Optimal homotopy analysis and control of error for solutions to the non-local Whitham equation.

*(English)*Zbl 1299.76025Summary: The Whitham equation is a non-local model for nonlinear dispersive water waves. Since this equation is both nonlinear and non-local, exact or analytical solutions are rare except for in a few special cases. As such, an analytical method which results in minimal error is highly desirable for general forms of the Whitham equation. We obtain approximate analytical solutions to the non-local Whitham equation for general initial data by way of the optimal homotopy analysis method, through the use of a partial differential auxiliary linear operator. A method to control the residual error of these approximate solutions, through the use of the embedded convergence control parameter, is discussed in the context of optimal homotopy analysis. We obtain residual error minimizing solutions, using relatively few terms in the solution series, in the case of several different kernels and associated initial data. Interestingly, we find that for a specific class of initial data, there exists an exact solution given by the first term in the homotopy expansion. A specific example of initial data which satisfies the condition producing an exact solution is included. These results demonstrate the applicability of optimal homotopy analysis to equations which are simultaneously nonlinear and non-local.

##### MSC:

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

35C10 | Series solutions to PDEs |

35Q35 | PDEs in connection with fluid mechanics |

##### Keywords:

non-local Whitham equation; approximate solution; control of error; optimal homotopy analysis method##### Software:

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\textit{K. Mallory} and \textit{R. A. Van Gorder}, Numer. Algorithms 66, No. 4, 843--863 (2014; Zbl 1299.76025)

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##### References:

[1] | Debnath, L.: Nonlinear Partial Differential Equations for Scientists and Engineers. Springer, Boston (2005) · Zbl 1069.35001 |

[2] | Naumkin, P.I., Shishmarev, I.A.: Nonlinear Nonlocal Equations in the Theory of Waves. American Mathematical Society, Providence (1994) · Zbl 0802.35002 |

[3] | Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience, New York (1974) · Zbl 0373.76001 |

[4] | Fornberg, B; Whitham, GB, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci, 289, 373-404, (1978) · Zbl 0384.65049 |

[5] | Whitham, GB, Variational methods and applications to water waves, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 299, 6-25, (1967) · Zbl 0163.21104 |

[6] | Constantin, A; Escher, J, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243, (1998) · Zbl 0923.76025 |

[7] | Zhou, J; Tian, L, A type of bounded traveling wave solutions for the fornberg-Whitham equation, J. Math. Anal. Appl, 346, 255-261, (2008) · Zbl 1146.35025 |

[8] | Kaikina, EI, Nonlinear nonlocal Whitham equation on a segment, Nonlinear Anal. Theor. Method. Appl, 59, 55-83, (2004) · Zbl 1064.45008 |

[9] | Liu, H, Wave breaking in a class of nonlocal dispersive wave equations, J. Nonlinear Math. Phys, 13, 441-466, (2006) · Zbl 1110.35069 |

[10] | Gupta, PK; Singh, M, Homotopy perturbation method for fractional fornberg-Whitham equation, Comput. Math. Appl, 61, 250-254, (2011) · Zbl 1211.65138 |

[11] | Abidi, F; Omrani, K, The homotopy analysis method for solving the fornberg-Whitham equation and comparison with adomian’s decomposition method, Comput. Math. Appl, 59, 2743-2750, (2010) · Zbl 1193.65179 |

[12] | Liao, S.J.: On the proposed homotopy analysis techniques for nonlinear problems and its application. Ph.D. dissertation, Shanghai Jiao Tong University (1992) · Zbl 0923.76025 |

[13] | Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton (2003) · Zbl 1051.76001 |

[14] | Liao, SJ, An explicit, totally analytic approximation of Blasius viscous flow problems, Int. J. Nonlinear Mech., 34, 759-778, (1999) · Zbl 1342.74180 |

[15] | Liao, SJ, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput, 147, 499-513, (2004) · Zbl 1086.35005 |

[16] | Liao, SJ; Tan, Y, A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math., 119, 297-354, (2007) |

[17] | Liao, SJ, Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul, 14, 983-997, (2009) · Zbl 1221.65126 |

[18] | Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press, Heidelberg (2012) · Zbl 1253.35001 |

[19] | Gorder, RA; Vajravelu, K, On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach, Commun. Nonlinear Sci. Numer. Simul, 14, 4078-4089, (2009) · Zbl 1221.65208 |

[20] | Liao, S, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul, 15, 2315-2332, (2010) |

[21] | Abbasbandy, S, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360, 109-113, (2006) · Zbl 1236.80010 |

[22] | Abbasbandy, S, Homotopy analysis method for heat radiation equations, Int. Commun. Heat Mass Transf., 34, 380-387, (2007) |

[23] | Liao, SJ; Su, J; Chwang, AT, Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, Int. J. Heat Mass Transf., 49, 2437-2445, (2006) · Zbl 1189.76549 |

