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Application of \(\tan (\Phi (\xi )/2)\)-expansion method to solve some nonlinear fractional physical model. (English) Zbl 1454.35322

Summary: Based on the \(\tan (\Phi (\xi )/2)\)-expansion method, five nonlinear fractional physical models for obtaining the solutions containing three types hyperbolic function, trigonometric function and rational function solutions are investigated. These equations are the time fractional biological population model, time fractional Burgers, time fractional Cahn-Hilliard, space-time fractional Whitham-Broer-Kaup, space-time fractional Fokas equations. The fractional derivative is described in the Caputo sense. We obtained the exact solutions for the aforementioned nonlinear fractional equations. A generalized fractional complex transform is appropriately used to convert these equations to ordinary differential equations which subsequently resulted into number of exact solutions.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
35Q91 PDEs in connection with game theory, economics, social and behavioral sciences
35Q51 Soliton equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)
35C09 Trigonometric solutions to PDEs
35R11 Fractional partial differential equations
34A34 Nonlinear ordinary differential equations and systems
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[1] Kilbas, Aa; Trujillo, Jj, Differential equations of fractional order: methods, results problems, Appl Anal, 78, 153-192 (2001) · Zbl 1031.34002
[2] Zhang, S.; Zhang, Hq, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys Lett A, 375, 1069-1073 (2011) · Zbl 1242.35217
[3] El-Sayed, Ama; Rida, Sz; Arafa, Aam, Exact solutions of fractional-order biological population model, Commun Theor Phys Beijing China, 52, 992-996 (2009) · Zbl 1184.92038
[4] Meng, F., A new approach for solving fractional partial differential equations, J Appl Math (2013) · Zbl 1266.35140 · doi:10.1155/2013/256823
[5] Singh, J.; Kumar, D.; Kiliçman, A., Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations, Abs Appl Anal (2014) · Zbl 1474.65415 · doi:10.1155/2014/535793
[6] Bekir, A.; Güner, Ö., Exact solutions of nonlinear fractional differential equations by \((G^{\prime }\)/G)-expansion method, Chin Phys B, 22, 110202 (2013)
[7] Choo, Sm; Chung, Sk; Lee, Yj, A conservative difference scheme for the viscous Cahn-Hilliard equation with a nonconstant gradient energy coefficient, Appl Numer Math, 51, 207-219 (2004) · Zbl 1112.65078
[8] Mohebbi, A.; Asgari, Z.; Dehghan, M., Numerical solution of nonlinear Jaulent-Miodek and Whitham-Broer-Kaup equations, Commun Nonlinear Sci Numer Simul, 17, 4602-4610 (2012) · Zbl 1266.65176
[9] Wang, L.; Gao, Yt; Gai, Xl, Gauge transformation, elastic and inelastic interactions for the Whitham-Broer-Kaup shallow-water model, Commun Nonlinear Sci Numer Simul, 17, 2833-2844 (2012) · Zbl 1335.35223
[10] Wang, Gw; Xu, Tz, The modified fractional sub-equation method and its applications to nonlinear fractional partial differential equations, Rom J Phys, 59, 636-645 (2014)
[11] Kilbas, Aa; Srivastava, Hm; Trujillo, Jj, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003
[12] Miller, Ks; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), New York: Wiley, New York · Zbl 0789.26002
[13] Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (1999), New York: Academic Press, New York · Zbl 0924.34008
[14] Hirota, R., The Direct Method in Soliton Theory (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1099.35111
[15] Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer Methods Partial Differ Equ J, 26, 448-479 (2010) · Zbl 1185.65187
[16] Dehghan, M.; Manafian, J.; Saadatmandi, A., The solution of the linear fractional partial differential equations using the homotopy analysis method, Z Naturforsch, 65a, 935-949 (2010)
