Thermal Science

International Scientific Journal

Find this paper on

Chun-Fu Wei

TWO-SCALE TRANSFORM FOR 2-D FRACTAL HEAT EQUATION IN A FRACTAL SPACE

ABSTRACT

A 2-D fractal heat conduction in a fractal space is considered by He’s fractal derivative. The two-scale transform is adopted to convert the fractal model to its differential partner. The homotopy perturbation method is used to find the approximate analytical solution.

KEYWORDS

fractal derivative, heat equation, two-scale transform, homotopy perturbation method

PAPER SUBMITTED: 2019-09-18

PAPER REVISED: 2020-06-14

PAPER ACCEPTED: 2020-06-15

PUBLISHED ONLINE: 2021-03-27

DOI REFERENCE: https://doi.org/10.2298/TSCI190918124W

CITATION EXPORT: view in browser or download as text file

THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE No. 3, PAGES [2339 - 2345]

REFERENCES

[1] Metzler, R., Klafter, J., The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach, Physics Reports-Review Section of Physics Letters, 339 (2000), 1, pp. 1-77, 10.1016/s0370-1573(00)00070-3

[2] He, J. H., The Simpler, the Better: Analytical Methods for Non-linear Oscillators and Fractional Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1252-1260, 10.1177/1461348419844145

[3] He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718, 10.1007/s10773-014-2123-8

[4] Wang, K. L., Wang, K. J., A Modification of the Reduced Differential Transform Method for Fractional Calculus, Thermal Science, 22 (2018), 4, pp. 1871-1875, 10.2298/tsci1804871w

[5] Wang, K. L., Yao, S. W., A Fractal Variational Principle for the Telegraph Equation with Fractal Derivatives, Fractals, 28 (2020), 4, ID 20500589, 10.1142/s0218348x20500589

[6] He, J. H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276, 10.1016/j.rinp.2018.06.011

[7] Wang, Y., et al., A fractal Derivative Model for Snow's Thermal Insulation Property, Thermal Science, 23 (2019), 4, pp. 2351-2354, 10.2298/tsci1904351w

[8] Liu, H. Y., et al., A Fractal Rate Model for Adsorption Kinetics at Solid/Solution Interface, Thermal Science, 23 (2019), 4, pp. 2477-2480, 10.2298/tsci1904477l

[9] Wang, Q. L., et al., Fractal Calculus and its Application to Explanation of Biomechanism of Polar Hairs (Vol. 26, 1850086, 2018), Fractals, 27 (2019), 5, ID 1992001

[10] Wang, Q. L., et al., Fractal Calculus and Its Application to Explanation of Biomechanism of Polar Hairs (Vol. 26, 1850086, 2018), Fractals, 26 (2018), 6, ID 1850086

[11] He, J. H., A Fractal Variational Theory for One-Dimensional Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 2, ID 2050024, 10.1142/s0218348x20500243

[12] He, J. H., A Simple Approach to One-Dimensional Convection-Diffusion Equation and Its Fractional Modification for E Reaction Arising in Rotating Disk Electrodes, Journal of Electroanalytical Chemis-try, 854 (2019), Dec., ID 113565, 10.1016/j.jelechem.2019.113565

[13] Wei, C.-F., Solving Time-Space Fractional Fitzhugh-Nagumo Equation by Using He-Laplace Decomposition Method, Thermal Science, 22 (2018), 4, pp. 1723-1728, 10.2298/tsci1804723w

[14] Wei, C-F., Application of the Homotopy Perturbation Method for Solving Fractional Lane-Emden Type Equation, Thermal Science, 22 (2019), 4, pp. 2237-2244, 10.2298/tsci1904237w

[15] Shen, Y., He, J. H., Variational Principle for a Generalized KdV Equation in a Fractal Space, Fractals, 28 (2020), 4, 20500693, 10.1142/s0218348x20500693

[16] Wei, C-F., Local Frational Heat and Wave Equations with Laguerre Type Derivatives, Thermal Science, 24 (2020), 4, pp. 1-6

[17] Wang, K. L., et al., Physical Insight of Local Fractional Calculus and Its Application to Fractional Kdv-Burgers-Kuramoto Equation, Fractals, 27 (2019), 7, ID 1950122, 10.1142/s0218348x19501226

[18] Ain, Q. T., He, J. H., On Two-Scale Dimension and Its Application, Thermal Science, 23 (2019), 3B, pp. 1707-1712

