## THERMAL SCIENCE

International Scientific Journal

### VARIATIONAL APPROACH FOR THE FRACTIONAL EXOTHERMIC REACTIONS MODEL WITH CONSTANT HEAT SOURCE IN POROUS MEDIUM

**ABSTRACT**

In this paper, a new fractional exothermic reactions model with constant heat source in porous media considering the memory effect is proposed. Applying the fractional complex transform, the fractional model is converted into its partner. Then the variational principle of the problem is successfully established. Based on the obtained variational principle, the Ritz method is used to seek the solution of the fractional model. Finally, the correctness and effectiveness of the proposed method are illustrated by the numerical results with the aid of the MATLAB. The obtained results show that the proposed method is easy but effective, and is expected to shed a bright light on practical applications of fractional calculus.

**KEYWORDS**

PAPER SUBMITTED: 2022-09-22

PAPER REVISED: 2022-11-04

PAPER ACCEPTED: 2022-11-07

PUBLISHED ONLINE: 2023-01-07

**THERMAL SCIENCE** YEAR

**2023**, VOLUME

**27**, ISSUE

**Issue 4**, PAGES [2879 - 2885]

- Pochai, N., Jaisaardsuetrong, J., A numerical treatment of an exothermic reactions model with constant heat source in a porous medium using finite difference method, Advanced Studies in Biology, 4 (2012) 6, pp. 287-296
- Sharma, R. P., Jain, M., Kumar, D., Analytical solution of exothermic reactions model with constant heat source and porous medium, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 90 (2020) 2, pp. 239-243
- Mabood, F., Pochai, N., Optimal homotopy asymptotic solution for exothermic reactions model with constant heat source in a porous medium, Advances in Mathematical Physics, 2015 (2015), pp. 825683
- Subramanian, S., Balakotaiah, V., Convective instabilities induced by exothermic reactions occurring in a porous medium, Physics of Fluids, 6 (1994), 9, pp. 2907-2922
- Liu, H., Liu, C., Bai, G., et al. Influence of pore defects on the hardened properties of 3D printed concrete with coarse aggregate, Additive Manufacturing, 55 (2022), pp. 102843
- Wang, K. J., A new fractional nonlinear singular heat conduction model for the human head considering the effect of febrifuge, Eur. Phys. J. Plus, 135 (2020), pp. 871
- Kumar, D., Singh, J., Al Qurashi, M., Baleanu, D. A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying, Advances in Difference Equations, 278 (2019), pp. 1-19
- Baleanu, D., Mohammadi, H., Rezapour, S., Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative, Advances in Difference Equations, 2020 (2020), pp. 1-17
- Wang, K. J., Periodic solution of the time-space fractional complex nonlinear Fokas-Lenells equation by an ancient Chinese algorithm, Optik, 243 (2021), pp. 167461
- Wang, K. J., Investigation to the local fractional Fokas system on Cantor set by a novel technology, Fractals, 30 (2022), 6, pp. 2250112
- Goswami A, Singh J, Kumar D, et al. An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A, 524 (2019), pp. 563-575
- Wang, K. J., Research on the nonlinear vibration of carbon nanotube embedded in fractal medium, Fractals, 30 (2020) 1, pp. 2250016
- He, J. H., Jiao, M. L., He, C. H., Homotopy perturbation method for fractal Duffing oscillator with arbitrary conditions, Fractals, 2022, doi.org/10.1142/S0218348X22501651
- He, J. H., Kou, S. J., He, C. H., Fractal oscillation and its frequency-amplitude property. Fractals, 29 (2021) 4, pp. 2150105
- Wang, K. J., On a High-pass filter described by local fractional derivative, Fractals, 28 (2020), 3, pp. 2050031
- Yang, X. J., Machado J A T, Cattani C, et al. On a fractal LC-electric circuit modeled by local fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), pp. 200-206
- Wang K J, Shi F, Wang G D. Periodic wave structure of the fractal generalized fourth order Boussinesq equation travelling along the non-smooth boundary, Fractals, 30 (2022), 9, pp. 2250168
- He, J. H., Hou, W. F., He, C. H., Variational approach to fractal solitary waves, Fractals, 29 (2021), 7, pp. 2150199
- Wang, K. J., Bäcklund transformation and diverse exact explicit solutions of the fractal combined KdV-mKdV equation, Fractals, 30 (2022), 9, pp. 2250189
- He, J. H., Qie, N., He, C. H., Solitary waves travelling along an unsmooth boundary, Results in Physics, 24 (2021), pp. 104104
