## THERMAL SCIENCE

International Scientific Journal

### TWO-DIMENSIONAL HEAT TRANSFER WITH MEMORY PROPERTY IN A FRACTAL SPACE

**ABSTRACT**

This paper considers a temperature-dependent thermal conductivity with memory property in a fractal space. The two-scale fractal derivative is adopted to model the temperature field in the spatial dimensions, and Caputo fractional derivative is used to describe its memory property. The variational iteration method is employed to solve the mixed model with great success. This paper offers a new window for studying intractable problems arising in porous media or unsmooth boundaries.

**KEYWORDS**

PAPER SUBMITTED: 2022-01-22

PAPER REVISED: 2023-05-21

PAPER ACCEPTED: 2023-05-21

PUBLISHED ONLINE: 2024-05-18

**THERMAL SCIENCE** YEAR

**2024**, VOLUME

**28**, ISSUE

**Issue 3**, PAGES [1993 - 1998]

- Mendes, E. M. A. M., et al., Numerical Solution of Caputo Fractional Differential Equations with Infinity Memory Effect at Initial Condition, Communications in Non-linear Science and Numerical Simulation, 69 (2019), Apr., pp. 237-247
- Gundogdu, H., Gozukizil, O. F., On the Approximate Numerical Solutions of Fractional Heat Equation with Heat Source and Heat Loss, Thermal Science, 26 (2022), 5A, pp. 3773-3786
- Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
- Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Academic Press, Pittsburgh, Penn., USA, 2015
- Wang, K. L., et al., Physical Insight of Local Fractional Calculus and Its Application to Fractional KdV-Burgers-Kuramoto Equation, Fractals, 27 (2019), 7, 1950122
- Kuo, P. H., et al., Novel Fractional-Order Convolutional Neural Network Based Chatter Diagnosis Approach in Turning Process with Chaos Error Mapping, Non-linear Dynamics, 111 (2023), 8, pp. 7547-7564
- He, J.-H., A Tutorial Review on Fractal Space-Time and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
- He, J.-H., Fractal Calculus and Its Geometrical Explanation, Results in Physics, 10 (2018), Sept., pp. 272-276
- Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, 1950134
- He, J.-H., et al., Solitary Waves Travelling Along an Unsmooth Boundary, Results in Physics, 24 (2021), 104104
- He, C. H., Liu, C., Fractal Approach to the Fluidity of a Cement Mortar, Non-linear Engineering, 11 (2022), 1, pp. 1-5
- He, C. H., et al., A Fractal Model for the Internal Temperature Response of a Porous Concrete, Applied and Computational Mathematics, 21 (2022), 1, pp. 71-77
- He, C. H., Liu, C., Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete's Strength, Facta Universitatis Series: Mechanical Engineering, 21 (2023), 1, pp. 137-150
- He, J.-H., et al., Pull-in Stability of a Fractal System and Its Pull-in Plateau, Fractals, 30 (2022), 9, 2250185
- He, J.-H.; et al. Periodic Property and Instability of a Rotating Pendulum System. Axioms, 10 (2021), 3, 191
- Tian, D., et al., Fractal N/MEMS: from Pull-in Instability to Pull-in Stability, Fractals, 29 (2021), 2, 2150030
- Tian, D., He, C. H., A Fractal Micro-Electromechanical System and Its Pull-in Stability, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, pp. 1380-1386
- He, C. H., A Variational Principle for a Fractal Nano/Microelectromechanical (N/MEMS) System, International Journal of Numerical Methods for Heat & Fluid Flow, 33 (2023), 1, pp. 351-359
- He, J.-H., El-Dib, Y. O., A Tutorial Introduction to the Two-Scale Fractal Calculus and Its Application to the Fractal Zhiber-Shabat Oscillator, Fractals, 29 (2021), 8, 2150268
- Kochubei, A. N., et al., On Fractional Heat Equation, Fractional Calculus and Applied Analysis, 24 (2021), 1, pp. 73-87
- Povstenko, Y. Z., Fractional Heat Conduction Equation and Associated Thermal Stress, Journal of Thermal Stresses, 28 (2005), 1, pp. 83-102
- Liu, F. J., et al., Thermal Oscillation Arising in a Heat Shock of a Porous Hierarchy and Its Application, Facta Universitatis Series: Mechanical Engineering, 20 (2022), 3, pp. 633-645
- He, C. H., et al., Taylor Series Solution for Fractal Bratu-Type Equation Arising in Electrospinning Process, Fractals, 28 (2020), 1, 2050011
- Kaur, P., Singh, S., Convective-Radiative Moving Porous fin with Temperature-Dependent Thermal Conductivity, Heat Transfer Coefficient and Wavelength-Dependent Surface Emissivity, Multidiscipline Modeling in Materials and Structures, 19 (2023), 2, pp. 176-196
- Verhoest, N., Troch, P. A., Some Analytical Solutions of the Linearized Boussinesq Equation with Recharge for a Sloping Aquifer, Water Resources Research, 36 (2000), 3, pp.793-800
- Abdou, M. A., et al., New Application of Exp-Function Method for Improved Boussinesq Equation, Physics Letters A, 369 (2007), 5, pp. 469-475
- Abassy, T. A., et al., Modified Variational Iteration Method for Boussinesq Equation, Computers and Mathematics with Applications, 54 (2007), 7, pp. 955-965
- He, J.-H., A Short Remark on Fractional Variational Iteration Method, Physics Letters A, 375 (2011), 38, pp. 3362-3364
- Wang, S. Q., He, J. H., Variational Iteration Method for Solving Integro-Differential Equations, Physics letters A, 367 (2007), 3, pp. 188-191
- Deng, S. X., Ge, X. X., The Variational Iteration Method for Whitham-Broer-Kaup System with Local Fractional Derivatives, Thermal Science, 26 (2022), 3B, pp. 2419-2426
- Wang, S. Q., A Variational Approach to Non-linear Two-Point Boundary Value Problems, Computers & Mathematics with Applications, 58 (2009), 11, pp. 2452-2455
- Sun, J. S., Approximate Analytic Solution of the Fractal Fisher's Equation via Local Fractional Variational Iteration Method, Thermal Science, 26 (2022), 3B, pp. 2699-2705