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Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. (English) Zbl 1422.35165

Summary: This work presents the homotopy perturbation transform method for nonlinear fractional partial differential equations of the Caputo-Fabrizio fractional operator. Perturbative expansion polynomials are considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding the exact solutions of the fractional equations proposed, for the special case when the limit of the integral order of the time derivative is considered.

MSC:

35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals

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[1] Cattani, C, Srivastava, HM, Yang, XJ: Fractional Dynamics. de Gruyter, Berlin (2016)
[2] Yao, JJ; Kumar, A; Kumar, S, A fractional model to describe the Brownian motion of particles and its analytical solution, Adv. Mech. Eng., 7, 1687814015618874, (2015) · doi:10.1177/1687814015618874
[3] Gómez-Aguilar, JF; Miranda-Hernández, M; López-López, MG; Alvarado-Martínez, VM; Baleanu, D, Modeling and simulation of the fractional space-time diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 30, 115-127, (2016) · doi:10.1016/j.cnsns.2015.06.014
[4] Kumar, D; Singh, J; Baleanu, D, A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dyn., 87, 511-517, (2017) · Zbl 1371.35326 · doi:10.1007/s11071-016-3057-x
[5] Kumar, D; Singh, J; Baleanu, D, Numerical computation of a fractional model of differential-difference equation, J. Comput. Nonlinear Dyn., 11, (2016) · doi:10.1115/1.4033899
[6] Singh, J; Kumar, D; Nieto, JJ, A reliable algorithm for a local fractional Tricomi equation arising in fractal transonic flow, Entropy, 18, 206, (2016) · doi:10.3390/e18060206
[7] Kumar, D; Singh, J; Kumar, S; Singh, BP, Numerical computation of nonlinear shock wave equation of fractional order, Ain Shams Eng. J., 6, 605-611, (2015) · doi:10.1016/j.asej.2014.10.015
[8] Kumar, D; Singh, J; Kılıçman, A, An efficient approach for fractional Harry Dym equation by using sumudu transform, Abstr. Appl. Anal., 2013, (2013) · Zbl 1275.65086 · doi:10.1155/2013/608943
[9] Yang, XJ; Tenreiro Machado, JA; Baleanu, D; Cattani, C, On exact traveling-wave solutions for local fractional Korteweg-de Vries equation, Chaos, Interdiscip. J. Nonlinear Sci., 26, (2016) · Zbl 1378.35329 · doi:10.1063/1.4960543
[10] Yang, XJ; Baleanu, D; Khan, Y; Mohyud-Din, ST, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Rom. J. Phys., 59, 36-48, (2014)
[11] Baleanu, D; Srivastava, HM; Yang, XJ, Local fractional variational iteration algorithms for the parabolic Fokker-Planck equation defined on Cantor sets, Prog. Fract. Differ. Appl., 1, 1-11, (2015)
[12] Yang, XJ: Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems. (2016) arXiv:1612.03202
[13] Xiao-Jun, XJ; Srivastava, HM; Machado, JT, A new fractional derivative without singular kernel, Therm. Sci., 20, 753-756, (2016) · doi:10.2298/TSCI151224222Y
[14] Hristov, J, Transient heat diffusion with a non-singular fading memory: from the Cattaneo constitutive equation with jeffrey’s kernel to the Caputo-fabrizio time-fractional derivative, Therm. Sci., 20, 757-762, (2016) · doi:10.2298/TSCI160112019H
[15] Spasic, DT; Kovincic, NI; Dankuc, DV, A new material identification pattern for the fractional Kelvin-Zener model describing biomaterials and human tissues, Commun. Nonlinear Sci. Numer. Simul., 37, 193-199, (2016) · doi:10.1016/j.cnsns.2016.01.004
[16] Ma, HC; Yao, DD; Peng, XF, Exact solutions of non-linear fractional partial differential equations by fractional sub-equation method, Therm. Sci., 19, 1239-1244, (2015) · doi:10.2298/TSCI1504239M
[17] Guo, S; Mei, L, Exact solutions of space-time fractional variant Boussinesq equations, Adv. Sci. Lett., 10, 700-702, (2012) · doi:10.1166/asl.2012.3388
