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Z. Naturforsch. 67a, 355 – 362 (2012)
doi:10.5560/ZNA.2012-0025
On the Hybrid of Fourier Transform and Adomian Decomposition Method for the Solution of Nonlinear Cauchy Problems of the Reaction-Diffusion Equation
Salman Nourazar1, Akbar Nazari-Golshan2, Ahmet Yıldırım3,4, and Maryam Nourazar5
1 Mechanical Engineering Department, Amirkabir University of Technology, Tehran, Iran
2 Department of Physics, Amirkabir University of Technology, Tehran, Iran
3 University of South Florida, Department of Mathematics and Statistics, Tampa, FL 33620-5700 USA
4 Department of Mathematics, Science Faculty, Ege University, 35100 Bornova-Izmir, Turkey
5 Civil Engineering Department, Amirkabir University of Technology, Tehran, Iran
Received September 19, 2011 / revised January 15, 2012 / published online July 3, 2012
Reprint requests to: S. S. N.; E-mail: icp@aut.ac.ir
The physical science importance of the Cauchy problem of the reaction-diffusion equation appears in the modelling of a wide variety of nonlinear systems in physics, chemistry, ecology, biology, and engineering. A hybrid of Fourier transform and Adomian decomposition method (FTADM) is developed for solving the nonlinear non-homogeneous partial differential equations of the Cauchy problem of reaction-diffusion. The results of the FTADM and the ADM are compared with the exact solution. The comparison reveals that for the same components of the recursive sequences, the errors associated with the FTADM are much lesser than those of the ADM. We show that as time increases the results of the FTADM approaches 1 with only six recursive terms. This is in agreement with the physical property of the density-dependent nonlinear diffusion of the Cauchy problem which is also in agreement with the exact solution.

The monotonic and very rapid convergence of the results of the FTADM towards the exact solution is shown to be much faster than that of the ADM.

Key words: Cauchy Reaction-Diffusion; Fourier Transformation; Adomian Decomposition Method; Non-Homogeneous Partial Differential Equation.
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