Solving nonlinear fractional partial differential equations using the homotopy analysis method.

*(English)*Zbl 1185.65187Summary: The homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional Korteweg-de Vries (KdV), \(K(2,2)\), Burgers, Benjamin-Buna-Mahony (BBM)-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like \(B(m,n)\) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

##### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35R11 | Fractional partial differential equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

35C10 | Series solutions to PDEs |

##### Keywords:

analytical solution; coupled KdV and Boussinesq-like \(B(m; n)\) equations; fractional KdV; \(K(2,2)\); Burgers; BBM-Burgers; cubic Boussinesq; fractional partial differential equations; homotopy analysis method; series solution; numerical examples
PDF
BibTeX
XML
Cite

\textit{M. Dehghan} et al., Numer. Methods Partial Differ. Equations 26, No. 2, 448--479 (2010; Zbl 1185.65187)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Ablowitz, Solitons, nonlinear evolution equations and inverse scattering (1991) · Zbl 0762.35001 |

[2] | Miller, An introduction to the fractional calculus and fractional differential equations (1993) · Zbl 0789.26002 |

[3] | Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999) · Zbl 0924.34008 |

[4] | Samko, Fractional integrals and derivatives: Theory and applications (1993) · Zbl 0818.26003 |

[5] | West, Physics of fractal operators (2003) |

[6] | Abbasbandy, An approximation solution of a nonlinear equation with Riemann-Liouville’s fractional derivatives by He’s variational iteration method, J Computat Appl Math 207 pp 53– (2007) · Zbl 1120.65133 |

[7] | Caputo, Linear models of dissipation whose Q is almost frequency independent, J R Astronomic Soc 13 pp 529– (1967) · Zbl 1210.65130 |

[8] | Debanth, Recents applications of fractional calculus to science and engineering, Int J Math Math Sci 54 pp 3413– (2003) |

[9] | Diethelm, A predictor-corrector approach for the numerical solution of fractional differential equation, Nonlinear Dyn 29 pp 3– (2002) · Zbl 1009.65049 |

[10] | Hayat, Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model, Appl Math Comput 151 pp 153– (2004) · Zbl 1161.76436 |

[11] | Jafari, Homotopy Analysis Method for solving linear and nonlinear fractional diffusion-wave equation, Commun Nonlinear Sci Numer Simul 14 pp 2006– (2009) · Zbl 1221.65278 |

[12] | Jafari, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun Nonlinear Sci Numer Simul 14 pp 1962– (2009) · Zbl 1221.35439 |

[13] | S. Kemple and H. Beyer, Global and causal solutions of fractional differential equations, Transform methods and special functions: Varna 96, Proceedings of 2nd international workshop (SCTP), Vol. 96, Singapore, 1997, pp. 210-216. |

[14] | Kilbas, Differential equations of fractional order: methods, results problems, Appl Anal 78 pp 153– (2001) · Zbl 1031.34002 |

[15] | Kiryakova, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J Comput Appl Math 118 pp 441– (2000) · Zbl 0966.33011 |

[16] | Momani, Decomposition method for solving fractional Riccati differential equations, Appl Math Comput 182 pp 1083– (2006) · Zbl 1107.65121 |

[17] | Oldham, The fractional calculus (1974) · Zbl 0206.46601 |

[18] | Adomian, A review of the decomposition method in applied mathematics, J Math Anal Appl 135 pp 501– (1988) · Zbl 0671.34053 |

[19] | Dehghan, The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Physica Scripta 78 pp 1– (2008) · Zbl 1159.78319 |

[20] | Dehghan, Solution of a semilinear parabolic equation with an unknown control function using the decomposition procedure of Adomian, Numer Methods Partial Differential Eq 23 pp 499– (2007) · Zbl 1119.65088 |

[21] | Tatari, Numerical solution of Laplace equation in a disk using the Adomian decomposition method, Physica Scripta 72 pp 345– (2005) · Zbl 1128.65311 |

[22] | Dehghan, The solution of coupled Burgers’ equations using Adomian-Pade technique, Appl Math Comput 189 pp 1034– (2007) |

