International Scientific Journal

Authors of this Paper

External Links


In order to solve the local fractional differential equations, we couple the fractional residual method with the Adomian decomposition method via the local fractional calculus operator. Several examples are given to illustrate the solution process and the reliability of the method.
PAPER REVISED: 2021-10-01
PAPER ACCEPTED: 2021-10-01
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2022, VOLUME 26, ISSUE Issue 3, PAGES [2667 - 2675]
  1. Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006
  2. He, J. H., When Mathematics Meets Thermal Science, the Simpler is the Better, Thermal Science, 25 (2021), 3, pp. 2039-2042
  3. He, J. H., Seeing with a Single Scale is Always Unbelieving: From Magic to Two-Scale Fractal, Thermal Science, 25 (2021), 2, pp. 1217-1219
  4. He, J. H., A Tutorial Review on Fractal Space Time and Fractional Calculus, Int. J. Theor. Phys., 53 (2014), June, pp. 3698-718
  5. He, J. H., Fractal Calculus and Its Geometrical Explanation, Result in Physics, 10 (2018), Sept., pp. 272-276
  6. Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Academic Press, Hong Kong, 2015
  7. He, J.-H., et al., Periodic Property and Instability of a Rotating Pendulum System, Axioms, 10 (2021), 3, 10030191
  8. Feng, G. Q., He's Frequency Formula to Fractal Undamped Duffing Equation, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 4, pp. 1671-1676
  9. Yang, X. J. Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Lim-ited, Hong Kong, China, 2011
  10. Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
  11. Li, X. X., He, J. H., Along the Evolution Process: Kleiber's 3/4 Law Makes Way for Rubner's Surface Law: A Fractal Approach, Fractals, 27 (2019), 2, 1950015
  12. Tian, D, et al., Hall-Petch Effect and Inverse Hall-Petch Effect: A Fractal Unification, Fractals, 26 (2018), 6, 1850083
  13. Baleanu, D., et al., A Modified Fractional Variational Iteration Method for Solving Non-linear Gas Dy-namic and Coupled KdV Equations Involving Local Fractional Operator, Thermal Science, 22 (2018), Suppl. 1, pp. S165-S175
  14. Anjum, N., He, J. H., Analysis of Non-linear Vibration of Nano/Microelectromechanical System Switch Induced by Electromagnetic Force Under Zero Initial Conditions, Alexandria Engineering Journal, 59 (2020), 6, pp. 4343-4352
  15. He, J. H., et al., Approximate Periodic Solutions to Microelectromechanical System Oscillator Subject to Magnetostatic Excitation, Mathematical Methods in Applied Sciences, On-line first,, 2020
  16. Yang, Y. J., The Fractional Residual Method for Solving the Local Fractional Differential Equations, Thermal Science, 24 (2020), 4, pp. 2535-2542
  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
  18. He, J.-H., et al., Homotopy Perturbation Method for the Fractal Toda Oscillator, Fractal Fract., 5 (2021), 5030093
  19. Li, X. X., He, C. H., Homotopy Perturbation Method Coupled with the Enhanced Perturbation Method, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1399-1403
  20. He, J. H., El-Dib, Y. O., Periodic Property of the Time-Fractional Kundu-Mukherjee-Naskar Equation, Results in Physics, 19 (2020), Dec., 103345
  21. Yang. Y. J. Wang, S. Q., Local Fractional Fourier Series Method for Solving Nonlinear Equations with Local Fractional Operators, Mathematical Problems in Engineering, 2015 (2015), ID104481905
  22. Yang, Y. J., et al., Local fractional Fourier method for solving modified diffusion equations with local fractional derivative, Journal of Non-linear Sciences and Applications, 9 (2016), 12, pp. 6153-6160
