## THERMAL SCIENCE

International Scientific Journal

### APPROXIMATE ANALYTICAL SOLUTION OF TIME-FRACTIONAL NON-LINEAR HEAT EQUATION VIA FRACTIONAL POWER SERIES METHOD

**ABSTRACT**

A time-fractional heat equation arising in a quiescent medium is established, and its approximate analytical solution is obtained by the fractional power series method. The results show that the method performs extremely well in terms of efficiency and simplicity.

**KEYWORDS**

PAPER SUBMITTED: 2020-10-05

PAPER REVISED: 2021-10-02

PAPER ACCEPTED: 2021-10-02

PUBLISHED ONLINE: 2022-07-16

**THERMAL SCIENCE** YEAR

**2022**, VOLUME

**26**, ISSUE

**Issue 3**, PAGES [2637 - 2643]

- El-Ajou, A., et al., New Results on Fractional Power Series: Theories and Applications, Entropy, 15 (2013), 12, pp. 5305-5323
- Rincon, M. A, et al., A Non-Linear Heat Equation with Temperature Dependent Parameters, Mathematical Physics Electronic Journal, 5 (2004), 58, pp. 601-615
- Leszczyński, H., On a Non-Linear Heat Equation with Functional Dependence, Applicable Analysis, 74 (2000), 3-4, pp. 233-251
- Borukhov, V. T., Zayats, G. M., Identification of a Time-Dependent Source Term in Non-Linear Hyperbolic or Parabolic Heat Equation, International Journal of Heat and Mass Transfer, 91 (2015), Dec., pp. 1106-1113
- Deng, S. X., Ge, X. X., Fractional Fokker-Planck Equation in a Fractal Medium, Thermal Science, 24 (2020), 4, pp. 2589-2595
- Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, N. Y., USA, 2012
- He, J. H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- Li, Y., He, C. H., A Short Remark on Kalaawy's Variational Principle for Plasma, International Journal of Numerical Methods for Heat & Fluid Flow, 27 (2017), 10, pp. 2203-2206
- Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950134
- Wang, K. L, et al., Physical Insight of Local Fractional Calculus and Its Application to Fractional KdVBurgers-Kuramoto Equation, Fractals, 27 (2019), 7, 1950122
- Wang K J, Generalized Variational Principle and Periodic Wave Solution to the Modified Equal Width-Burgers Equation in Nonlinear Dispersion Media, Physics Letters A, 419 (2021), Dec., 127723
- 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 (2022), 6, 62001
- He, J. H., et al., Variational Approach to Fractal Solitary Waves, Fractals, 29 (2021), 7, 2150199
- 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
- Nadeem, M., He, J. H. He-Laplace Variational Iteration Method for Solving the Non-linear Equations Arising in Chemical Kinetics and Population Dynamics, Journal of Mathematical Chemistry, 59 (2021), 5, pp. 1234-1245
- Lu, F. J., An Analytical Approach to Fractional Bousinesq-Burges Equations, Thermal Science, 24 (2020), 4, pp. 2581-2588
- Yang, Y. J., The Local Fractional Variational Iteration Method a Promising Technology for Fractional Calculus, Thermal Science, 24 (2020), 4, pp. 2605-2614
- He, J. H., Maximal Thermo-Geometric Parameter in a Non-linear Heat Conduction Equation, Bulletin of the Malaysian Mathematical Sciences Society, 39 (2016), 2, pp. 605-608
- He, J. H., El-Dib, Y. O., Periodic Property of the Time-Fractional Kundu-Mukherjee-Naskar Equation , Results in Physics, 19 (2020), Dec., 103345
- He, J.‐H., et al., Homotopy Perturbation Method for the Fractal Toda Oscillator, Fractal and Fractional, 5 (2021), 93, 5030093
- Tian, Y., Liu, J., Direct Algebraic Method for Solving Fractional Fokas Equation, Thermal Science, 25 (2021), 3, pp. 2235-2244
- Tian, Y., Wan, J. X., Exact Solutions of Space-Time Fractional 2+1 Dimensional Breaking Soliton Equation, Thermal Science, 25 (2021), 2, pp. 1229-1235
- Tian, Y, Liu, J., A Modified Exp-Function Method for Fractional Partial Differential Equations, Thermal Science, 25 (2021), 2, pp. 1237-1241
- Han, C., et al., Numerical Solutions of Space Fractional Variable-Coefficient KdV-Modified KdV Equa-tion by Fourier Spectral Method，Fractals, 29 (2021), 8, 21502467
- 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
- Abu-Gdairi, R., et al., An Expansion Iterative Technique for Handling Fractional Differential Equations Using Fractional Power Series Scheme, Journal of Mathematics & Statistics, 11 (2015), 2, pp. 29-38
- Jena, R. M., Chakraverty, S., Residual Power Series Method for Solving Time-Fractional Model of Vi-bration Equation of Large Membranes, Journal of Applied and Computational Mechanics, 4 (2019), 5, pp. 603-615
- Lee, K., Continued Fractions for Linear Fractional Transformations of Power Series, Finite Fields and Thr Applications, 1 (2004), 11, pp. 45-55
- Alquran, M., et al., Analytical Solutions of Fractional Population Diffusion Model: Residual Power Se-ries, Non-linear Studies, 1 (2015), 22, pp. 31-39
- Jafari, H., Jassim, H. K., Local Fractional Series Expansion Method for Solving Laplace and Schroding-er Equations on Cantor Sets within Local Fractional Operators, International Journal of Mathematics and Computer Research, 11 (2014), 2, pp. 736-744
- Yang, Y. J.,The Fractional Residual Method for Solving the Local Fractional Differential Equations, Thermal Science, 24 (2020), 4, pp. 2535-2542
- He, J. H., Taylor Series Solution for a Third Order Boundary Value Problem Arising in Architectural Engineering, Ain Shams Engineering Journal, 11 (2020), 4, pp. 1411-1414