- Wang K J，A fractal modification of the unsteady korteweg-de vries model and its generalized fractal variational principle and diverse exact solutions, Fractals,30 (2022), 9, pp. 2250192
- Wang, K. J., Wang, G. D., Solitary waves of the fractal regularized long wave equation travelling along an unsmooth boundary, Fractals, 30 (2022), 1, pp. 2250008
- Khater, M. M. A., Attia, R. A. M., Abdel-Aty, A. H., Abundant analytical and numerical solutions of the fractional microbiological densities model in bacteria cell as a result of diffusion mechanisms. Chaos, Solitons & Fractals, 136 (2020), pp. 109824
- Wang, K. L., A novel perspective to the local fractional bidirectional wave model on Cantor sets, Fractals, 30 (2022), 6, pp. 2250107
- Sun, W. B., Liu, Q., Hadamard type local fractional integral inequalities for generalized harmonically convex functions and applications, Math Meth Appl Sci. 43 (2020), 9, pp. 5776-5787
- Wang, K. J, Si, J., On the non-differentiable exact solutions of the (2+1)-dimensional local fractional breaking soliton equation on Cantor sets, Mathematical Methods in the Applied Sciences, 2022, doi.org/10.1002/mma.8588.
- Liu, J. G., Yang, X. J., Feng, Y. Y., Iqbal, M., On group analysis to the time fractional nonlinear wave equation, International Journal of Mathematics, 31 (2020), 4, pp. 20500299
- Wang, K. L., A novel perspective to the local fractional Zakharov-Kuznetsov-modified equal width dynamical model on Cantor sets, Mathematical Methods in the Applied Sciences, 2022, doi.org/10.1002/mma.8533
- Kumar, D., Singh, J., Baleanu, D., A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Thermal Science, 22 (2018), 6B, pp. 2791-2802
- He, J. H., A Tutorial Review on Fractal Spacetime 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), pp. 272-276
- He, J. H., Ji, F.Y. Two-scale mathematics and fractional calculus for thermodynamics. Thermal science. 23 (2019) 4, pp. 2131-2134
- Ain, Q. T., He, J. H., On two-scale dimension and its applications, Thermal Science, 23 (2019), 3, pp. 1707-1712
- Wang, S. Q., A variational approach to nonlinear two-point boundary value problems, Computers & Mathematics with Applications, 58 (2009), 11, pp. 2452-2455
- Wang, S. Q., Variational iteration method for solving integro-differential equations, Physics letters A, 367 (2007), 3, pp. 188-191
- Liu F J, Li Z B, Zhang S, et al. He's fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy. Thermal Science, 2015, 19(4): 1155-1159.
- He, J, H., Elagan, S. K., Li, Z. B., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Physics letters A, 376 (2012), 4, 257-259
- Ain, Q. T., He, J. H., Anjum, N., The fractional complex transform: A novel approach to the time-fractional Schrödinger equation, Fractals, 28 (2020), 7, pp. 2050141
- He, J. H., Qie, N., He, C., On a strong minimum condition of a fractal variational principle, Applied Mathematics Letters, 119 (2021), pp. 107199
- He, C. H., A variational principle for a fractal nano/microelectromechanical (N/MEMS) system, International Journal of Numerical Methods for Heat & Fluid Flow, 2022, DOI10.1108/HFF-03-2022-0191
- Wang, K. J., Variational principle and diverse wave structures of the modified Benjamin-Bona-Mahony equation arising in the optical illusions field, Axioms, 11 (2022), 9, pp. 445
- Wang, K. J., Wang, G. D., Variational theory and new abundant solutions to the (1+2)-dimensional chiral nonlinear Schrödinger equation in optics, Physics Letters A, 412 (2021), 7, pp. 127588
- He, J. H., Sun, C., A variational principle for a thin film equation, Journal of Mathematical Chemistry, 57 (2019), 9, pp. 2075-2081
- He, J. H., Lagrange crisis and generalized variational principle for 3D unsteady flow, International Journal of Numerical Methods for Heat & Fluid Flow, 30 (3) (2019) 1189-1196
- Wang, K. J., Variational principle and diverse wave structures of the modified Benjamin-Bona-Mahony equation arising in the optical illusions field, Axioms, 11 (2022), 9, pp. 445
- He, J. H., Asymptotic Methods for Solitary Solutions and Compactons, Abstract and Applied Analysis, 2012 (2012), pp. 916793
- He, J. H., Variational approach for nonlinear oscillators, Chaos, Solitons & Fractals, 34 (2007) 5, pp. 1430-1439
- Liu, H. Y., Si, N., He, J. H., A short remark on Chie's variational principle of maximum power losses for viscous fluids, International Journal of Numerical Methods for Heat & Fluid Flow, 26 (2016), 3, pp. 694-697