R. Hasankhani Gavabaria1, D.D. Ganjib2,*, A.Bozorgic3

Davood Domiri Ganji; R. Hasankhanigavabari; A. Bozorgi

Applications of the two-dimensional differential transform and least square method for solving nonlinear wave equations

Yıl 2014, Cilt: 2 Sayı: 2, 95 - 105, 01.08.2014

Davood Domiri Ganji R. Hasankhanigavabari A. Bozorgi

Öz

The differential transform and least square are analytical methods for solving differential equations. In this article, twoDimensional Differential Transform Method (2D DTM) and Least Square Method (LSM) are applied to obtaining the analytic solution of the two- dimensional non- linear wave equations. We demonstrate that the differential transform method and least square are very effective and convenient for achieving the analytical solutions of linear or nonlinear partial differential equations. Also, three examples are given to demonstrate the exactness of the methods. Results of these methods are compared with the exact solution

Anahtar Kelimeler

Two-Dimensional Transform Method (2D DTM), wave equation, analytical solution, Least Square Method (LSM) and exact solution

Kaynakça

J.K. Zhou, Differential Transformation Method and Its Application for Electrical Circuits, Hauzhang University Press, Wuhan, China, 1986.
F. Ayaz, On the two- dimensional differential transform method, Applied Mathematics and Computation 143 (2-3) (2003) 361-374.
C.K. Chen, S.H. Ho, Solving partial differential equation by two- dimensional differential equation, Applied Mathematics and Computation 106 (1999) 171-179.
M.J. Jang, C.L. Chen, Y.C. Liu, Two-dimensional differential transform for partial differential Equations, Applied Mathematics and Computation 121 (2001) 261–270.
B. Shiri, A note on using the Differential Transformation Method for the Integro-Differential equations, Applied Mathematics and Computation 219 (2013) 7306-7309.
A. Arikoglu, I. Ozkol, Solution of boundary value problem for integro-differential equations by using differential transform method, Applied Mathematics and Computation 168 (2005) 1145-1158.
Z. M. Odibat, Differential transform method for solving Volterra integral equations with separable kernels, Mathematics and Computation Model 48 (7-8) (2008) 1144-1149.
S. Momani, V. S. Erturk, Solutions of non-linear oscillators by the modified differential transform method, Computers and Mathematics with Applications 55 (2008) 833-842.
S. Momani, Z. Obidat, Generalized Differential Transform Method for solving a space-and time-fractional diffusion-wave equation, Physics Letters A 370 (5-6) (2007) 379-387.
S. Ghafoori, M. Motevalli, M.G. Nejad, F. Shakeri, D.D. Ganji, M. Jalaal, Efficiency of differential transformation method for nonlinear oscillator: Comparison with HPM and VIM, Current Applied Physics 11 (2011) 965-971.
Hessameddin. Yaghoobi, Mohsen. Torabi, The application of differential transformation method to nonlinear equations arising in heat transfer, International Communication in Heat and Mass Transfer 38 (2011) 815-820.
M.J. Jang, Y.L. Yeh, C.L. Chen, W.C. Yeh, Differential transformation approach to thermal conductive problems with discontinuous boundary condition, Applied Mathematics and Computation 216 (2010) 2339–2350.
O. Özkan, Numerical implementation of differential transformations method for integro-differential equations, International Journal of Computer Mathematics 87 (2010) 2786–2797.
A. Borhanifar, R. Abazari, Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method, Optics Communications 283 (2010) 2026–2031.
I.H. Abdel-Halim Hassan, Application to differential transformation method for solving systems of differential equations, Applied Mathematical Modeling 32 (2008) 2552–2559.
A.H. Hassan, Differential transformation technique for solving higher-order initial value problems, Applied Mathematics and Computation 154 (2004) 299-311.
A.S.V. Ravikanth, K. Aruna, Differential transform method for solving the linear and nonlinear Klein-Gordon Equation, Computer physics Communication 180 (5) (2009) 708-711.
A. Gokdogan, M. Marden, A. Yildirin, The modified algorithm for the differential transform method to solution of Genesio system, Communication in Nonlinear Science and Numerical Simulation 17 (1) (2012) 45-51.
BouazizMN, Aziz A, Simple and accurate solution for convective– radiative fin with temperature dependent thermal conductivity using double optimal linearization, Energy Conversion and Management 51(12) (2010) 76-82.
Aziz A, Bouaziz MN, A least squares method for a longitudinal fin with temperature dependent integral heat generation and thermal conductivity, Energy conversion and Management 52 (8-9) (2011) 2876-2882.
M. Hatami, D.D. Ganji, thermal and flow analysis of micro channel heat sink (MCHS) cooled by Cu-water nanofluid using porous media approach and least square method, Energy Conversion and management 78 (2014) 347-358.
P.L. Ndlovu, R.J. Moitsheki, Application of the two- dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins, Commun Nonlinear Sci Number Simulate 18 (2013) 2689-2698.
A. Tari, M. Y. Rahimi, S. Shahmorad, F. Talati, Solving class of two- dimensional linear and nonlinear Volterra integral equations by the differential transform method, Journal of Computational and Applied Mathematics 228 (2009) 70-76.
S. Momani, Z. Obdibat, A novel method for nonlinear fractional partial differential equation: Combination of DTM and generalized Taylor’s formula, Journal of Computational and Applied Mathematics 220 (2008) 85-95.
A. Tari, S. Shahmorad, Differential transform method for the system of two- dimensional nonlinear Volterraintegro- differential equations, Computers and Mathematics with Applications 61 (2001) 2621-2629.
M. Ghasemi, M.T. Kajani, A. Davari, Numerical simulation of two- dimensional nonlinear differential equation by homotoy perturbation method, applied Mathematics and Computation 189 (20007) 341-345.
J. Biazar, M. Eslami, A new technique for non-linear two- dimensional wave equations, ScientiaIranica B 9(2013) 20 (2) 359-363.