## THERMAL SCIENCE

International Scientific Journal

### SOLUTION OF DIFFUSION EQUATION WITH LOCAL DERIVATIVE WITH NEW PARAMETER

**ABSTRACT**

A local derivative with new parameter was used to model diffusion. The modified equation was solved iteratively. Stability of the used method together with the uniqueness of the special solution was studied. An algorithm was proposed to derive the special solution.

**KEYWORDS**

PAPER SUBMITTED: 2014-11-15

PAPER REVISED: 2015-02-02

PAPER ACCEPTED: 2015-03-04

PUBLISHED ONLINE: 2015-08-02

**THERMAL SCIENCE** YEAR

**2015**, VOLUME

**19**, ISSUE

**Supplement 1**, PAGES [S231 - S238]

- Kilbas, A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, 2006
- Li, B., Q., Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Springer-Verlag, London, 2006
- Versteeg, H., Malalasekera, W., An Introduction to Computational Fluid Dynamics, Prentice Hall, Upper Saddle River, N. J. USA, 2007, pp. 262-263
- Smoluchowski, M. v., About Brownian Motion under the Action of External Forces and the Relationship with the Generalized Diffusion Equation (in German), Ann. Phys. 353 (1915), pp. 1103-1112
- Yang, X. J. et al., Cantor-Type Cylindrical-Coordinate Method for Differential Equations with Local Fractional Derivatives, Physics Letters A, 377 (2013), 28-30, pp. 1696-1700
- Florian C., The History of Notations of the Calculus, Annals of Mathematics, 25 (1923), 1, pp. 1-46
- Leonid, P., Cloud, M. J., Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Princeton University Press, Princeton, N. J., USA, 2004
- Apostol, T. M., One-Variable Calculus with an Introduction to Linear Algebra, Wiley, New York, USA, 1967
- Atangana, A., Aydin, S., A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions, Abstract and Applied Analysis, 2013 (2013), ID 279681
- Caputo, M., Linear Models of Dissipation Whose Q is Almost Frequency Independent, Part II, Geophysical Journal International, 13 (1967), 5, pp. 529-539
- Atangana, A., Botha, J. F., Generalized Groundwater Flow Equation Using the Concept of Variable Order Derivative, Boundary Value Problems, 2013 (2013), pp. 53
- Magin, R. L., Fractional Calculus in Bioengineering, Begell House, Redding, Conn., USA, 2006
- Chechkin, A. V. et al., Fractional Diffusion in Inhomogeneous Media, Journal of Physics A, 38 (2005), 42, pp. L679-L684
- Granville, S., The Numerical Solution of Ordinary and Partial Differential Equations, Academic Press Professional, Inc., San Diego, Cal., USA, 1988
- Yang, X. J., Local Fractional Integral Transforms, Progress in Non-linear Science, 4 (2011), pp. 1-225
- Merdan, M., Mohyud-Din, S. T., A New Method for Time-Fractional Coupled-KDV Equations with Modified Riemann-Liouville Derivative, Studies in Non-linear Science, 2 (2011), 2, pp. 77-86
- Tan, Y., Abbasbandy S., Homotopy Analysis Method for Quadratic Riccati Differential Equation, Communications in Non-linear Science and Numerical Simulation, 13 (2008), 3, pp. 539-546
- Atangana, A., Drawdown in Prolate Spheroidal-Spherical Coordinates Obtained via Green's Function and Perturbation Methods, Communications in Non-linear Science and Numerical Simulation, 19 (2014), 5, pp. 1259-1269
- Atangana, A., A Note on the Triple Laplace Transform and its Applications to some Kind of Third- Order Differential Equation, Abstract and Applied Analysis, 2013 (2013), ID 769102
- Duffy, D. G., Transform Methods for Solving Partial Differential Equations, CRC Press, New York, USA, 2004
- Brychkov, Y. A., Multidimensional Integral Transformations, Gordon and Breach Science Publishers, Philadelphia, Penn., USA, 1992
- Kilicman, A., Gadain, H. E., On the Applications of Laplace and Sumudu Transforms, Journal of the Franklin Institute, 347 (2010), 5, pp. 848-862
- Yang, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628
- Chen, Y. Q., Moore, K. L., Discretization Schemes for Fractional-Order Differentiators and Integrators, IEEE Transactions on Circuits and Systems I, 49 (2002), 3, pp. 363-367