A Comparative Study of Adomain Decompostion Method and He-Laplace Method
Author(s)
Badradeen A. A. Adam1,2
Affiliation(s)
1 Department of Mathematics, Faculty of Education, University of Khartoum, Omdurman, Sudan. 2 Department of Mathematics, Northwest Normal University, Lanzhou, China.
1 Department of Mathematics, Faculty of Education, University of Khartoum, Omdurman, Sudan. 2 Department of Mathematics, Northwest Normal University, Lanzhou, China.
ABSTRACT
In this paper, we present a comparative study between the He-Laplace and Adomain decomposition method. The study outlines the significant features of two methods. We use the two methods to solve the nonlinear Ordinary and Partial differential equations. Laplace transformation with the homotopy method is called He-Laplace method. A comparison is made among Adomain decomposition method and He-Laplace. It is shown that, in He-Laplace method, the nonlinear terms of differential equation can be easy handled by the use He’s polynomials and provides better results.
In this paper, we present a comparative study between the He-Laplace and Adomain decomposition method. The study outlines the significant features of two methods. We use the two methods to solve the nonlinear Ordinary and Partial differential equations. Laplace transformation with the homotopy method is called He-Laplace method. A comparison is made among Adomain decomposition method and He-Laplace. It is shown that, in He-Laplace method, the nonlinear terms of differential equation can be easy handled by the use He’s polynomials and provides better results.
KEYWORDS
Adomain Decomposition Method, He-Laplace Transform Method, Homotopy Perturbation Method, Ordinary Differential Equation, Partial Differential Equations, He’s Polynomials
Adomain Decomposition Method, He-Laplace Transform Method, Homotopy Perturbation Method, Ordinary Differential Equation, Partial Differential Equations, He’s Polynomials
Cite this paper
Adam, B. (2014) A Comparative Study of Adomain Decompostion Method and He-Laplace Method. Applied Mathematics, 5, 3353-3364. doi: 10.4236/am.2014.521312.
Adam, B. (2014) A Comparative Study of Adomain Decompostion Method and He-Laplace Method. Applied Mathematics, 5, 3353-3364. doi: 10.4236/am.2014.521312.
References
[1] Adomian, G. (1988) A Review of the Decomposition Method in Applied Mathematics. Journal of Mathematical Analysis and Applications, 135, 501-544.
http://dx.doi.org/10.1016/0022-247X(88)90170-9 [2] Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston. [3] Wazwaz, A.M. (1997) Necessary Conditions for the Appearance of Noise Terms in Decomposition Solution Series. Applied Mathematics and Computation, 81, 265-274.
http://dx.doi.org/10.1016/S0096-3003(95)00327-4 [4] Wazwaz, A.M. (1997) A First Course in Integral Equations. World Scientific, Singapore.
http://dx.doi.org/10.1142/3444 [5] Wazwaz, A.M. (1999) Analytical Approximations and Padé’s Approximants for Volterra’s Population Model. Applied Mathematics and Computation, 100, 13-25.
http://dx.doi.org/10.1016/S0096-3003(98)00018-6 [6] Wazwaz, A.M. (2000) A New Technique for Calculating Adomian Polynomials for Nonlinear Polynomials. Applied Mathematics and Computation, 111, 33-51.
http://dx.doi.org/10.1016/S0096-3003(99)00063-6 [7] Wazwaz, A.M. (2000) A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators. Applied Mathematics and Computation, 111, 53-69. [8] Wazwaz, A.M. (2000) The Decomposition Method for Solving the Diffusion Equation Subject to the Classification of Mass, Internat. Applied Mathematics and Computation, 3, 25-34. [9] Wazwaz, A.M. (2002) Partial Differential Equations: Methods and Applications. Balkema Publishers, The Netherlands. [10] Wazwaz, A.M. (2002) A New Method for Solving singular Initial Value Problems in the Second Order Differential Equations. Applied Mathematics and Computation, 128, 47-57.
http://dx.doi.org/10.1016/S0096-3003(01)00021-2 [11] Ghorbani, A. (2009) Beyond Adomian’s Polynomials: He’s Polynomials. Chaos, Solitons Fractals, 39, 1486-1492.
http://dx.doi.org/10.1016/j.chaos.2007.06.034 [12] Lyapunov, A.M. (1992) The General Problem of the Stability of Motion. Taylor & Francis, London. [13] Saberi-Nadjafi, J. and Ghorbani, A. (2009) He’s Homotopy Per-turbation Method: An Effective Tool for Solving Nonlinear Integral and Integro-Differential Equations. Computers and Mathematics with Applications, 58, 1345-1351. [14] Sweilam, N.H. and Khadar, M.M. (2009) Exact Solution of Some Coupled Nonlinear Partial Differential Equations Using the Homotopy Perturbation Method. Computers and Mathematics with Applications, 58, 2134-2141.