[24] | Liao, SJ; Campo, A, Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. Fluid Mech., 453, 411-425, (2002) · Zbl 1007.76014 |

[25] | Liao, SJ, An explicit, totally analytic approximation of Blasius viscous flow problems, Int. J. Nonlinear Mech., 34, 759-778, (1999) · Zbl 1342.74180 |

[26] | Liao, SJ, A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. Fluid Mech., 385, 101-128, (1999) · Zbl 0931.76017 |

[27] | Liao, SJ, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. Fluid Mech., 488, 189-212, (2003) · Zbl 1063.76671 |

[28] | Akyildiz, FT; Vajravelu, K; Mohapatra, RN; Sweet, E; Gorder, RA, Implicit differential equation arising in the steady flow of a sisko fluid, Appl. Math. Comput., 210, 189-196, (2009) · Zbl 1160.76002 |

[29] | Hang, X; Lin, ZL; Liao, SJ; Wu, JZ; Majdalani, J, Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls, Phys. Fluids, 22, 053601, (2010) · Zbl 1190.76132 |

[30] | Sajid, M; Hayat, T; Asghar, S, Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt, Nonlinear Dyn., 50, 27-35, (2007) · Zbl 1181.76031 |

[31] | Hayat, T; Sajid, M, On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A, 361, 316-322, (2007) · Zbl 1170.76307 |

[32] | Turkyilmazoglu, M, Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer, Phys. Fluids, 21, 106104, (2009) · Zbl 1183.76529 |

[33] | Abbasbandy, S; Zakaria, FS, Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear Dyn., 51, 83-87, (2008) · Zbl 1170.76317 |

[34] | Wu, W; Liao, SJ, Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos Solitons Fractals, 26, 177-185, (2005) · Zbl 1071.76009 |

[35] | Sweet, E; Gorder, RA, Analytical solutions to a generalized drinfel’d-Sokolov equation related to DSSH and kdv6, Appl. Math. Comput, 216, 2783-2791, (2010) · Zbl 1195.35271 |

[36] | Wu, Y; Wang, C; Liao, SJ, Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos Solitons Fractals, 23, 1733-1740, (2005) · Zbl 1069.35060 |

[37] | Cheng, J; Liao, SJ; Mohapatra, RN; Vajravelu, K, Series solutions of nano-boundary-layer flows by means of the homotopy analysis method, J. Math. Anal. Appl., 343, 233-245, (2008) · Zbl 1135.76016 |

[38] | Gorder, RA; Sweet, E; Vajravelu, K, Nano boundary layers over stretching surfaces, Commun. Nonlinear Sci. Numer. Simul, 15, 1494-1500, (2010) · Zbl 1221.76024 |

[39] | Bataineh, AS; Noorani, MSM; Hashim, I, Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method, Phys. Lett. A, 371, 72-82, (2007) · Zbl 1209.65104 |

[40] | Bataineh, AS; Noorani, MSM; Hashim, I, Homotopy analysis method for singular IVPs of Emden-Fowler type, Commun. Nonlinear Sci. Numer. Simul, 14, 1121-1131, (2009) · Zbl 1221.65197 |

[41] | Gorder, RA; Vajravelu, K, Analytic and numerical solutions to the Lane-Emden equation, Phys. Lett. A, 372, 6060-6065, (2008) · Zbl 1223.85004 |

[42] | Liao, S, A new analytic algorithm of Lane-Emden type equations, Appl. Math. Comput., 142, 1-16, (2003) · Zbl 1022.65078 |

[43] | Gorder, RA, Analytical method for the construction of solutions to the Föppl - von Kármán equations governing deflections of a thin flat plate, Int. J. Non-Linear Mech., 47, 1-6, (2012) |

[44] | Gorder, RA, Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function \(H\) (\(x\)) in the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul, 17, 1233-1240, (2012) · Zbl 1243.35163 |

[45] | Ghoreishi, M; Ismail, AIB; Alomari, AK; Sami Bataineh, A, The comparison between homotopy analysis method and optimal homotopy asymptotic method for nonlinear age-structured population models, Commun. Nonlinear Sci. Numer. Simul, 17, 1163-1177, (2012) · Zbl 1239.92075 |

[46] | Abbasbandy, S; Shivanian, E; Vajravelu, K, Mathematical properties of h-curve in the frame work of the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul, 16, 4268-4275, (2011) · Zbl 1222.65060 |

[47] | Gorder, RA, Control of error in the homotopy analysis of semi-linear elliptic boundary value problems, Numer. Algor, 61, 613-629, (2012) · Zbl 1257.65063 |

[48] | Mallory, K., Van Gorder, R.A.: Control of error in the homotopy analysis of solutions to the Zakharov system with dissipation. Numer. Algor. (2013) in press. doi:10.1007/s11075-012-9683-6 · Zbl 1283.65090 |

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