[23] | Wazwaz, The variational iteration method for rational solutions for K(2,2), Burgers, and cubic Boussinesq equations, J Comput Appl Math 207 pp 18– (2007) · Zbl 1119.65102 |

[24] | Peter, Solitary wave solutions to a system of Boussinesq-like equations, Chaos, Solitons Fractals 2 pp 529– (1992) · Zbl 0749.35042 |

[25] | Rosenau, Compactons: Solitons with finite wavelengths, Phys Rev Lett 70 pp 564– (1993) · Zbl 0952.35502 |

[26] | T. J. Priestly and P. A. Clarkson, Symmetries of a generalized Boussinesq equation, IMS Technical Report, UKC/IMS/ 59, 1996. |

[27] | Kaya, Explicit solutions of generalized Boussinesq equations, J Appl Math 1 pp 29– (2001) · Zbl 0976.35066 |

[28] | Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl Math Comput 81 pp 265– (1997) · Zbl 0882.65132 |

[29] | Wazwaz, A new method for solving singular initial value problems in the second order differential equations, Appl Math Comput 128 pp 47– (2002) · Zbl 1030.34004 |

[30] | Al-Khaled, An approximate solution for a fractional diffusion-wave equation using the decomposition method, Appl Math Comput 165 pp 473– (2005) · Zbl 1071.65135 |

[31] | Ray, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl Math Comput 167 pp 561– (2005) · Zbl 1082.65562 |

[32] | Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl Math Comput 131 pp 241– (2002) · Zbl 1029.34003 |

[33] | Song, Solving the fractional BBM-Burgers equation using the homotopy analysis method, Chaos, Solitons Fractals 40 pp 1616– (2009) · Zbl 1198.65205 |

[34] | Zhu, Existence and convergence of solutions for the generalized BBM-Burgers equation with dissipative term, Nonlinear Anal 28 pp 1835– (1997) |

[35] | Gao, Ion-acoustic shocks in space and laboratory dusty plasmas: Two-dimensional and non-traveling-wave observable effects, Phys Plasma 8 pp 3146– (2001) |

[36] | Osborne, The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves, Chaos, Solitons Fractals 5 pp 2623– (1995) · Zbl 1080.86502 |

[37] | Yan, New families of solitons with compact support for Boussinesq-like B(m,n) equations with fully nonlinear dispersion, Chaos, Solitons Fractals 14 pp 1151– (2002) · Zbl 1038.35082 |

[38] | Fermi 2 (1965) |

[39] | Zhu, Exact solitary solutions with compact support for the nonlinear dispersive Boussinesq-like B(m,n) equations, Chaos, Solitons Fractals 26 pp 407– (2005) · Zbl 1070.35047 |

[40] | Dehghan, The use of He’s variational iteration method for solving a Fokker-Planck equation, Phys Scripts 74 pp 310– (2006) · Zbl 1108.82033 |

[41] | Dehghan, Identifying an unknown function in a parabolic equation with overspecified data via He’s variational iteration method, Chaos, Solitons Fractals 36 pp 157– (2008) · Zbl 1152.35390 |

[42] | Dehghan, Application of He’s variational iteration method for solving the Cauchy reaction-diffusion problem, J Comput Appl Math 214 pp 435– (2008) · Zbl 1135.65381 |

[43] | Shakeri, Numerical solution of the Klein-Gordon equation via He’s variational iteration method, Nonlinear Dyn 51 pp 89– (2008) · Zbl 1179.81064 |

[44] | Tatari, Solution of problems in calculus of variations via He’s variational iteration method, Phys Lett A 362 pp 401– (2007) · Zbl 1197.65112 |

[45] | Tatari, On the convergence of Hes variational iteration method, J Comput Appl Math 207 pp 121– (2007) · Zbl 1120.65112 |

[46] | Tatari, He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Choas, Solitons and Fractals 33 pp 671– (2007) · Zbl 1131.65084 |

[47] | Dehghan, Numerical solution of a biological population model using He’s variational iteration method, Comput Math Applic 54 pp 1197– (2007) · Zbl 1137.92033 |

