### Solving fractional difference equations by discrete Adomian decomposition method

#### Öz

In this paper, we propose the discrete Adomian decomposition method(DADM) to solve linear as well as nonlinear fractional partial difference equations and provide few examples to illustrate the applicability of proposed method. The results show that DADM is efficient, accurate and can be applied to other fractional difference equations.

#### Tam Metin:

PDF#### Referanslar

Abdeljawad T., On Riemann and Caputo fractional diﬀerences, Computers and Mathematics with Applications, 62, 1602–1611, (2011).

Ablowitz M.J. and Ladik J.F., Nonlinear diﬀerential–diﬀerence equation and Fourier analysis, Journal of Mathematical Physics, 17, 1011–1018, (1976).

Ablowitz M.J. and Ladik J.F., A nonlinear diﬀerence scheme and inverse scattering, Studies in Applied Mathematics, 55 , 213–229, (1976).

Adomian G., A Review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications, 135, 501–544, (1988).

Adomian G., Solving frontier problems of physics: the decomposition method, Boston: Kluwer Academic Publishers, (1994).

Adomain G., Solution of coupled nonlinear partial differential equations by decomposition, Computer and Mathematics with Applications, 31, 117-120, (1996).

Agarwal, R.P., Diﬀerence equations and inequalities, Marcel Dekker, Newyork, (1992).

Anastassiou G.A., About discrete fractional calculus with inequalities, intelligent mathematics: computational analysis, Intelligent Systems Reference Library, 5, 575-585, (2011).

Atici F.M. and Eloe P.W., A transform method in discrete fractional calculus, International Journal of Difference Equations, 2, 165–176, (2007).

Atici F.M. and Eloe P.W., Initial value problems in discrete fractional calculus, Proceedings of the American Mathematical Society, 137, 981–989, (2009).

Atici F.M. and Sengül, S., Modeling with fractional diﬀerence equations, Journal of Mathematical Analysis and Applications, 369, 1–9, (2010).

Bratsos A., Ehrhardt M. and Famelis I.T., A discrete Adomian decomposition method for discrete nonlinear Schrödinger equations, Applied Mathematics and Computation, 197, 190–205, (2008).

Bulut H. and Evans D.J., On the solution of the Riccati equation by the decomposition method, International Journal of Computer Mathematics, 79 (1), 103-109, (2002).

Caputo M., Linear models of dissipition whose Q is almost independent- II, Geophysical Journal of the Royal Astronomical Society, 13, 529–539, (1967).

Dhaigude D.B. and Birajdar G.A., Numerical solutions of fractional partial diﬀerential equations by discrete Adomian decomposition method, Advences in Applied Mathematics Mechanics, 6, 107–119, (2014).

Ergüt M. and Bulut H., The numerical solution of 2-dimensional partial differential equations by the decomposition method, Journal of Natural ScienceMathematics, 42 (1), 07-20, (2002).

Evans D.J. and Bulut H., A new approach to the gas dynamics equations: an application of the decomposition method, International Journal of Computer Mathematics, 79 (7), 817-822, (2002).

He J.H., Variational iteration method-a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics, 34, 699-708, (1999).

He J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 257-262, (1999).

Liao S.J., The proposed homotopy analysis technique for the solution of nonlinear problems., PhD Thesis, Shanghai Jiao Tong University, (1992).

Liao S.J., Beyond perturbation: introduction to homotopy analysis method, Chapman and Hall/CRC Press, Boca Raton, (2003).

Mickens R.E., Nonstandard finite diﬀerence models of diﬀerential equations, World Publishing Company, Singapore, (1994).

Özpınar F., Applying discrete homotopy analysis method for solving fractional partial differential equations, Entropy, 20(5), 332, (2018).

Podlubny I., Fractional diﬀerential equations, Academic Press: San Diego, (1999).

Sharma D., Singh P., Chauhan S., Homotopy perturbation Sumudu transform method with He’s polynomial for solutions of some fractional nonlinear partial differential equations, International Journal of Nonlinear Science, 21(2), 91-97, (2016).

Wazwaz A.M., A reliable modiﬁcation of Adomian’s decomposition method, Applied Mathematics and Computation, 102, 77–86, (1999).

Wazwaz A.M., A reliable technique for solving linear and nonlinear Schrödinger equations by Adomian decomposition method, Bulletin of Institute of Mathematics, 29, 125– 134, (2001).

Wazwaz A.M. A reliable analysis for nonlinear Schrodinger equations with cubic nonlinearity and a power law nonlinearity, Mathematical and Computer Modelling, 43, 178–184, (2006).

Wazwaz A.M., A comparison between the variational iteration method and Adomaian decompostion method, Journal of Computational and Applied Mathematics, 207, 129-136, (2007).

Wu G.C., Balenau D., Zeng S.D., Deng Z.G., Discrete fractional diffusion equation, Nonlinear Dynamics, 80, 281-286, (2015).

### Refback'ler

- Şu halde refbacks yoktur.

Telif Hakkı (c) 2018 Figen Özpınar

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

ISSN: 1301-7985 fbedergi@balikesir.edu.tr http://fbed.balikesir.edu.tr/