Thermal Science

International Scientific Journal

Find this paper on

Jian-She Sun

APPROXIMATE ANALYTIC SOLUTION FOR MULTI-DIMENSIONAL FRACTIONAL WAVE-LIKE EQUATION

ABSTRACT

The fractional power series method is used to solve 2- and 3-D fractional wave-like models with variable coefficients. The fractional derivatives are described in the Caputo sense. Two examples are considered to show the effectiveness and convenience of the method.

KEYWORDS

fractional power series, fractional series expansion method, fractal calculus, fractal derivative

PAPER SUBMITTED: 2019-04-04

PAPER REVISED: 2019-10-20

PAPER ACCEPTED: 2019-10-20

PUBLISHED ONLINE: 2020-06-21

DOI REFERENCE: https://doi.org/10.2298/TSCI2004645S

CITATION EXPORT: view in browser or download as text file

THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE No. 4, PAGES [2645 - 2652]

REFERENCES

[1] Caputo, M., Linear Models of Dissipation Whose Q is almost Frequency Independent Part II, Geophysical Journal International, 13 (1967), 5, pp. 529-539, 10.1111/j.1365-246x.1967.tb02303.x

[2] Li, C., et al., On Riemann-Liouville and Caputo Derivatives, Discrete Dynamics in Nature and Society, 2011 (2011), 1, pp. 309-323, 10.1155/2011/562494

[3] Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Academic Press, New York, USA, 2015, 10.1016/c2014-0-04768-5

[4] Yan, J. P., Li, C.P., On Chaos Synchronization of Fractional Differential Equations, Chaos, Solitons Fractals, 32 (2007), 2, pp. 725-735, 10.1016/j.chaos.2005.11.062

[5] Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006, 10.1016/s0304-0208(06)x8001-5

[6] Jumarie, G., Table of Some Basic Fractional Calculus Formulae Derived from a Modified Riemann-Liouville Derivative for Non-Differentiable Functions, Applied Mathematics Letters, 22 (2009), 3, pp. 378-385, 10.1016/j.aml.2008.06.003

[7] Molliq, Y. R., et al., Variational Iteration Method for Fractional Heat- and Wave-Like Equations, Nonlinear Analysis: Real World Applications, 10 (2009), 3, pp. 1854-1869, 10.1016/j.nonrwa.2008.02.026

[8] Momani, S., Analytical Approximate Solution for Fractional Heat-Like and Wave-Like Equations with Variable Coefficients Using the Decomposition Method, Applied Mathematics and Computation, 165 (2005), 2, pp. 459-472, 10.1016/j.amc.2004.06.025

[9] Al-Hayani,W., Daftardar-Jafari Method for Fractional Heat-Like and Wave-Like Equations with Variable Coefficients, Applied Mathematics, 8 (2017), 2, pp. 215-228, 10.4236/am.2017.82018

[10] Andrezei, H., Multi-Dimensional Solutions of Space-Time-Fractional Diffusion Equations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 458 (2002), 2018, pp. 429-450, 10.1098/rspa.2001.0893

[11] El-Ajou, A., et al., New Results on Fractional Power Series: Theories and Applications, Entropy, 15 (2013), 12, pp. 5305-5323, 10.3390/e15125305

[12] Shou, D. H., He, J. H., Beyond Adomian Methods: The Variational Iteration Method for Solving Heat-Like and Wave-Like Equations with Variable Coefficients, Physics Letters A, 73 (2007), 1, pp. 1-5, 10.1016/j.physleta.2007.07.011

[13] Adomian, G., A Review of the Decomposition Method in Applied Mathematics, Journal of mathematical analysis and applications, 135 (1988), 2, pp. 501-544, 10.1016/0022-247x(88)90170-9

[14] Podlubny, I., Fractional Differential Equations, Academic Press New York, NY, USA, 1999, 10.3390/books978-3-7258-4742-6

[15] He, J. H., Ji, F. Y., Taylor Series Solution for Lane-Emden Equation, Journal of Mathematical Chemistry, 57 (2019), 8, pp. 1932-1934, 10.1007/s10910-019-01048-7

[16] He, J. H., The Simplest Approach to Nonlinear Oscillators, Results in Physics, 15 (2019), 102546, 10.1016/j.rinp.2019.102546

[17] Yang, A. M., et al., Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets, Abstract and Applied Analysis, 2013 (2013), ID 351057, 10.1155/2013/351057

[18] Li, Z. B., Zhu, W. H., Fractional Series Expansion Method for Fractional Differential Equations, International Journal of Numerical Methods for Heat & Fluid Flow, 25 (2015), 7, pp. 1525-1530, 10.1108/hff-05-2014-0160

[19] He, J. H., et al., A New Fractional Derivative and Its Application to Explanation of Polar Bear Hairs, Journal of King Saud Universe Science, 28 (2016), 2, pp. 190-192, 10.1016/j.jksus.2015.03.004

[20] He, J. H., Li, Z. B., A Fractional Model for Dye Removal, Journal of King Saud Universe Science, 28 (2016), 1, pp. 14-16, 10.1016/j.jksus.2015.07.001

[21] Liu, H. Y., et al., Fractional Calculus for Nanoscale Flow and Heat Transfer, International Journal of Numerical Methods for Heat & Fluid Flow, 24 (2014), 6, pp. 1227-1250, 10.1108/hff-07-2013-0240

[22] Liu, H. Y., et al., A Fractional Nonlinear System for Release Oscillation of Silver Ions from Hollow Fibers, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 1, pp. 88-92, 10.1177/1461348418814122

[23] Ren, Z. F., et al., He's Multiple Scales Method for Nonlinear Vibrations, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1708-1712, 10.1177/1461348419861450

[24] Yang, J. J., Wang, S. Q., An Improved Homotopy Perturbation Method for Solving Local Fractional Non-linear Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 918-927, 10.1177/1461348418820676

[25] 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

[26] 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 Chemistry, 854 (2019), 113565, 10.1016/j.jelechem.2019.113565

[27] Wang, Y., et al., A Variational Formulation for Anisotropic Wave Traveling in a Porous Medium, Fractals, 27 (2019), 4, 1950047, 10.1142/s0218348x19500476

[28] Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, ID 1950134, 10.1142/s0218348x19501342

[29] Wang, Q. L., et al., Fractal Calculus and Its Application to Explanation of Biomechanism of Polar Bear Hairs, Fractals, 26 (2018), 6, ID 1850086, 10.1142/s0218348x1850086x

[30] Wang, Y., Deng, Q., Fractal Derivative Model For Tsunami Travelling, Fractals, 27 (2019), 1, 1950017, 10.1142/s0218348x19500178

[31] Sun, J. S., Analytical Approximate Solutions Of (N+1)-Dimensional Fractal Harry Dym Equations, Fractals, 26 (2018), 6, ID 1850094, 10.1142/s0218348x18500949

[32] Sun, J. S., Approximate Analytic Solutions of Multi-Dimensional Fractional Heat-Like Models with Variable Coefficients, Thermal Science, 23 (2019), 6B, pp. 3725-3729, 10.2298/tsci180612256s

PDF VERSION [DOWNLOAD]

Approximate analytic solution for multi-dimensional fractional wave-like equation

© 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