[48] | Dehghan, Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He’s variational iteration technique, Communications in Numerical Methods in Engineering (2008) |

[49] | Dehghan, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy 13 pp 53– (2008) |

[50] | Shakeri, Solution of a model describing biological species living together using the variational iteration method, Mathematical and Computer Modelling 48 pp 685– (2008) · Zbl 1156.92332 |

[51] | Dehghan, Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos, Solitons Fractals 41 pp 1448– (2009) · Zbl 1198.65202 |

[52] | S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992. |

[53] | Liao, Series solutions of unsteady boundary-layer flows over a stretching flat plate, Studies Appl Math 117 pp 2529– (2006) · Zbl 1145.76352 |

[54] | Liao, An approximate solution technique which does not depend upon small parameters: a special example, Int J Nonlinear Mech 30 pp 371– (1995) |

[55] | Liao, An approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int J Nonlinear Mech 32 pp 815– (1997) · Zbl 1031.76542 |

[56] | Liao, An explicit, totally analytic approximation of Blasius viscous flow problems, Int J Nonlinear Mech 34 pp 759– (1999) · Zbl 1342.74180 |

[57] | Liao, Beyond perturbation: introduction to the homotopy analysis method (2003) · Zbl 1051.76001 |

[58] | Liao, On the homotopy analysis method for nonlinear problems, Appl Math Comput 147 pp 499– (2004) · Zbl 1086.35005 |

[59] | Liao, Analytic solutions of the temperature distribution in Blasius viscous flow problems, J Fluid Mech 453 pp 411– (2002) · Zbl 1007.76014 |

[60] | Liao, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J Fluid Mech 488 pp 189– (2003) · Zbl 1063.76671 |

[61] | Liao, An analytic approximate approach for free oscillations of self-excited systems, Int J Nonlinear Mech 39 pp 271– (2004) · Zbl 1348.34071 |

[62] | Abbasbandy, Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem Eng J 136 pp 144– (2008) |

[63] | Hayat, Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Int J Eng Sci 45 pp 393– (2007) · Zbl 1213.76137 |

[64] | Song, Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation, Phys Lett A 367 pp 88– (2007) · Zbl 1209.65115 |

[65] | Xu, Analysis of a time fractional wave-like equation with the homotopy analysis method, Phys Lett A 372 pp 1250– (2008) · Zbl 1217.35111 |

[66] | Dehghan, Use of He’s homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media, Journal of Porous Media 11 pp 765– (2008) |

[67] | Shakeri, Solution of the delay differential equations via homotopy perturbation method, Mathematical and Computer Modelling 48 pp 486– (2008) · Zbl 1145.34353 |

[68] | Dehghan, Solution of an integro-differential equation arising in oscillating magnetic fields using He’s homotopy perturbation method, Progress in Electromagnetic Research, PIER 78 pp 361– (2008) |

[69] | Saadatmandi, Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems, Nonlinear Analysis: Real World Applications 10 pp 1912– (2009) |

[70] | Dehghan, The numerical solution of the second Painleve equation, Numer Methods Partial Differential Eq (2009) |

[71] | Chowdhury, Comparison of homotopy analysis method and homotopy perturbation method for purely nonlinear fin-type problems, Commun Nonlinear Sci Numer Simul 14 pp 371– (2009) · Zbl 1221.80021 |

[72] | Sajid, Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations, Nonlinear Anal Real World Appl 9 pp 2296– (2008) · Zbl 1156.76436 |

[73] | Lu, Exact soliton solutions of some nonlinear physical models, Phys Lett A 255 pp 249– (1999) · Zbl 0936.35164 |

[74] | El-Wakil, An improved variational iteration method for solving coupled KdV and Boussinesqlike B(m,n) equations, Chaos, Solitons Fractals 39 pp 1324– (2009) |

[75] | Momani, Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method, Appl Math Comput 165 pp 459– (2005) · Zbl 1070.65105 |

[76] | Kaya, Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation, Appl Math Comput 151 pp 775– (2004) · Zbl 1048.65096 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.