Numerical Study of Some Coupled PDEs by using Differential Transformation Method
In this paper, the two-dimension differential transformation method (DTM) is employed to obtain the closed form solutions of the three famous coupled partial differential equation with physical interest namely, the coupled Korteweg-de Vries(KdV) equations, the coupled Burgers equations and coupled nonlinear Schrödinger equation. We begin by showing that how the differential transformation method applies to a linear and non-linear part of any PDEs and apply on these coupled PDEs to illustrate the sufficiency of the method for this kind of nonlinear differential equations. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.
[1] A. Borhanifar, Reza Abazari, An unconditionally stable parallel difference
scheme for Telegraph equation, Math. Prob. Eng, vol. 2009, Article ID
969610, 17pages, 2009. http://doi:10.1155/2009/969610.
[2] A. Borhanifar, Reza Abazari, Exact solutions for non-linear Schr¨odinger
equations by differential transformation method, J. Appl. Math. Comput,
http://doi:10.1007/s12190-009-0338-2.
[3] A. Borhanifar, Reza Abazari, Numerical study of nonlinear Schr¨odinger
and coupled Schr¨odinger equations by differential transformation method,
Opt. Commun, http://doi:10.1016/j.optcom.2010.01.046.
[4] Hirota. R, Satsuma. J, Soliton solutions of a coupled Kortewege-de Vries
equation, Phys. Lett. A, 85 (1981), 407-408.
[5] Reza Abazari, A. Borhanifar, Numerical study of Burgers- and coupled
Burgers- equations by differential transformation method, Comput. Math.
Appl, http://doi:10.1016/j.camwa.2010.01.039.
[6] Chen. R, Wu. Z, Solving partial differential equation by using multiquadric
quasi-interpolation, Appl.Math.Comput, 186 (2007), 1502-1510.
[7] Xu. Y, Shu. C-W, Local discontinuous Galerkin methods for the
Kuramoto- Sivashinsky equations and the Ito-type coupled equations,
Comput. Methods. Appl. Mech, 195 (2006), 3430-3447.
[8] Kaya. D, Inan. EI, Exact and numerical traveling wave solutions for
nonlinear coupled equations using symbolic computation, Appl. Math.
Comput, 151 (2004), 775-787.
[9] J. Biazar, H. Aminikhah, Exact and numerical solutions for non-linear
Burgers- equation by VIM, Math. Comput. Modelling, 49 (2009) 1394-
1400.
[10] Zayed EME, Zedan HA, Gepreel KA, On the solitary wave solutions
for nonlinear Hirota Satsuma coupled-KdV of equations, Chaos. Solitons.
Fractals, 22 (2004), 285-303.
[11] Caom DB, Yan JR, Zang Y, Exact solutions for a new coupled MKdV
equations and a coupled KdV equations, J. Phys. Lett. A, 297 (2002),
68-74.
[12] Zhang JL, Wang ML, Feng ZD, The improved F-expansion method and
its applications, Phys. Lett. A, 350 (2006), 103-109.
[13] Khater AH, Temsah RS, Hassan MM, A Chebyshev spectral collocation
method for solving Burgers type equations, J. Appl. Math. Comput,
http://doi:10.1016/j.cam.2007.11.007.
[14] Ganji DD, Rafei M, Solitary wave equations for a generalized Hirotasatsuma
coupled-KdV equation by homotopy perturbation method, Phys.
Lett. A, 356 (2006), 131-137.
[15] J.K. Zhou, Differential Transformation and its Application for Electrical
Circuits, Huazhong University Press, Wuhan, China, 1986.
[16] M. Mossa Al-sawalha, M.S.M. Noorani , A numeric-analytic method
for approximating the chaotic Chen system, Chaos. Soliton. Fract. 42
(2009) 1784-1791.
[17] F. Ayaz, Application of differential transform method to differential-
algebraic equations, Appl. Math. Comput. 152 (2004) 649-657.
[18] A. Arikoglu, I. Ozkol, Solution of difference equations by using differential
transform method, Appl. Math. Comput. 174 (2006) 1216-1228.
[19] A. Arikoglu, I. Ozkol, Solution of differential-difference equations by
using differential transform method, Appl. Math. Comput. 181 (2006)
153-162.
[20] Figen Kangalgil, Fatma Ayaz, Solitary wave solutions for the KdV and
mKdV equations by differential transform method, Chaos. Soliton. Fract.
41 (2009) 464-472.
[21] A. Arikoglu, I. Ozkol, Solution of fractional differential equations by
using differential transform method, Chaos. Soliton. Fract. 34 (2007)
1473-1481.
[22] Nemat Abazari, Reza Abazari, Solution of nonlinear second-order
pantograph equations via differential transform method, International
Conference on Computer and Applied Mathematics, October 28-30, 2009,
Venice, Italy (Available at http://www.waset.org/journals/waset/v58).
[23] A. Arikoglu, I. Ozkol, Solution of boundary value problems for integro-
differential equations by using differential transform method, Appl. Math.
Comput. 168 (2005) 1145-1158.
[24] Reza Abazari, Solution of Riccati types matrix differential equations
using matrix differential transform method, J. Appl. Math. & Informatics.
27 (2009), pp. 1133-1143.
[1] A. Borhanifar, Reza Abazari, An unconditionally stable parallel difference
scheme for Telegraph equation, Math. Prob. Eng, vol. 2009, Article ID
969610, 17pages, 2009. http://doi:10.1155/2009/969610.