[18] Mohyud-Din, ST, Nawaz, T, Azhar, E, Akbar, MA: Fractional sub-equation method to space-time fractional Calogero-Degasperis and potential Kadomtsev-Petviashvili equations. Journal of Taibah University for Science (2015)
[19] Jafari, H; Ghorbani, M; Ghasempour, S, A note on exact solutions for nonlinear integral equations by a modified homotopy perturbation method, New Trends Math. Sci., 1, 22-26, (2013)
[20] Nino, UF; Leal, HV; Khan, Y; Díaz, DP; Sesma, AP; Fernández, VJ; Orea, JS, Modified nonlinearities distribution homotopy perturbation method as a tool to find power series solutions to ordinary differential equations, Nova Scientia, 12, 13-38, (2014) · doi:10.21640/ns.v6i12.22
[21] Sayevand, K; Jafari, H, On systems of nonlinear equations: some modified iteration formulas by the homotopy perturbation method with accelerated fourth-and fifth-order convergence, Appl. Math. Model., 40, 1467-1476, (2016) · doi:10.1016/j.apm.2015.06.030
[22] Das, N; Singh, R; Wazwaz, AM; Kumar, J, An algorithm based on the variational iteration technique for the bratu-type and the Lane-Emden problems, J. Math. Chem., 54, 527-551, (2016) · Zbl 1349.65238 · doi:10.1007/s10910-015-0575-6
[23] Mistry, PR; Pradhan, VH, Approximate analytical solution of non-linear equation in one dimensional instability phenomenon in homogeneous porous media in horizontal direction by variational iteration method, Proc. Eng., 127, 970-977, (2015) · doi:10.1016/j.proeng.2015.11.445
[24] Wu, GC; Baleanu, D; Deng, ZG, Variational iteration method as a kernel constructive technique, Appl. Math. Model., 39, 4378-4384, (2015) · doi:10.1016/j.apm.2014.12.032
[25] Wu, GC; Baleanu, D, Variational iteration method for the burgers’ flow with fractional derivatives-new Lagrange multipliers, Appl. Math. Model., 37, 6183-6190, (2013) · doi:10.1016/j.apm.2012.12.018
[26] Wu, GC; Baleanu, D, New applications of the variational iteration method-from differential equations to q-fractional difference equations, Adv. Differ. Equ., 2013, (2013) · Zbl 1365.39006 · doi:10.1186/1687-1847-2013-21
[27] Duan, JS; Rach, R; Wazwaz, AM, Higher order numeric solutions of the Lane-Emden-type equations derived from the multi-stage modified Adomian decomposition method, Int. J. Comput. Math., 94, 197-215, (2015) · Zbl 1366.65069 · doi:10.1080/00207160.2015.1100299
[28] Kumar, D; Singh, J; Kumar, S, Analytic and approximate solutions of space-time fractional telegraph equations via Laplace transform, Walailak J. Sci. Technol., 11, 711-728, (2013)
[29] Kumar, D; Singh, J; Kumar, S, Numerical computation of nonlinear fractional Zakharov-Kuznetsov equation arising in ion-acoustic waves, J. Egypt. Math. Soc., 22, 373-378, (2014) · Zbl 06363304 · doi:10.1016/j.joems.2013.11.004
[30] Duan, JS; Rach, R; Wazwaz, AM, Steady-state concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method, J. Math. Chem., 53, 1054-1067, (2015) · Zbl 1323.34024 · doi:10.1007/s10910-014-0469-z
[31] Tsai, PY; Chen, COK, Free vibration of the nonlinear pendulum using hybrid Laplace Adomian decomposition method, Int. J. Numer. Methods Biomed. Eng., 27, 262-272, (2011) · Zbl 1370.70025 · doi:10.1002/cnm.1304
[32] Tsai, PY, An approximate analytic solution of the nonlinear Riccati differential equation, J. Franklin Inst., 347, 1850-1862, (2010) · Zbl 1210.34016 · doi:10.1016/j.jfranklin.2010.10.005
[33] Lu, L; Duan, J; Fan, L, Solution of the magnetohydrodynamics Jeffery-Hamel flow equations by the modified Adomian decomposition method, Adv. Appl. Math. Mech., 7, 675-686, (2015) · doi:10.4208/aamm.2014.m543
[34] Yousefi, SA; Dehghan, M; Lotfi, A, Generalized Euler-Lagrange equations for fractional variational problems with free boundary conditions, Comput. Math. Appl., 62, 987-995, (2011) · Zbl 1228.49016 · doi:10.1016/j.camwa.2011.03.064