[17] He, Jh, Variational iteration method a kind of non-linear analytical technique: some examples, Int J Nonlinear Mech, 34, 699-708 (1999) · Zbl 1342.34005
[18] Dehghan, M.; Manafian, J.; Saadatmandi, A., Application of semi-analytic methods for the Fitzhugh-Nagumo equation, which models the transmission of nerve impulses, Math Methods Appl Sci, 33, 1384-1398 (2010) · Zbl 1196.35025
[19] Dehghan, M.; Manafian, J., The solution of the variable coefficients fourth-order parabolic partial differential equations by homotopy perturbation method, Z Naturforsch, 64a, 420-430 (2009)
[20] Barik, Rn; Dash, Gc; Rath, Pk, Homotopy perturbation method (HPM) solution for flow of a conducting visco-elastic fluid through a porous medium, Proc Natl Acad Sci India Sect A Phys Sci, 84, 55-61 (2015) · Zbl 1284.76044
[21] Wazwaz, Am, Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method, Appl Math Comput, 177, 755-760 (2006) · Zbl 1099.65095
[22] Manafian Heris, J.; Lakestani, M., Solitary wave and periodic wave solutions for variants of the KdV-Burger and the K(n, n)-Burger equations by the generalized tanh-coth method, Commun Numer Anal, 2013, 1-18 (2013)
[23] Menga, Xh; Liua, Wj; Zhua, Hw; Zhang, Cy; Tian, B., Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation, Phys A Stat Mech Appl, 387, 97-107 (2008)
[24] Fazli Aghdaei, M.; Manafianheris, J., Exact solutions of the couple Boiti-Leon-Pempinelli system by the generalized \((\frac{{\rm G}^{\prime }}{{\rm G}})\)-expansion method, J Math Ext, 5, 91-104 (2011) · Zbl 1254.35043
[25] Younis, M.; Rizvi, Str, Dispersive dark optical soliton in (2+1)-dimensions by \(G^{prime}\)/G-expansion with dual-power law nonlinearity, Opt Int J Light Electron Opt, 126, 5812-5814 (2015)
[26] Dehghan, M.; Manafian, J.; Saadatmandi, A., Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics, Int J Numeric Methods Heat Fluid Flow, 21, 736-753 (2011)
[27] Dehghan, M.; Manafian, J.; Saadatmandi, A., Analytical treatment of some partial differential equations arising in mathematical physics by using the Exp-function method, Int J Mod Phys B, 25, 2965-2981 (2011) · Zbl 1333.35234
[28] Manafian Heris, J.; Bagheri, M., Exact solutions for the modified KdV and the generalized KdV equations via Exp-function method, J Math Ext, 4, 77-98 (2010) · Zbl 1233.35173
[29] Jawad, Ajm; Petkovic, Md; Biswas, A., Modified simple equation method for nonlinear evolution equations, Appl Math Comput, 217, 869-877 (2010) · Zbl 1201.65119
[30] Malfliet, W., Solitary wave solutions of nonlinear wave equations, Am J Phys, 60, 650-654 (1992) · Zbl 1219.35246
[31] Malfliet, W.; Hereman, W., The tanh method: II. Perturbation technique for conservative systems, Phys Scr, 54, 569-575 (1996) · Zbl 0942.35035
[32] Naher, H.; Abdullah, Fa, New approach of \((G^{\prime }\)/G)-expansion method and new approach of generalized \((G^{\prime }\)/G)-expansion method for nonlinear evolution equation, AIP Adv, 3, 032116 (2013) · doi:10.1063/1.4794947
[33] Manafian Heris, J.; Lakestani, M., Exact solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota system by the analytical methods, Int Sch Res Not, 2014, 1-8 (2014) · Zbl 1490.35382
[34] Chand, F.; Malik, Ak, Exact traveling wave solutions of some nonlinear equations using \((G^\prime \)/G)-expansion method, Int J Nonlinear Sci, 14, 416-424 (2012) · Zbl 1394.35094
[35] Wang, M.; Li, X.; Zhang, J., The \((\frac{G^{\prime }}{G})\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys Lett A, 372, 417-423 (2008) · Zbl 1217.76023
[36] Zhang, J.; Wei, X.; Lu, Y., A generalized \((\frac{G^{\prime }}{G})\)-expansion method and its applications evolution equations in mathematical physics, Phys Lett A, 372, 3653-3658 (2008) · Zbl 1220.37070
[37] Fan, E., Extended tanh-function method and its applications to nonlinear equations, Phys Lett A, 277, 212-218 (2000) · Zbl 1167.35331
