THERMAL SCIENCE

International Scientific Journal

Ai-Min YangORCID

,

Yu-Zhu ZhangORCID

,

Weijie LiORCID

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
local fractional Laplace variational iteration method, diffusion equation, non-differentiable solution, local fractional derivative
PAPER SUBMITTED: 2014-11-26
PAPER REVISED: 2015-01-10
PAPER ACCEPTED: 2015-02-09
PUBLISHED ONLINE: 2015-08-02
DOI REFERENCE: https://doi.org/10.2298/TSCI15S1S63Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement, PAGES [163 - 168]
REFERENCES
[1] Crank, J., The Mathematics of Diffusion, Oxford University Press, Oxford, UK, 1975, 10.1016/0032-3861(75)90130-5
[2] Hilfer, R., Fractional Diffusion Based on Riemann-Liouville Fractional Derivatives, The Journal of Physical Chemistry B, 104 (2000), 16, pp. 3914-3917, 10.1021/jp9936289
[3] Sandev, T., et al., Fractional Diffusion Equation with a Generalized Riemann-Liouville Time Fractional Derivative, Journal of Physics A, 44 (2011), 25, 255203, 10.1088/1751-8113/44/25/255203
[4] 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
[5] Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), 2, pp. 291-316, 10.2298/tsci1002291h
[6] Hristov, J., Approximate Solutions to Fractional Subdiffusion Equations, The European Physical Journal - Special Topics, 193 (2011), 1, pp. 229-243, 10.1140/epjst/e2011-01394-2
[7] 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, 10.48550/arxiv.1012.2530
[8] 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, 10.1016/j.jcp.2005.08.008
[9] Mandelbrot, B. B., et al., Fractal Character of Fracture Surfaces of Metals, Nature, 308 (1984), pp. 721- 722, 10.1038/308721a0
[10] Ganti, S., Bhushan, B., Generalized Fractal Analysis and its Applications to Engineering Surfaces, Wear, 180 (1995), 1, pp. 17-34, 10.1016/0043-1648(94)06545-4
[11] Neimark, A., A New Approach to the Determination of the Surface Fractal Dimension of Porous Solids, Physica A, 191 (1992), 1, pp. 258-262, 10.1016/0378-4371(92)90536-y
[12] Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
[13] 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
[14] 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, 10.1155/2013/754248
[15] 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, 10.2298/tsci120826075l
[16] 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, 10.1155/2014/914725
[17] Li, Y., et al., Local Fractional Laplace Variational Iteration Method for Fractal Vehicular Traffic Flow, Advances in Mathematical Physics, 2014 (2014), ID 649318, 10.1155/2014/649318
[18] 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, 10.1155/2014/365981
[19] 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, 10.1016/j.physleta.2013.04.012
[20] Yang, X. J., et al., Systems of Navier-Stokes Equations on Cantor Sets, Mathematical Problems in Engineering, 2013 (2013), ID 769724, 10.1155/2013/769724
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Laplace variational iteration method for the two-dimensional diffusion equation in homogeneous materials

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence