35] N.I. Okposo, M.O. Adewole, E.N. Okposo, H.I. Ojarikre and F.A. Abdullah, A mathematical study on a fractional COVID-19 transmission model within the framework of nonsingular and nonlocal kernel, Chaos Solitons Fractals 152 (2021), 111427.
[36] N.I. Okposo, A.J. Jonathan, E.N. Okposo and M. Ossaiugbo, Existence of solutions and stability analysis for a fractional helminth transmission model within the framework of Mittag-Leffler kernel, Nigerian J. Sci. Envir. 19 (2021), 67–80.
[37] N.I. Okposo, P. Veeresha and E.N. Okposo, Solutions for time-fractional coupled nonlinear Schr¨odinger equations arising in optical solitons, Chinese J. Phys. 77 (2022), 965–984.
[38] L.A. Peletier and V. Rottschafer, Large time behaviour of solutions of the swift-hohenberg equation, R. Acad. Sci. Paris Ser I 336 (2003), 225–30.
[39] Y. Pomeau and S. Zaleski, Dislocation motion in cellular structures, Phys. Rev. A 27 (1983), 2710–2726.
[40] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[41] D.G. Prakasha, P. Veeresha and H.M. Baskonus, Residual power series method for fractional Swift–Hohenberg equation, Fractal and Fractional. 3 (2019), 9.
[42] S. Rashid, R. Ashraf and F.S. Bayones, A novel treatment of fuzzy fractional Swift-Hohenberg equation for a hybrid transform within the fractional derivative operator, Fractal Fract. 5 (2021), 209.
[43] N. Rashid, A. Nasir, Z. Laiq, S.N. Kottakkkaran, M.R. Alharthi and J. Wasim, Extension of natural transform method with Daftardar-Jafari polynomials for fractional order differential equations, Alexandria Engin. J. 60 (2021), 3205–3217.
[44] P.N. Ryabov and N.A. Kudryashov. Nonlinear waves described by the generalized swift-hohenberg equation, J. Phys. Conf. Ser. 788 (2017), 012032.
[45] T.D. Seyma, H. Bulut and F.B.M. Belgacem. Sumudu transform method for analytical solutions of fractional type ordinary differential equations, Math. Prob. Engin. 2015 (2015), Article ID 131690.
[46] S.R. Moosavi Noori and N. Taghizadeh, Study of convergence of reduced differential transform method for different classes of differential equations, Int. J. Differ. Equ. 2021 (2021), Article ID 6696414.
[47] K. Shah, H. Khalil and R. A. Khan, Analytical solutions of fractional order diffusion equations by natural transform method, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018), no. 3, 1479–1490.
[48] J. Singh, D. Kumar and S. Kumar, An efficient computational method for local fractional transport equation occurring in fractal porous media, Comput. Appl. Math. 39 (2020).
[49] S. Soradi-Zeid, M. Mesrizadeh and C. Cattani, Numerical solutions of fractional differential equations by using Laplace transformation method and quadrature rule Fractal Fract. 5 (2021), 111.
[50] J. B. Swift and P. C. Hohenberg, Hydrodynamics fluctuations at the convective instability, Phys. Rev. A. 15 (1977), 319–328.
[51] P. Veeresha, D. G. Prakasha and D. Baleanu, Analysis of fractional Swift-Hohenberg equation using a novel computational technique, Math. Meth. Appl. Sci. 43 (2019), 1970–1987.
[52] P. Veeresha, D. G. Prakasha and S. Kumar, A fractional model for propagation of classical optical solitons by using nonsingular derivative, Math. Methods Appl. Sci. (2020), 1–15, https://doi.org/10.1002/mma.6335
[53] K. Vishal, S. Das, S.H. Ong and P. Ghosh, On the solutions of fractional Swift-Hohenberg equation with dispersion, Appl. Math. Comput. 219 (2013), 5792–5801.
[54] K. Vishal, S. Kumar and S. Das, Application of homotopy analysis method for fractional Swift-Hohenberg equation - Revisited, Appl. Math. Model, 36 (2012), 3630-3637.
[55] A.-M. Wazwaz, Partial Differential Equations And Solitary Waves Theory, Springer Science and Business Media: Berlin/Heidelberg, Germany, 2010.
[56] J. Xu, H. Khan, R. Shah, A.A. Alderremy, S. Aly and D. Baleanu, The analytical analysis of nonlinear fractional-order dynamical models, AIMS Math. 6 (2021), 6201–6219.
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Volume 13, Issue 2
July 2022Pages 2669-2684
- Receive Date: 11 April 2022
- Accept Date: 05 July 2022