## THERMAL SCIENCE

International Scientific Journal

### A MULTIPOLE FAST ASYMPTOTIC ALGORITHM FOR A CLASS OF EQUATIONS BASED ON THE FLOW FUNCTION METHOD WITH FRACTIONAL ORDER LAPLACE TRANSFORM

**ABSTRACT**

As a mature and reliable method, this study is based on the flow function method for mathematical modeling and establishes a class of mathematical models that are approximately realistic, flexible, and easy to calculate. According to the characteristics of fractional order calculus, the initial boundary conditions are modified and optimized to reduce the model error of this class of equations. According to the minimum energy principle and linearized integral calculation method, the multi-field multi-parameter non-linear coupling problem in the calculation process is solved, and the rapid calculation of the initial boundary model is realized. The accuracy of the model is tested by numerical simulation and simulation validation of different processes. A reliable theoretical and technical support is provided for the calculation of this type of equations.

**KEYWORDS**

PAPER SUBMITTED: 2022-08-05

PAPER REVISED: 2023-07-02

PAPER ACCEPTED: 2023-08-09

PUBLISHED ONLINE: 2024-05-18

**THERMAL SCIENCE** YEAR

**2024**, VOLUME

**28**, ISSUE

**Issue 3**, PAGES [2361 - 2370]

- Saichev, A. I., Zaslavsky, G. M., Fractional Kinetic Equations: Solutions and Applications, Chaos, 7 (1997), 1, pp. 753-764
- Alzahrani, S. S., Khaliq, A. Q. M., Fourier Spectral Exponential Time Differencing Methods for Multi-Dimensional Space-Fractional Reaction-Diffusion Equations, Journal of Computational and Applied Mathematics, 27 (2019), 4, pp. 423-436
- Feynman, R. P., Hibbs, A. R., Quantum Mechanics and Path Integral, McGraw-Hill, New York, USA, 1965
- Laskin, N. Fractional Quantum Mechanics and Levy Path Integrals, Phys. Lett. A, 268 (2010), 3, pp. 298-305
- Gray, P., Scott, S. K., Sustained Oscillations and other Exotic Patterns of Behavior in Isothermal Reactions, J. Phys. Chem., 89 (1985), 7, pp. 22-32
- Yang, X. J., et al., A New General Fractional-Order Derivative with Rabotnov Fractional-Exponential Kernel Applied to Model the Anomalous Heat Transfer, Thermal Science, 23 (2019), 3A, pp. 1677-1681
- Yang, X. J., et al., Fundamental Solutions of the General Fractional-Order Diffusion Equations, Math. Methods Appl. Sci. 41 (2017), 18, pp. 9312-9320
- Yang, X. J., et al., New Mathematical Models in Anomalous Viscoelasticity from the Derivative with Respect to Another Function View Point, Thermal Science, 23 (2016), 3A, pp. 1555-1561
- He, J.-H., The Simpler, the Better: Analytical Methods for Non-linear Oscillators and Fractional Oscillators, Journal of Low Frequency Noise Vibration and Active Control, 38 (2019), 3-4, pp. 1252-1260
- He, J.-H., Ji, F.-Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Science, 23 (2019), 4, pp. 2131-2133
- 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), 12, pp. 113-121
- Magin, R. L., et al., Anomalous Diffusion Expressed through Fractional Order Differential Operators in the Bloch-Torrey Equation, J. Magn. Reson., 190 (2008), 7, pp. 255-270
- Kilbas, A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, Boston, Mass., USA, 2006
- Zhao, X., et al., Adaptive Finite Element Method for Fractional Differential Equations Using Hierarchical Matrices, Comput. Methods. Appl. Mech. Engrg., 325 (2017), 1, pp. 56-76
- Li, D., et al., Analysis of L1-Galerkin FEMs for Time-Fractional Non-Linear Parabolic Problems, Commun. Comput. Phys., 24 (2018), 1, pp. 86-103
- Jannelli, A., et al., Exact and Numerical Solutions of Time-Fractional Advection Diffusion Equation with a Non-Linear Source Term by Means of the Lie Symmetries, Non-linear Dyn., 92 (2018), 3, pp. 543-555