THERMAL SCIENCE

International Scientific Journal

Yong-Ju YangORCID

THE LOCAL FRACTIONAL VARIATIONAL ITERATION METHOD A PROMISING TECHNOLOGY FOR FRACTIONAL CALCULUS

ABSTRACT
In order to make the local variational iteration algorithm converge faster and more effective, the Sumudu transform is adopted and a proper initial solution is chosen. Some examples are given to show that the presented method is reliable, efficient and easy to implement from a computational viewpoint.
KEYWORDS
local fractional derivative, local fractional integral, variational iteration method, Sumudu transform
PAPER SUBMITTED: 2019-04-25
PAPER REVISED: 2019-10-25
PAPER ACCEPTED: 2019-10-25
PUBLISHED ONLINE: 2020-06-21
DOI REFERENCE: https://doi.org/10.2298/TSCI2004605Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE No. 4, PAGES [2605 - 2614]
REFERENCES
[1] Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999, 10.3390/books978-3-7258-4742-6
[2] Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006, 10.1016/s0304-0208(06)x8001-5
[3] Baleanu, D., et al., Fractional calculus models and numerical methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass., USA, 2012, 10.1142/10044#t=toc
[4] 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
[5] Wang, Y., et al., A Variational Formulation for Anisotropic Wave Traveling in a Porous Medium, Fractals, 27 (2019), 4, 1950047, 10.1142/s0218348x19500476
[6] Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, ID 1950134, 10.1142/s0218348x19501342
[7] He, J. H., A Short Remark on Fractional Variational Iteration Method, Physics Letters A, 375 (2011), 38, pp. 3362-3364, 10.1016/j.physleta.2011.07.033
[8] 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
[9] Yu, D. N., et al., Homotopy Perturbation Method With An Auxiliary Parameter For Nonlinear Oscillators, Journal of Low Frequency Noise, Vibration & Active Control, 38 (2019), 3-4, pp. 1540-1554, 10.1177/1461348418811028
[10] Ul Rahman, J., et al., He-Elzaki Method for Spatial Diffusion of Biological Population, Fractals, 27 (2019), 5, 1950069, 10.1142/s0218348x19500695
[11] Wang, Q. L., et al., Fractal Calculus and Its Application to Explanation of Biomechanism of Polar Bear hairs, Fractals, 26 (2018), 6, 1850086, 10.1142/s0218348x1850086x
[12] 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
[13] 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
[14] 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
[15] 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
[16] Wang, Q. L., et al., Fractal Calculus and its Application to Explanation of Biomechanism of Polar Bear Hairs, Fractals, 26 (2018), 6, ID 1850086
[17] Yang, Y. J., Wang, S. Q., An Improved Homotopy Perturbation Method for Solving Local Fractional Nonlinear Oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38 (2019), 3-4, pp. 918-927, 10.1177/1461348418820676
[18] Wang, Y., An, J. Y., Amplitude-frequency Relationship to a Fractional Duffing Oscillator Arising in Microphysics and Tsunami Motion, J. Low Freq. Noise V. A., 38 (2019), 3-4, pp. 1008-1012, 10.1177/1461348418795813
[19] 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
[20] Yang, X. J., Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Limited, Hong Kong, China 2011
[21] Yang, X. J, Advanced Local Fractional Calculus and Its Applications, World Science publisher, New York, USA, 2012
[22] Srivastava, H. M., et al., Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets, Abstract and Applied Analysis, 2014 (2014), ID 620529, 10.1155/2014/620529
[23] He, J. H., Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Me-dia, Computer Methods in Applied Mechanics and Engineering, 167 (1998), 1-2, pp. 57-68, 10.1016/s0045-7825(98)00108-x
[24] He, J. H., A Short Remark on Fractional Variational Iteration Method, Physics Letters A, 375 (2011), 38, pp. 3362-3364
[25] Yang, X. J., et al., A Local Fractional Variational Iteration Method for Laplace Equation Within Local Fractional Operators, Abstract and Applied Analysis, 2013 (2013), ID 202650, 10.1155/2013/202650
[26] Anjum, N., He, J. H., Laplace Transform: Making the Variational Iteration Method Easier, Applied Mathematics Letters, 92 (2019), June, pp. 134-138, 10.1016/j.aml.2019.01.016
[27] He, J. H., et al., Geometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus, Physics Letters A, 376 (2012), 4, pp. 257-259, 10.1016/j.physleta.2011.11.030
[28] He, J. H., Some Asymptotic Methods for Strongly Nonlinear Equations, International Journal of Modern Physics B, 20 (2006), 10, pp. 1141-1199, 10.1142/s0217979206033796
[29] 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
[30] Ain, Q. T., He, J. H., On Two-Scale Dimension and Its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712, 10.2298/tsci190408138a
[31] 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
[32] He, J. H., The Simplest Approach to Nonlinear Oscillators, Results in Physics, 15 (2019), Dec., ID 102546, 10.1016/j.rinp.2019.102546
PDF VERSION [DOWNLOAD]
The local fractional variational iteration method a promising technology for fractional calculus

© 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