## THERMAL SCIENCE

International Scientific Journal

### LAPLACE VARIATIONAL ITERATION METHOD FOR THE TWO-DIMENSIONAL DIFFUSION EQUATION IN HOMOGENEOUS MATERIALS

**ABSTRACT**

In this paper, we suggest the local fractional Laplace variational iteration method to deal with the two-dimensional diffusion in homogeneous materials. The operator is considered in local fractional sense. The obtained solution shows the non-differentiable behavior of homogeneous materials with fractal characteristics.

**KEYWORDS**

PAPER SUBMITTED: 2014-11-26

PAPER REVISED: 2015-01-10

PAPER ACCEPTED: 2015-02-09

PUBLISHED ONLINE: 2015-08-02

**THERMAL SCIENCE** YEAR

**2015**, VOLUME

**19**, ISSUE

**Supplement 1**, PAGES [S163 - S168]

- Crank, J., The Mathematics of Diffusion, Oxford University Press, Oxford, UK, 1975
- Hilfer, R., Fractional Diffusion Based on Riemann-Liouville Fractional Derivatives, The Journal of Physical Chemistry B, 104 (2000), 16, pp. 3914-3917
- Sandev, T., et al., Fractional Diffusion Equation with a Generalized Riemann-Liouville Time Fractional Derivative, Journal of Physics A, 44 (2011), 25, 255203
- Huang, F., Liu, F., The Space-Time Fractional Diffusion Equation with Caputo Derivatives, Journal of Applied Mathematics and Computing, 19 (2005), 1-2, pp. 179-190
- Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), 2, pp. 291-316
- Hristov, J., Approximate Solutions to Fractional Subdiffusion Equations, The European Physical Journal - Special Topics, 193 (2011), 1, pp. 229-243
- Hristov, J., A Short-Distance Integral-Balance Solution to a Strong Subdiffusion Equation: a Weak Power- Law Profile, International Review of Chemical Engineering, 2 (2010), 5, pp. 555-563
- Tadjeran, C., et al., A Second-Order Accurate Numerical Approximation for the Fractional Diffusion Equation, Journal of Computational Physics, 213 (2006), 1, pp. 205-213
- Mandelbrot, B. B., et al., Fractal Character of Fracture Surfaces of Metals, Nature, 308 (1984), pp. 721- 722
- Ganti, S., Bhushan, B., Generalized Fractal Analysis and its Applications to Engineering Surfaces, Wear, 180 (1995), 1, pp. 17-34
- Neimark, A., A New Approach to the Determination of the Surface Fractal Dimension of Porous Solids, Physica A, 191 (1992), 1, pp. 258-262
- Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
- Yang, X. J., et al., Approximation Solutions for Diffusion Equation on Cantor Time-Space, Proceeding of the Romanian Academy A, 14 (2013), 2, pp. 127-133
- Hao, Y. J., et al., Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates, Advances in Mathematical Physics, 2013 (2013), ID 754248
- Liu, C. F., et al., Reconstructive Schemes for Variational Iteration Method within Yang-Laplace Transform with Application to Fractal Heat Conduction Problem, Thermal Science, 17 (2013), 3, pp. 715-721
- Xu, S., et al., Local Fractional Laplace Variational Iteration Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow, Mathematical Problems in Engineering, 2014 (2014), ID 914725
- Li, Y., et al., Local Fractional Laplace Variational Iteration Method for Fractal Vehicular Traffic Flow, Advances in Mathematical Physics, 2014 (2014), ID 649318
- Yang, A. M., et al., The Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivatives, Discrete Dynamics in Nature and Society, 2014 (2014), ID 365981
- Yang, X. J., et al., Cantor-Type Cylindrical-Coordinate Method for Differential Equations with Local Fractional Derivatives, Physics Letters A, 377 (2013), 28, pp. 1696-1700
- Yang, X. J., et al., Systems of Navier-Stokes Equations on Cantor Sets, Mathematical Problems in Engineering, 2013 (2013), ID 769724