[19] He, J. H., Ji, F. Y. Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133, 10.2298/tsci1904131h

[20] He, J. H., Ain, Q. T., New Promises and Future Challenges of Fractal Calculus: From Two-Scale Thermodynamics to Fractal Variational Principle, Thermal Science, 24 (2020), 2A, pp. 659-681, 10.2298/tsci200127065h

[21] He, J. H., Homotopy Perturbation Technique, Computer Methods in Applied Mechanics and Engineering, 178 (1999), 3-4, pp. 262-257, 10.1016/s0045-7825(99)00018-3

[22] Yu, D. N., et al., Homotopy Perturbation Method with an Auxiliary Parameter for Non-Linear Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1540-1554, 10.1177/1461348418811028

[23] Kuang, W. X., et al., Homotopy Perturbation Method with an Auxiliary Term for the Optimal Design of a Tangent Non-linear Packaging System, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1075-1080, 10.1177/1461348418821204

[24] Yao, S. W., Cheng, Z. B., The Homotopy Perturbation Method for a Non-linear Oscillator with a Damping, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1110-1112, 10.1177/1461348419836344

[25] He, J. H., Jin, X., A Short Review on Analytical Methods for the Capillary Oscillator in a Nanoscale Deformable Tube, Mathematical Methods in the Applied Sciences, On-line first, [doi.org/10.](https://doi.org/10 1002/mma.6321, , 2020, 10.1002/mma.6321

[26] Zhang, J. J., et al., Some Analytical Methods for Singular Boundary Value Problem in a Fractal Space, Appl. Comput. Math., 18 (2019), 3, pp. 225-235

[27] Wang, K. L., Wei, C.-F., New Analytical Approach for Fractal K(p,q) Model, Fractals, On-line first, , 2021, 10.1142/S0218348X21501164

[28] Wang, K. L., He's Frequency Formulation for Fractal Nonlinear Oscillator Arising in a Microgravity Space, Numerical Methods for Partial Differential Equations, 37 (2020), 2, pp. 1374-1384, 10.1002/num.22584

[29] Wang, K. L., A New Fractal Transform Frequency Formulation for Fractal Nonlinear Oscillators, Fractals, On-line first, , 2020, 10.1142/S0218348X21500626

[30] Wang, K. L., Effect of Fangzhu's Nanoscale Surface Morphology on Water Collection, Mathematical Method in the Applied Sciences, On-line first, , 2020, 10.1002/mma.6569

[31] Wang, K. J., Wang, K. L., Variational Principles for Fractal Whitham-Broer-Kaup Equations in Shallow Water, Fractals, On-line first, , 2020, 10.1142/S0218348X21500286

[32] Wang, K. J., A New Fractional Nonlinear Singular Heat Conduction Model for the Human Head Considering the Effect of Febrifuge, Eur. Phys. J. Plus, 871 (2020), Nov., ID 871, 10.1140/epjp/s13360-020-00891-x

[33] Wang, K. J., Variational Principle and Approximate Solution for the Generalized Burgers-Huxley Equation with Fractal Derivative, Fractals, On-line first, , 2020, 10.1142/S0218348X21500444

[34] Wang, K. J., Wang, G .D., Solitary and Periodic Wave Solutions of the Generalized Fourth Order Boussinesq Equation via He's Variational Methods, Mathematical Methods in the Applied Sciences, 44 (2021), 7, pp. 5617-5625, 10.1002/mma.7135

[35] Wang, K. J., Wang, G. D., Variational Principle and Approximate, Solution for the Fractal Generalized Benjamin-Bona-Mahony-Burgers Equation in Fluid Mechanics, Fractals, On-line first, [doi.org/](https://doi.org/ , 2021, 10.1142/S0218348X21500754

[36] Wang, K. L., A Novel Perspective for the Fractal Schrödinger Equation, Fractals, On-line first, , 2020, 10.1142/S0218348X21500936

[37] Wang, K. L., Variational Principle for Nonlinear Oscillator Arising in a Fractal Nano/Microelectromechanical System, Mathematical Methods in the Applied Sciences, On-line first, [doi.org/](https://doi.org/ , 2020, 10.1002/mma.6726

PDF VERSION [DOWNLOAD]

Two-scale transform for 2-D fractal heat equation in a fractal space

© 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