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Homotopy perturbation transform method for nonlinear equations using He’s polynomials. (English) Zbl 1219.65119
Summary: A combined form of the Laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations. This method is called the homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He’s polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.
MSC:
65M99Numerical methods for IVP of PDE
35F25Initial value problems for first order nonlinear PDE
35L60Nonlinear first-order hyperbolic equations
References:
[1]Lyapunov, A. M.: The general problem of the stability of motion, (1992) · Zbl 0786.70001
[2]He, J. H.: Homotopy perturbation technique, Computer methods in applied mechanics and engineering 178, 257-262 (1999)
[3]Sweilam, N. H.; Khader, M. M.: Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method, Computers mathematics with applications 58, 2134-2141 (2009) · Zbl 1189.65259 · doi:10.1016/j.camwa.2009.03.059
[4]Saberi-Nadjafi, J.; Ghorbani, A.: He’s homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Computers mathematics with applications 58, 1345-1351 (2009) · Zbl 1189.65173 · doi:10.1016/j.camwa.2009.03.032
[5]Karmishin, A. V.; Zhukov, A. I.; Kolosov, V. G.: Methods of dynamics calculation and testing for thin-walled structures, (1990)
[6]Hirota, R.: Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons, Physical review letters 27, 1192-1194 (1971) · Zbl 1168.35423 · doi:10.1103/PhysRevLett.27.1192
[7]Wazwaz, A. M.: On multiple soliton solutions for coupled KdV–mkdv equation, Nonlinear science letters A 1, 289-296 (2010)
[8]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[9]Wu, G. C.; He, J. H.: Fractional calculus of variations in fractal spacetime, Nonlinear science letters A 1, 281-287 (2010)
[10]He, J. H.: Variational iteration method–a kind of nonlinear analytical technique: some examples, International journal of nonlinear mechanics 34, 699-708 (1999)
[11]He, J. H.; Wu, X. H.: Variational iteration method: new development and applications, Computers mathematics with applications 54, 881-894 (2007) · Zbl 1141.65372 · doi:10.1016/j.camwa.2006.12.083
[12]He, J. H.; Wu, G. C.; Austin, F.: The variational iteration method which should be followed, Nonlinear science letters A 1, 1-30 (2009)
[13]Soltani, L. A.; Shirzadi, A.: A new modification of the variational iteration method, Computers mathematics with applications 59, 2528-2535 (2010) · Zbl 1193.65150 · doi:10.1016/j.camwa.2010.01.012
[14]N. Faraz, Y. Khan, A. Yildirim, Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II, Journal of King Saud University (2010) doi:10.1016/j.jksus.2010.06.010.
[15]Wu, G. C.; Lee, E. W. M.: Fractional variational iteration method and its application, Physics letters A (2010)
[16]Hesameddini, E.; Latifizadeh, H.: Reconstruction of variational iteration algorithms using the Laplace transform, International journal of nonlinear sciences and numerical simulation 10, 1377-1382 (2009)
[17]Chun, C.: Fourier-series-based variational iteration method for a reliable treatment of heat equations with variable coefficients, International journal of nonlinear sciences and numerical simulation 10, 1383-1388 (2009)
[18]Adomian, G.: Solution of physical problems by decomposition, Computers mathematics with applications 2, 145-154 (1994) · Zbl 0803.35020 · doi:10.1016/0898-1221(94)90132-5
[19]Wazwaz, A. M.: A comparison between the variational iteration method and Adomian decomposition method, Journal of computational and applied mathematics 207, 129-136 (2007) · Zbl 1119.65103 · doi:10.1016/j.cam.2006.07.018
[20]Abdou, M. A.; Soliman, A. A.: New applications of variational iteration method, Physica D: Nonlinear phenomena 211, 1-8 (2005) · Zbl 1084.35539 · doi:10.1016/j.physd.2005.08.002
[21]Dehghan, M.: Weighted finite difference techniques for the one-dimensional advection–diffusion equation, Applied mathematics and computation 147, 307-319 (2004) · Zbl 1034.65069 · doi:10.1016/S0096-3003(02)00667-7
[22]Ganji, D. D.; Sadighi, A.: Application of he’s homotopy perturbation method to nonlinear coupled systems of reaction diffusion equations, International journal of nonlinear sciences and numerical simulation 7, 411-418 (2006)
[23]Khan, Y.; Austin, F.: Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations, Zeitschrift fuer naturforschung 65a, 1-5 (2010)
[24]Mohyud-Din, S. T.; Yildirim, A.: Homotopy perturbation method for advection problems, Nonlinear science letters A 1, 307-312 (2010)
