## THERMAL SCIENCE

International Scientific Journal

### APPROXIMATE SOLUTION OF THE NON-LINEAR DIFFUSION EQUATION OF MULTIPLE ORDERS

**ABSTRACT**

In this paper, fractional diffusion equation of multiple orders is approximately solved. The equation is given in the equivalent integral form. The Adomian polynomial is adopted and analytical solutions are obtained. The result contains two parameters that can have more space for fitting the experiment data.

**KEYWORDS**

PAPER SUBMITTED: 2015-12-29

PAPER REVISED: 2016-02-13

PAPER ACCEPTED: 2016-03-15

PUBLISHED ONLINE: 2016-09-24

**THERMAL SCIENCE** YEAR

**2016**, VOLUME

**20**, ISSUE

**Supplement 3**, PAGES [S683 - S687]

- Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, Cal., USA, 1999
- Metzler, R., et al., The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach, Physics Reports, 339 (2000), 1, pp. 11-77
- Yang, X. J., et al., An Asymptotic Perturbation Solution for a Linear Oscillator of Free Damped Vibrations in Fractal Medium Described by Local Fractional Derivatives, Communications in Non-Linear Science and Numerical Simulation, 29 (2015), 1-3, pp. 499-504
- Yang, X. J., et al., Fractal Heat Conduction Problem Solved by Local Fractional Variational Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628
- Das, S., et al., Fractional Diffusion Equations in the Presence of Reaction Terms, Journal of Computational Complexity and Applications, 1 (2015), 1, pp. 15-21
- Wu, G. C., et al., Lattice Fractional Diffusion Equation in Terms of A Riesz-Caputo Difference, Phys. A, 438 (2015), NOV., pp. 335-3398
- Wu, G. C., et al., Discrete Fractional Logistic Map and Its Chaos, Non-Linear Dynamics, 75 (2014), 1, pp. 283-287
- Guo, B. B., et al., Numerical Application for Volterra's Population Growth Model with Fractional Order by the Modified Reproducing Kernel Method, J. Computational Complexity and Applications, 1 (2015), 1, pp. 1-9
- Zhao, D. F., et al., A Note on Riemann Diamond Integrals on Time Scales, Journal of Computational Complexity and Applications, 2 (2016), 2, pp. 64-72
- Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method, Springer, N. Y., USA, 1994
- Daftardar-Gejji, V., et al., Adomian Decomposition: A Tool for Solving A System of Fractional Differential Equations, Journal of Mathematical Analysis and Applications, 301 (2005), 2, pp. 508-518
- Saha, R. S., et al., An Approximate Solution of a Non-Linear Fractional Differential Equation by Adomian Decomposition Method, Applied Mathematics and Computation, 167 (2005), 1, pp. 561-571
- Duan, J. S., Recurrence Triangle for Adomian Polynomials, Applied Mathematics and Computation, 216 (2010), 4, pp. 1235-1241
- Duan, J. S., et al., Higher-Order Numeric Wazwaz-El-Sayed Modified Adomian Decomposition Algorithms, Computers & Mathematics with Applications, 63 (2012), 11, pp. 1557-1568
- Duan, J. S., et al., Solutions of the Initial Value Problem for Non-Linear Fractional Ordinary Differential Equations by the Rach-Adomian-Meyers Modified Decomposition Method, Applied Mathematics and Computation, 218 (2012), 17, pp. 8370-8392
- Yang, X. J., et al., Approximate Solutions for Diffusion Equations on Cantor Space-Time, Proceedings, Romanian Academy, Series A, 14 (2013), 2, pp. 127-133
- Yang, X. J., et al., Fractal Boundary Value Problems for Integral and Differential Equations with Local Fractional Operators, Thermal Science, 3 (2015), 19, pp. 959-966
- Yang, X. J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New York, USA, 2015