[38] Wazwaz, Am, The tanh-coth method for new compactons and solitons solutions for the \(K(n, n)\) and the \(K(n + 1, n + 1)\) equations, Chaos Solitons Fractals, 188, 1930-1940 (2007) · Zbl 1119.65101
[39] Wazwaz, Am, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl Math Comput, 184, 1002-1014 (2007) · Zbl 1115.65106
[40] Kumar A, Kumar S (2017) A modified analytical approach for fractional discrete KdV equations arising in particle vibrations. Proc Nat Acad Sci Sect A Phys Sci 1-12 · Zbl 1393.34016
[41] Singh, J.; Kumar, D.; Swroop, R.; Kumar, S., An efficient computational approach for time-fractional Rosenau-Hyman equation, Neural Comput Appl (2017) · doi:10.1007/s00521-017-2909-8
[42] Kumar, S.; Kumar, A.; Odibat, Zm, A nonlinear fractional model to describe the population dynamics of two interacting species, Math Methods Appl Sci, 40, 11, 4134-4148 (2017) · Zbl 1368.34011
[43] Kumar, S.; Kumar, D.; Singh, J., Fractional modelling arising in unidirectional propagation of long waves in dispersive media, Adv Nonlinear Anal, 5, 4, 2013-2033 (2016)
[44] Prakash, A.; Kumar, M.; Sharma, Kk, Numerical method for solving coupled Burgers equation, Appl Math Comput, 260, 314-320 (2015) · Zbl 1410.65413
[45] Kumar, S.; Rashidi, Mm, New analytical method for gas dynamics equation arising in shock fronts, Comput Phys Commun, 185, 17, 1947-1954 (2014) · Zbl 1351.35253
[46] Prakash, A.; Kumar, M., Numerical method for fractional dispersive partial differential equations, Commun Numer Anal, 2017, 1, 1-18 (2017)
[47] Sakar, Mg; Ergoren, H., Alternative variational iteration method for solving the time-fractional Fornberg-Whitham equation, Appl Math Model, 39, 5, 3972-3979 (2015) · Zbl 1443.65278
[48] Prakash, A.; Kumar, M., Numerical solution of two dimensional time fractional order biological population model, Open Phys, 14, 177-186 (2016)
[49] Prakash, A.; Kumar, M., He’s variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation, J Appl Anal Comput, 6, 3, 738-748 (2016) · Zbl 1463.35305
[50] Kumar, A.; Kumar, S., A modified analytical approach for fractional discrete KdV equations arising in particle vibrations, Proc Natl Acad Sci Sect A Phys Sci (2017) · Zbl 1393.34016 · doi:10.1007/s40010-017-0369-2
[51] Prakash, A.; Kaur, H., Numerical solution for fractional model of Fokker-Planck equation by using q-HATM, Chaos Solitons Fractals, 105, 99-110 (2017) · Zbl 1380.35154
[52] Caputo, M., Linear models of dissipation whose Q is almost frequency independent, J R Astron Soc, 13, 529-539 (1967)
[53] Debanth, L., Recent applications of fractional calculus to science and engineering, Int J Math Math Sci, 54, 3413-3442 (2003) · Zbl 1036.26004
[54] Jafari, H.; Seifi, S., Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun Nonlinear Sci Numer Simul, 14, 1962-1969 (2009) · Zbl 1221.35439
[55] Kemple S, Beyer H (1997) Global and causal solutions of fractional differential equations, Transform methods and special functions, Varna 96, In: Proceedings of 2nd international workshop (SCTP), Singapore, vol 96, pp 210-216
[56] Momani, S.; Shawagfeh, Nt, Decomposition method for solving fractional Riccati differential equations, Appl Math Comput, 182, 1083-1092 (2006) · Zbl 1107.65121
[57] Oldham, Kb; Spanier, J., The Fractional Calculus (1974), New York: Academic Press, New York · Zbl 0292.26011
[58] Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput Math Appl, 51, 1367-1376 (2006) · Zbl 1137.65001
[59] Manafian, J.; Lakestani, M.; Bekir, A., Study of the analytical treatment of the (2+1)-dimensional Zoomeron, the Duffing and the SRLW equations via a new analytical approach, Int J Appl Comput Math (2015) · Zbl 1420.35061 · doi:10.1007/s40819-015-0058-2
[60] Manafian, J.; Lakestani, M., New improvement of the expansion methods for solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Int J Eng Math (2015) · Zbl 1342.35308 · doi:10.1155/2015/107978
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