[2] A. Borhanifar, Reza Abazari, Exact solutions for non-linear Schr¨odinger
equations by differential transformation method, J. Appl. Math. Comput,
http://doi:10.1007/s12190-009-0338-2.
[3] A. Borhanifar, Reza Abazari, Numerical study of nonlinear Schr¨odinger
and coupled Schr¨odinger equations by differential transformation method,
Opt. Commun, http://doi:10.1016/j.optcom.2010.01.046.
[4] Hirota. R, Satsuma. J, Soliton solutions of a coupled Kortewege-de Vries
equation, Phys. Lett. A, 85 (1981), 407-408.
[5] Reza Abazari, A. Borhanifar, Numerical study of Burgers- and coupled
Burgers- equations by differential transformation method, Comput. Math.
Appl, http://doi:10.1016/j.camwa.2010.01.039.
[6] Chen. R, Wu. Z, Solving partial differential equation by using multiquadric
quasi-interpolation, Appl.Math.Comput, 186 (2007), 1502-1510.
[7] Xu. Y, Shu. C-W, Local discontinuous Galerkin methods for the
Kuramoto- Sivashinsky equations and the Ito-type coupled equations,
Comput. Methods. Appl. Mech, 195 (2006), 3430-3447.
[8] Kaya. D, Inan. EI, Exact and numerical traveling wave solutions for
nonlinear coupled equations using symbolic computation, Appl. Math.
Comput, 151 (2004), 775-787.
[9] J. Biazar, H. Aminikhah, Exact and numerical solutions for non-linear
Burgers- equation by VIM, Math. Comput. Modelling, 49 (2009) 1394-
1400.
[10] Zayed EME, Zedan HA, Gepreel KA, On the solitary wave solutions
for nonlinear Hirota Satsuma coupled-KdV of equations, Chaos. Solitons.
Fractals, 22 (2004), 285-303.
[11] Caom DB, Yan JR, Zang Y, Exact solutions for a new coupled MKdV
equations and a coupled KdV equations, J. Phys. Lett. A, 297 (2002),
68-74.
[12] Zhang JL, Wang ML, Feng ZD, The improved F-expansion method and
its applications, Phys. Lett. A, 350 (2006), 103-109.
[13] Khater AH, Temsah RS, Hassan MM, A Chebyshev spectral collocation
method for solving Burgers type equations, J. Appl. Math. Comput,
http://doi:10.1016/j.cam.2007.11.007.
[14] Ganji DD, Rafei M, Solitary wave equations for a generalized Hirotasatsuma
coupled-KdV equation by homotopy perturbation method, Phys.
Lett. A, 356 (2006), 131-137.
[15] J.K. Zhou, Differential Transformation and its Application for Electrical
Circuits, Huazhong University Press, Wuhan, China, 1986.
[16] M. Mossa Al-sawalha, M.S.M. Noorani , A numeric-analytic method
for approximating the chaotic Chen system, Chaos. Soliton. Fract. 42
(2009) 1784-1791.
[17] F. Ayaz, Application of differential transform method to differential-
algebraic equations, Appl. Math. Comput. 152 (2004) 649-657.
[18] A. Arikoglu, I. Ozkol, Solution of difference equations by using differential
transform method, Appl. Math. Comput. 174 (2006) 1216-1228.
[19] A. Arikoglu, I. Ozkol, Solution of differential-difference equations by
using differential transform method, Appl. Math. Comput. 181 (2006)
153-162.
[20] Figen Kangalgil, Fatma Ayaz, Solitary wave solutions for the KdV and
mKdV equations by differential transform method, Chaos. Soliton. Fract.
41 (2009) 464-472.
[21] A. Arikoglu, I. Ozkol, Solution of fractional differential equations by
using differential transform method, Chaos. Soliton. Fract. 34 (2007)
1473-1481.
[22] Nemat Abazari, Reza Abazari, Solution of nonlinear second-order
pantograph equations via differential transform method, International
Conference on Computer and Applied Mathematics, October 28-30, 2009,
Venice, Italy (Available at http://www.waset.org/journals/waset/v58).
[23] A. Arikoglu, I. Ozkol, Solution of boundary value problems for integro-
differential equations by using differential transform method, Appl. Math.
Comput. 168 (2005) 1145-1158.
[24] Reza Abazari, Solution of Riccati types matrix differential equations
using matrix differential transform method, J. Appl. Math. & Informatics.
27 (2009), pp. 1133-1143.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58185", author = "Reza Abazari and Rasool Abazari", title = "Numerical Study of Some Coupled PDEs by using Differential Transformation Method", abstract = "In this paper, the two-dimension differential transformation method (DTM) is employed to obtain the closed form solutions of the three famous coupled partial differential equation with physical interest namely, the coupled Korteweg-de Vries(KdV) equations, the coupled Burgers equations and coupled nonlinear Schrödinger equation. We begin by showing that how the differential transformation method applies to a linear and non-linear part of any PDEs and apply on these coupled PDEs to illustrate the sufficiency of the method for this kind of nonlinear differential equations. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.
", keywords = "Coupled Korteweg-de Vries(KdV) equation, Coupled Burgers equation, Coupled Schrödinger equation, differential transformation method.", volume = "4", number = "6", pages = "686-8", }