[25]He, J. H.: Homotopy perturbation method: a new nonlinear analytical technique, Applied mathematics and computation 135, 73-79 (2003) · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[26]He, J. H.: Comparison of homotopy perturbation method and homotopy analysis method, Applied mathematics and computation 156, 527-539 (2004) · Zbl 1062.65074 · doi:10.1016/j.amc.2003.08.008
[27]He, J. H.: The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied mathematics and computation 151, 287-292 (2004) · Zbl 1039.65052 · doi:10.1016/S0096-3003(03)00341-2
[28]He, J. H.: Homotopy perturbation method for bifurcation of nonlinear problems, International journal of nonlinear sciences and numerical simulation 6, 207-208 (2005)
[29]He, J. H.: Some asymptotic methods for strongly nonlinear equation, International journal of modern physics 20, 1144-1199 (2006) · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[30]He, J. H.: Homotopy perturbation method for solving boundary value problems, Physics letters A 350, 87-88 (2006) · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005
[31]Rafei, M.; Ganji, D. D.: Explicit solutions of helmhotz equation and fifth-order KdV equation using homotopy perturbation method, International journal of nonlinear sciences and numerical simulation 7, 321-328 (2006)
[32]Siddiqui, A. M.; Mahmood, R.; Ghori, Q. K.: Thin film flow of a third grade fluid on a moving belt by he’s homotopy perturbation method, International journal of nonlinear sciences and numerical simulation 7, 7-14 (2006)
[33]Ganji, D. D.: The applications of he’s homotopy perturbation method to nonlinear equation arising in heat transfer, Physics letters A 335, 337-3341 (2006)
[34]Xu, L.: He’s homotopy perturbation method for a boundary layer equation in unbounded domain, Computers mathematics with applications 54, 1067-1070 (2007)
[35]He, J. H.: An elementary introduction of recently developed asymptotic methods and nanomechanics in textile engineering, International journal of modern physics 22, 3487-4578 (2008) · Zbl 1149.76607 · doi:10.1142/S0217979208048668
[36]He, J. H.: Recent developments of the homotopy perturbation method, Topological methods in nonlinear analysis 31, 205-209 (2008)
[37]Hesameddini, E.; Latifizadeh, H.: An optimal choice of initial solutions in the homotopy perturbation method, International journal of nonlinear sciences and numerical simulation 10, 1389-1398 (2009)
[38]Hesameddini, E.; Latifizadeh, H.: A new vision of the he’s homotopy perturbation method, International journal of nonlinear sciences and numerical simulation 10, 1415-1424 (2009)
[39]Biazar, J.; Porshokuhi, M. Gholami; Ghanbari, B.: Extracting a general iterative method from an Adomian decomposition method and comparing it to the variational iteration method, Computers mathematics with applications 59, 622-628 (2010) · Zbl 1189.65245 · doi:10.1016/j.camwa.2009.11.001
[40]Khuri, S. A.: A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of applied mathematics 1, 141-155 (2001) · Zbl 0996.65068 · doi:10.1155/S1110757X01000183
[41]Yusufoglu, E.: Numerical solution of Duffing equation by the Laplace decomposition algorithm, Applied mathematics and computation 177, 572-580 (2006) · Zbl 1096.65067 · doi:10.1016/j.amc.2005.07.072
[42]Khan, Yasir: An effective modification of the Laplace decomposition method for nonlinear equations, International journal of nonlinear sciences and numerical simulation 10, 1373-1376 (2009)
[43]Khan, Yasir; Faraz, Naeem: A new approach to differential difference equations, Journal of advanced research in differential equations 2, 1-12 (2010)
[44]Islam, S.; Khan, Y.; Faraz, N.; Austin, F.: Numerical solution of logistic differential equations by using the Laplace decomposition method, World applied sciences journal 8, 1100-1105 (2010)
[45]Madani, M.; Fathizadeh, M.: Homotopy perturbation algorithm using Laplace transformation, Nonlinear science letters A 1, 263-267 (2010)
[46]M.A. Noor, S.T. Mohyud-Din, Variational homotopy perturbation method for solving higher dimensional initial boundary value problems, Mathematical Problems in Engineering 2008 (2008) 11. Article ID 696734, doi:10.1155/2008/696734. · Zbl 1155.65082 · doi:10.1155/2008/696734
[47]Ghorbani, A.; Saberi-Nadjafi, J.: He’s homotopy perturbation method for calculating Adomian polynomials, International journal of nonlinear sciences and numerical simulation 8, 229-232 (2007)
[48]Ghorbani, A.: Beyond Adomian’s polynomials: he polynomials, Chaos solitons fractals 39, 1486-1492 (2009) · Zbl 1197.65061 · doi:10.1016/j.chaos.2007.06.034
[49]Mohyud-Din, S. T.; Noor, M. A.; Noor, K. I.: Traveling wave solutions of seventh-order generalized KdV equation using he’s polynomials, International journal of nonlinear sciences and numerical simulation 10, 227-233 (2009)