  23. Zhang, Y., Solving Initial-Boundary Value Problems for Local Fractional Differential Equation by Local Fractional Fourier Series Method, Abstract and Applied Analysis, 2014 (2014), ID 912464
  24. Yang, Y. J., et al., The Yang Laplace Transform - DJ Iteration Method for Solving the Local Fractional Differential Equation, J. Nonlinear Sci. Appl., 10 (2017), 6, pp. 3023-3029
  25. Habib, S., et al., Study of Non-linear Hirota-Satsuma Coupled KdV and Coupled mKdV System with Time Fractional Derivative, Fractals, 29 (2021), 5, 2150108
  26. He, J. H., et al., Non-linear Instability of Two Streaming-Superposed Magnetic Reiner-Rivlin Fluids by He-Laplace Method, Journal of Electroanalytical Chemistry, 895 (2021), Aug., 115388
  27. Anjum, N., et al., Two-Scale Fractal Theory for the Population Dynamics, Fractals, 29 (2021), 7, 21501826
  28. He, J. H., et al., A Fractal Modification of Chen-Lee-Liu Equation and its Fractal Variational Principle, International Journal of Modern Physics B, 35 (2021), 21, 2150214
  29. Yang, Y. J., A New Method Solving Local Fractional Differential Equations in Heat Transfer, Thermal Science, 23 (2019), 3, pp. 1663-1669
  30. Yang, X. J., et al., A New Numerical Technique for Solving the Local Frac-Tional Diffusion Equation: Two-Dimensional Extended Differential Transform Approach, Applied Mathematics and Computation 274 (2016), Feb., pp. 143-151
  31. Yang, X. J., Srivastava, H. M., An Asymptotic Perturbation Solution for a Linear Oscillator of Free Damped Vibrations in Fractal Medium Described by Local Fractional Derivatives, Communications in Non-linear Science and Numerical Simulation, 29 (2015), 1-3, pp. 499-504
  32. Gao, F., et al., A Coupling Method Involving the Sumudu Transform and the Variational Iteration Method for a Class of Local Fractional Diffusion Equations, Non-linear Sci. Appl., 9 (2016), 11, pp. 5830-5835
  33. Tian, Y., Liu, J., Direct Algebraic Method for Solving Fractional Fokas Equation, Thermal Science, 25 (2021), 3, pp. 2235-2244
  34. Tian, Y., Wan, J. X., Exact Solutions of Space-Time Fractional 2+1 Dimensional Breaking Soliton Equa-tion, Thermal Science, 25 (2021), 2, pp. 1229-1235
  35. Tian, Y., Liu, J., A Modified Exp-Function Method for Fractional Partial Differential Equations, Thermal Science, 25 (2021), 2, pp. 1237-1241
  36. Wang, K. J., On New Abundant Exact Traveling Wave Solutions to the Local Fractional Gardner Equation Defined on Cantor Sets, Mathematical Methods in the Applied Sciences, 45 (2021), 4, pp. 1904-1915
  37. Wang, K. J., Generalized Variational Principle and Periodic Wave Solution to the Modified Equal Width-Burgers Equation in Non-linear Dispersion Media, Physics Letters A, 419 (2021), Dec., 127723
  38. Wang, K. J., Zhang, P. L., Investigation of the Periodic Solution of the Time-Space Fractional Sasa-Satsuma Equation Arising in the Monomode Optical Fibers, EPL, 137 (2021), 6, 62001
  39. He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199
  40. Han, C., et al., Numerical Solutions of Space Fractional Variable-Coefficient KdV-Modified KdV Equa-tion by Fourier Spectral Method,Fractals, 29 (2021), 8, 2150246
  41. Dan, D. D., et al.,Using Piecewise Reproducing Kernel Method and Legendre Polynomial for Solving a Class of the Time Variable Fractional Order Advection-Reaction-Diffusion Equation, Thermal Science, 25 (2021), 2B, pp. 1261-1268
  42. Duan, J. S., Rach, R., A New Modification of the Adomian Decomposition Method for Solving Boundary Value Problems for Higher Order Nonlinear Differential Equations, Applied Mathematics and Computa-tion, 218 (2011), 8, pp. 4090-4118

© 2024 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