Homotopy Perturbation Method Using He’s Polynomial for Solving Nonlinear Differential Equations
[1]
Mubashra Saleem, Department of Mathematics, University of Wah, Wah, Pakistan.
[2]
Aqsa Mumtaz, Department of Mathematics, University of Wah, Wah, Pakistan.
[3]
Tahira Amir, Department of Mathematics, University of Wah, Wah, Pakistan.
[4]
Qazi Mahmood Ul-Hassan, Department of Mathematics, University of Wah, Wah, Pakistan.
[5]
Kamran Ayub, Department of Mathematics, Riphah International University, Islamabad, Pakistan.
[6]
Farhana Kanwal, Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan.
In this paper, for solving nonlinear equations a combined form of the Laplace transform method along with the Homotopy perturbation method is used. This method is called the Homotopy perturbation transform method (HPTM). He’s polynomials can be used to handle the nonlinear terms easily. This method avoids the round-off errors and finds the solution without any discretization or restrictive assumptions. Proposed method solves nonlinear problems without using Adomian’s polynomials can be thought as a vibrant advantage of this algorithm over the decomposition method.
Homotopy Perturbation Method, Nonlinear Advection Equations, He’s Polynomial, Laplace Transform Method
[1]
A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London, UK, 1992, English translation.
[2]
J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) 257–262.
[3]
N. H. Sweilam, M. M. Khader, Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method, Computers & Mathematics with Applications 58 (2009) 2134–2141.
[4]
J. Saberi-Nadjafi, A. Ghorbani, He’s Homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Computers & Mathematics with Applications 58 (2009) 1345–1351.
[5]
A. V. Karmishin, A. I. Zhukov, V. G. Kolosov, Methods of Dynamics Calculation and Testing for Thin-Walled Structures, Mashinostroyenie, Moscow,Russia, 1990.
[6]
R. Hirota, Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons, Physical Review Letters 27 (1971) 1192–1194.
[7]
A. M. Wazwaz, on multiple soliton solutions for coupled KdV–mkdV equation, Nonlinear Science Letters A 1 (2010) 289–296.
[8]
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acad. Publ., Boston, 1994.
[9]
G. C. Wu, J. H. He, Fractional calculus of variations in fractal spacetime, Nonlinear Science Letters A 1 (2010) 281–287.
[10]
J. H. He, Variational iteration method—a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics 34(1999) 699–708.
[11]
J. H. He, X. H. Wu, Variational iteration method: new development and applications, Computers & Mathematics with Applications 54 (2007) 881–894.
[12]
J. H. He, G. C. Wu, F. Austin, the variational iteration method which should be followed, Nonlinear Science Letters A 1 (2009) 1–30.
[13]
L. A. Soltani, A. Shirzadi, A new modification of the variational iteration method, Computers & Mathematics with Applications 59 (2010) 2528–2535.
[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]
G. C. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Physics Letters A (2010) doi: 10.1016/j.physleta.2010.04.034.
[16]
E. Hesameddini, H. Latifizadeh, Reconstruction of variational iteration algorithms using the Laplace transform, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009) 1377–1382.
[17]
C. Chun, 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 (2009) 1383–1388.
[18]
G. Adomian, Solution of physical problems by decomposition, Computers & Mathematics with Applications 2 (1994) 145–154.
[19]
A. M. Wazwaz, a comparison between the variational iteration method and adomian decomposition method, Journal of Computational and Applied Mathematics 207 (2007) 129–136.
[20]
M. A. Abdou, A. A. Soliman, New applications of variational iteration method, Physica D: Nonlinear Phenomena 211 (2005) 1–8.
[21]
M. Dehghan, weighted finite difference techniques for the one-dimensional advection–diffusion equation, Applied Mathematics and Computation147 (2004) 307–319.
[22]
D. D. Ganji, A. Sadighi, Application of He’s homotopy perturbation method to nonlinear coupled systems of reaction diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation 7 (2006) 411–418.
[23]
Y. Khan, F. Austin, Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations, Zeitschriftfuer Naturforschung 65a (2010) 1–5.
[24]
S. T. Mohyud-Din, A. Yildirim, Homotopy perturbation method for advection problems, Nonlinear Science Letters A 1 (2010) 307–312.
[25]
J. H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135 (2003) 73–79.
[26]
J. H. He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation 156 (2004) 527–539.
[27]
J. H. He, the homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation 151 (2004) 287–292.
[28]
J. H. He, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation 6 (2005) 207–208.
[29]
J. H. He, some asymptotic methods for strongly nonlinear equation, International Journal of Modern Physics 20 (2006) 1144–1199.
[30]
J. H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A 350 (2006) 87–88.
[31]
M. Rafei, D. D. Ganji, Explicit solutions of helmhotz equation and fifth-order KdV equation using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation 7 (2006) 321–328.
[32]
A. M. Siddiqui, R. Mahmood, Q. K. Ghori, 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 (2006) 7–14.
[33]
E. Hesameddini, H. Latifizadeh, An optimal choice of initial solutions in the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009) 1389–1398.
[34]
E. Hesameddini, H. Latifizadeh, A new vision of the He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009) 1415–1424.
[35]
A. Ghorbani, Beyond Adomian’s polynomials: He polynomials, Chaos Solitons Fractals 39 (2009) 1486–1492.
[36]
S. T. Mohyud-Din, M. A. Noor, K. I. Noor, traveling wave solutions of seventh-order generalized KdV equation using He’s polynomials, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009) 227–233.
