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Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

Published online by Cambridge University Press:  03 June 2015

D. B. Dhaigude  and
Gunvant A. Birajdar
D. B. Dhaigude
Affiliation:
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, (M.S.), India
Gunvant A. Birajdar*
Affiliation:
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad-431 004, (M.S.), India
*
*Corresponding author. Email: gabirajdar11@gmail.com
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Abstract

In this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger’s equation. The obtained solution is verified by comparison with exact solution when α = 1.

Keywords

35R11Discrete Adomian decomposition methodCaputo fractional derivativefractional discrete Schrodinger equationfractional discrete Burger’s equation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 1 , February 2014 , pp. 107 - 119
Copyright
Copyright © Global-Science Press 2014

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References

[1]Ablowitz, A. M. and Ladik, J. F., Nonlinear differential-difference equation and Fourier analysis, J. Math. Phys., 17 (1976), pp. 1011–1018.Google Scholar
[2]Ablowitz, A. M. and Ladik, J. F., A nonlinear difference scheme and inverse scattering, Stud. Appl. Math., 55 (1976), pp. 213–229.Google Scholar
[3]Abbasbandy, S. and Darvishi, M. T., A numerical solution of Burger’s equation by time discretization of Adomian’s decomposition method, Appl. Math. Comput., 170 (2005), pp. 95–102.Google Scholar
[4]Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, 1994.CrossRefGoogle Scholar
[5]Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), pp. 501–544.CrossRefGoogle Scholar
[6]Basto, M., Semiao, V. and Calheiros, F., Dynamics and synchronization of numerical solution of the Burger’s eqution, J. Comput. Appl. Math., 231 (2009), pp. 793–806.Google Scholar
[7]Bateman, H., Some recent researches in motion of fluids, Mon. Weather Rev., 43 (1915), pp. 163–170.Google Scholar
[8]Beatus, T., Tlusty, T. and Bar-Ziv, R., Burger’s shock waves and sound in 2D kicrofluidic droplets ensemble, Phys. Rev. Lett., 103 (2009), 114502.Google Scholar
[9]Bratsos, A., Ehrhardt, M. and Famelis, I. Th., A discrete Adomian decomposition method for discrete nonlinear Schrodinger equations, Appl. Math. Comput., 197 (2008), pp. 190–205.Google Scholar
[10]Burgers, J. M., A Mathematical model illustration the theory of turbulence, Advances in Applied Mechanics 1, Academic Press, New York, 1948, pp. 171–199.Google Scholar
[11]Caputo, M., Linear models of dissipition whose Q is almost independent, II, GeophyS. J. Roy. Astron., 13 (1967), pp. 529–5397.Google Scholar
[12]Daftardar-Gejji, V. AND Jafari, H., Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301 (2005), pp. 508–518.Google Scholar
[13]Daftardar-Gejji, V. and Jafari, H., An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl., 316 (2006), pp. 753–763.Google Scholar
[14]Daftardar-Gejji, V. and Bhalekar, Sachin, Solving multi-term linear and nonlinear diffusion-wave equations of fractional order by Adomian decomposition, Appl. Math. Comput., 202 (2008), pp. 113–120.Google Scholar
[15]Debnath, L., Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhauser, Boston, 2006.Google Scholar
[16]Dhaigude, C. D., Studies On Fractional Reaction Diffusion Problems With Applications, Ph.D thesis, Dr. Babasaheb Ambedkar Marathwada University, Aurangbad, India, 2012.Google Scholar
[17]Dhaigude, D. B., Birajdar, G. A. and Nikam, V. R., Adomian decomposition method for fractional Benjamin-Bona-Mahony-Burger’s equations, Int. J. Appl. Math. Mech., 8(12) (2012), pp. 42–51.Google Scholar
[18]Dhaigude, D. B. and Birajdar, G. A., Numerical solution of system of fractional partial differential equations by discrete Adomian decomposition method, J. Frac. Cal. Appl., 3(12) (2012), pp. 1–11.Google Scholar
[19]Dhaigude, D. B. and Dhaigude, C. D., Linear initial value problems for fractional partial differential equations, Bull. Marathwada Math. Soc., 13(2) (2012), pp. 17–38.Google Scholar
[20]Dhaigude, C. D. and Nikam, V. R., Solution of fractional partial differential equations using iterative method, Frac. Cal. Appl. Anal., 15(4) (2012), pp. 684–699.Google Scholar
[21]Gorenflo, R. and Minardi, F., Fractional calculas intrgrals and differentials equations of fractional order, in Carpinteri, A., Mainardi, F.(Eds), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wein and New York, (1997), pp. 223–276.Google Scholar
[22]Gorenflo, R., Minardi, F., Moretti, D. and Paradisi, P., Time fractional diffusion: a discrete random walk approach, Nonlinear Dyn., 29 (2002), pp. 129–143.Google Scholar
[23]Gorenflo, R. AND Abdel-Rehim, E. A., Approximation of Time Fractional Diffusion with Central Drift by Difference Schemes, Berlin Free University, 2003.Google Scholar
[24]He, Y., Burov, S., Metzler, R. and Barkai, E., Random time-scale invariant diffusion and transport coefficients, Phys. Rev. Lett., 101(5) (2008), 058101.Google Scholar
[25]Jafari, H. and Daftardar-Gejji, V., Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition, Appl. Math. Comput., 180 (2006), pp. 488–497.Google Scholar
[26]Kaya, D. and El-Sayed, S. M., On the solution of the coupled Schrodinger-KdV equation by the decomposition method, Phys. Lett. A, 313 (2003), pp. 82–88.Google Scholar
[27]Logan, J. D., An Introduction to Nonlinear Partial Differential Equations, Wiley-Interscience, New York, 1994.Google Scholar
[28]Luchko, Y. AND Gorenflo, R., An operational method for solving fractional differential equations with the Caputo derivative, Acta Math. Vietnam., 24 (1999), pp. 207–233.Google Scholar
[29]Mainardi, F., Fractional relaxation oscillation and fractional diffusion-wave phenomena, Chaos, Soliton Fract., 7(9) (1996), pp. 1461–1477.Google Scholar
[30]Podlubny, I., Fractional Differential Equations, Academic Press, San Diego,1999.Google Scholar
[31]Ray, S. S. and Bera, R. K., Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comput., 174 (2006), pp. 329–336.Google Scholar
[32]Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional Integral and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.Google Scholar
[33]Schneider, W. R. and Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), pp. 134–144.Google Scholar
[34]Shawagfeh, N. T., Analytical approximate solution for nonlinear fractional differential equations, Appl. Math. Comput., 131(2) (2002), pp. 517–529.Google Scholar
[35]Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon Press Oxford, 1978.Google Scholar
[36]Wazwaz, A. M., A reliable analysis for nonlinear Schrodinger equations with cubic nonlinearity and a power law nonlinearity, Math. Comput. Model., 43 (2006), pp. 178–184.CrossRefGoogle Scholar
[37]Wazwaz, A. M., Exact solutions for the fourth order nonlinear Schrodinger equations with cubic nonlinearity and a power law nonlinearity, Math. Comput. Model., 43 (2006), pp. 802–808.Google Scholar
[38]Wyss, W., The fractional diffusion equations, J. Math. Phys., 27 (1986), pp. 2782–2785.Google Scholar
[39]Wazwaz, A.M., The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model, Appl. Math. Comput., 110 (2000), pp. 251–264.Google Scholar
[40]Zhu, H., Shu, H. and Ding, M., Numerical solution of two-dimensional Burger’s equations by discrete Adomian decomposition method, Comput. Math. Appl., 60 (2010), pp. 840–848.Google Scholar