[35] Arqub, OA; El-Ajou, A, Solution of the fractional epidemic model by homotopy analysis method, J. King Saud Univ., Sci., 25, 73-81, (2013) · doi:10.1016/j.jksus.2012.01.003
[36] Vong, S; Wang, Z, A compact difference scheme for a two dimensional fractional Klein-Gordon equation with Neumann boundary conditions, J. Comput. Phys., 274, 268-282, (2014) · Zbl 1352.65273 · doi:10.1016/j.jcp.2014.06.022
[37] Atangana, A, On the new fractional derivative and application to nonlinear fisher’s reaction-diffusion equation, Appl. Math. Comput., 273, 948-956, (2016)
[38] Mohebbi, A; Abbaszadeh, M; Dehghan, M, High-order difference scheme for the solution of linear time fractional Klein-Gordon equations, Numer. Methods Partial Differ. Equ., 30, 1234-1253, (2014) · Zbl 1300.65060 · doi:10.1002/num.21867
[39] Atangana, A; Secer, A, A note on fractional order derivatives and table of fractional derivatives of some special functions, Abstr. Appl. Anal., 2013, (2013) · Zbl 1276.26010
[40] Caputo, M; Fabricio, M, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1, 73-85, (2015)
[41] Lozada, J; Nieto, JJ, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1, 87-92, (2015)
[42] Gómez-Aguilar, JF; Córdova-Fraga, T; Escalante-Martínez, JE; Calderón-Ramón, C; Escobar-Jiménez, RF, Electrical circuits described by a fractional derivative with regular kernel, Rev. Mex. Fis., 62, 144-154, (2016)
[43] Gómez-Aguilar, JF; Yépez-Martínez, H; Calderón-Ramón, C; Cruz-Orduña, I; Escobar-Jiménez, RF; Olivares-Peregrino, VH, Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel, Entropy, 17, 6289-6303, (2015) · Zbl 1338.70026 · doi:10.3390/e17096289
[44] Atangana, A; Alkahtani, BST, Extension of the resistance, inductance, capacitance electrical circuit to fractional derivative without singular kernel, Adv. Mech. Eng., 7, 1-6, (2015) · doi:10.1177/1687814015591937
[45] Liao, S: Homotopy Analysis Method in Nonlinear Differential Equations. Springer, Berlin (2012) · Zbl 1253.35001 · doi:10.1007/978-3-642-25132-0
[46] Yin, XB; Kumar, S; Kumar, D, A modified homotopy analysis method for solution of fractional wave equations, Adv. Mech. Eng., 7, 1687814015620330, (2015) · doi:10.1177/1687814015620330
[47] Kumar, S, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model., 38, 3154-3163, (2014) · doi:10.1016/j.apm.2013.11.035
[48] Kumar, S; Rashidi, MM, New analytical method for gas dynamics equation arising in shock fronts, Comput. Phys. Commun., 185, 1947-1954, (2014) · Zbl 1351.35253 · doi:10.1016/j.cpc.2014.03.025
[49] Mohamed, MS; Gepreel, KA; Al-Malki, FA; Al-Humyani, M, Approximate solutions of the generalized abel’s integral equations using the extension khan’s homotopy analysis transformation method, J. Appl. Math., 2015, (2015) · Zbl 1343.65149 · doi:10.1155/2015/357861
[50] Gupta, VG; Kumar, P, Approximate solutions of fractional linear and nonlinear differential equations using Laplace homotopy analysis method, Int. J. Nonlinear Sci., 19, 113-120, (2015)
[51] Khan, Y; Wu, Q, Homotopy perturbation transform method for nonlinear equations using he’s polynomials, Comput. Math. Appl., 61, 1963-1967, (2011) · Zbl 1219.65119 · doi:10.1016/j.camwa.2010.08.022
[52] Ghorbani, A, Beyond adomian’s polynomials: he’s polynomials, Chaos Solitons Fractals, 39, 1486-1492, (2009) · Zbl 1197.65061 · doi:10.1016/j.chaos.2007.06.034
[53] Madani, M; Fathizadeh, M; Khan, Y; Yildirim, A, On the coupling of the homotopy perturbation method and Laplace transformation, Math. Comput. Model., 53, 1937-1945, (2011) · Zbl 1219.65121 · doi:10.1016/j.mcm.2011.01.023
[54] Liao, S: Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC press, New York (2003) · Zbl 1051.76001 · doi:10.1201/9780203491164
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