The Combined Laplace-homotopy Analysis Method for Partial Differential Equations
- Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.
In this paper, the Laplace transform homotopy analysis method (LHAM) is employed to obtain
approximate analytical solutions of the linear and nonlinear differential equations. This method
is a combined form of the Laplace transform method and the homotopy analysis method. The
proposed scheme finds the solutions without any discretization or restrictive assumptions and is free
from round-off errors and therefore, reduces the numerical computations to a great extent. Some
illustrative examples are presented and the numerical results show that the solutions of the LHAM
are in good agreement with those obtained by exact solution.
- Homotopy analysis method
- Laplace transform method
- partial differential equation.
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Fundamental Theories of Physics, 60. Kluwer Academic Publishers Group, Dordrecht (1994)
A. K. Alomari , A novel solution for fractional chaotic Chen system, J. Nonlinear Sci. Appl., 8 (2015), 478-488.
J. Biazar, M. Gholami Porshokuhi, B. Ghanbari, Extracting a general iterative method from an adomian decomposition method and comparing it to the variational iteration method, Comput. Math. Appl., 59 (2010), 622-628.
C. Chun, Fourier-series-based variational iteration method for a reliable treatment of heat equations with variable coefficients, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), 1383-1388.
N. Faraz, Y. Khan, A. Yildirim, Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II, J. King Saud University, 1 (2011), 77-81.
J. H. He, Homotopy perturbation technique , Comput. Methods Appl. Mech. Eng., 178 (1999), 257-262.
J. H. He, Variational iteration method-a kind of nonlinear analytical technique: some examples, Int. J. Nonlinear Mech., 34 (1999), 699-708.
J. H. He, X. H. Wu, Variational iteration method: new development and applications, Comput. Math. Appl., 54 (2007), 881-894.
J. H. He, G. C. Wu, F. Austin, The variational iteration method which should be followed, Nonlinear Sci. Lett. A, 1 (2010), 1-30.
E. Hesameddini, H. Latifizadeh, Reconstruction of variational iteration algorithms using the Laplace transform, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), 1377-1382.
R. Hirota, Exact solutions of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27 (1971), 1192-1194.
S. Islam, Y. Khan, N. Faraz, F. Austin, Numerical solution of logistic differential equations by using the Laplace decomposition method, World Appl. Sci. J., 8 (2010), 1100-1105.
H. Jafari, C. Chun, S. Seifi, M. Saeidy, Analytical solution for nonlinear Gas Dynamic equation by Homotopy Analysis Method, Appl. Appl. Math., 4 (2009), 149-154.
H. Jafari, M. Saeidy, M. A. Firoozjaee, The Homotopy analysis method for solving higher dimensional initial boundary value problems of variable cofficients, Numer. Meth. Part. D. E., 26 (2010), 1021-1032.
A. V. Karmishin, A. I. Zhukov, V. G. Kolosov, Methods of dynamics calculation and testing for thin- walled structures, Mashinostroyenie, Moscow , Russia (1990)
S. A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, J. Appl. Math., 1 (2001), 141-155.
Y. Khan, An effective modification of the Laplace decomposition method for nonlinear equations, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), 1373-1376.
Y. Khan, N. Faraz, A new approach to differential difference equations, J. Adv. Res. Differ. Equations, 2 (2010), 1-12.
S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University (1992)
S. J. Liao, An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics, Int. J. Nonlinear Mech., 32 (1997), 815-822.
S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, CRC Press, Boca Raton: Chapman Hall, (2003)
A. M. Lyapunov, The general problem of the stability of motion, (English translation), Taylor & Francis, London, UK (1992)
J. Saberi Nadjafi, A. Ghorbani, He's homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential equations, Comput. Math. Appl., 58 (2009), 2379-2390.
L. A. Soltani, A. Shirzadi, , A new modification of the variational iteration method , Comput. Math. Appl., 59 (2010), 2528-2535.
A. M. Wazwaz, On multiple soliton solutions for coupled KdV-mkdV equation, Nonlinear Sci. Lett. A, 1 (2010), 289-296.
G. C. Wu, J. H. He, Fractional calculus of variations in fractal spacetime, Nonlinear Sci. Lett. A, 1 (2010), 281-287.
G. C. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett. A, 374 (2010), 2506-2509.
Z. Yao, Almost periodicity of impulsive Hematopoiesis model with infnite delay, J. Nonlinear Sci. Appl., 8 (2015), 856-865.
E. Yusufoglu, Numerical solution of Duffing equation by the Laplace decomposition algorithm, Appl. Math. Comput., 177 (2006), 572-580.