http://dx.doi.org/10.1016/j.camwa.2009.03.059 [15] Hirota, R. (1971) Exact Solutions of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Physics Review Letters, 27, 1192-1194. [16] Wazwaz, A.M. (2010) On Multiple Soliton Solution for Coupled KdV-mkdV Equation. Nonlinear Science Letter A, 1, 289-296. [17] Wu, G.C. and He, J.H. (2010) Fractional Calculus of Variations in Fractal Space Time. Nonlinear Science Letter A, 1, 281-287. [18] He, J.H. (2003) A Simple Perturbation Approach to Blasius Equation. Applied Mathematics and Computation, 140, 217-222.
http://dx.doi.org/10.1016/S0096-3003(02)00189-3 [19] Liu, G.L. (1995) Weighted Residual Decomposition Method in Nonlinear Applied Mathematics. Proceedings of 6th Congress of Modern Mathematics and Mechanics, Suzhou, 1995, 643-648. [20] He, J.H. (1997) A New Approach to Nonlinear Partial Differential Equations. Communications in Nonlinear and Numerical Simulation, 2, 230-235.
http://dx.doi.org/10.1016/S1007-5704(97)90007-1
[1] Adomian, G. (1988) A Review of the Decomposition Method in Applied Mathematics. Journal of Mathematical Analysis and Applications, 135, 501-544.
http://dx.doi.org/10.1016/0022-247X(88)90170-9 [2] Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston. [3] Wazwaz, A.M. (1997) Necessary Conditions for the Appearance of Noise Terms in Decomposition Solution Series. Applied Mathematics and Computation, 81, 265-274.
http://dx.doi.org/10.1016/S0096-3003(95)00327-4 [4] Wazwaz, A.M. (1997) A First Course in Integral Equations. World Scientific, Singapore.
http://dx.doi.org/10.1142/3444 [5] Wazwaz, A.M. (1999) Analytical Approximations and Padé’s Approximants for Volterra’s Population Model. Applied Mathematics and Computation, 100, 13-25.
http://dx.doi.org/10.1016/S0096-3003(98)00018-6 [6] Wazwaz, A.M. (2000) A New Technique for Calculating Adomian Polynomials for Nonlinear Polynomials. Applied Mathematics and Computation, 111, 33-51.
http://dx.doi.org/10.1016/S0096-3003(99)00063-6 [7] Wazwaz, A.M. (2000) A New Algorithm for Calculating Adomian Polynomials for Nonlinear Operators. Applied Mathematics and Computation, 111, 53-69. [8] Wazwaz, A.M. (2000) The Decomposition Method for Solving the Diffusion Equation Subject to the Classification of Mass, Internat. Applied Mathematics and Computation, 3, 25-34. [9] Wazwaz, A.M. (2002) Partial Differential Equations: Methods and Applications. Balkema Publishers, The Netherlands. [10] Wazwaz, A.M. (2002) A New Method for Solving singular Initial Value Problems in the Second Order Differential Equations. Applied Mathematics and Computation, 128, 47-57.
http://dx.doi.org/10.1016/S0096-3003(01)00021-2 [11] Ghorbani, A. (2009) Beyond Adomian’s Polynomials: He’s Polynomials. Chaos, Solitons Fractals, 39, 1486-1492.
http://dx.doi.org/10.1016/j.chaos.2007.06.034 [12] Lyapunov, A.M. (1992) The General Problem of the Stability of Motion. Taylor & Francis, London. [13] Saberi-Nadjafi, J. and Ghorbani, A. (2009) He’s Homotopy Per-turbation Method: An Effective Tool for Solving Nonlinear Integral and Integro-Differential Equations. Computers and Mathematics with Applications, 58, 1345-1351. [14] Sweilam, N.H. and Khadar, M.M. (2009) Exact Solution of Some Coupled Nonlinear Partial Differential Equations Using the Homotopy Perturbation Method. Computers and Mathematics with Applications, 58, 2134-2141.
http://dx.doi.org/10.1016/j.camwa.2009.03.059 [15] Hirota, R. (1971) Exact Solutions of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Physics Review Letters, 27, 1192-1194. [16] Wazwaz, A.M. (2010) On Multiple Soliton Solution for Coupled KdV-mkdV Equation. Nonlinear Science Letter A, 1, 289-296. [17] Wu, G.C. and He, J.H. (2010) Fractional Calculus of Variations in Fractal Space Time. Nonlinear Science Letter A, 1, 281-287. [18] He, J.H. (2003) A Simple Perturbation Approach to Blasius Equation. Applied Mathematics and Computation, 140, 217-222.
http://dx.doi.org/10.1016/S0096-3003(02)00189-3 [19] Liu, G.L. (1995) Weighted Residual Decomposition Method in Nonlinear Applied Mathematics. Proceedings of 6th Congress of Modern Mathematics and Mechanics, Suzhou, 1995, 643-648. [20] He, J.H. (1997) A New Approach to Nonlinear Partial Differential Equations. Communications in Nonlinear and Numerical Simulation, 2, 230-235.
http://dx.doi.org/10.1016/S1007-5704(97)90007-1