Fractional Order Dual-Phase-Lag Model of Heat Conduction in a Composite Spherical Medium

Kukla, Stanisław; Siedlecka, Urszula; Ciesielski, Mariusz

doi:10.3390/ma15207251

Cited by 12 publications

(7 citation statements)

References 37 publications

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“…Also, a TFDPL model with a single order of fractional differentiation was considered by several scholars. In [21], such a model was used for describing the heat conduction in a multi-layered spherical medium with azimuthal symmetry. In [22], a similar model with temperature jump boundary condition was utilized for numerical simulation of heat transfer in transistors.…”

Section: Of 14mentioning

confidence: 99%

A Semi-explicit Algorithm for Parameters Estimation in a Time-Fractional Dual-Phase-Lag Heat Conduction Model

Lukashchuk

2024

Preprint

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This paper presents a new semi-explicit algorithm for parameters estimation in a time-fractional generalization of dual-phase-lag heat conduction model with the Caputo fractional derivatives. It is shown that this model can be derived from a general linear constitutive relation for the heat transfer by conduction when the heat conduction relaxation kernel contains the Mittag-Leffler function. The model can be used to describe heat conduction phenomena in a material with power-law memory. The proposed algorithm of parameters estimation is based on the time integral characteristics method. The explicit representations of the thermal diffusivity and the fractional analogues of the thermal relaxation time and the thermal retardation are obtained via a Laplace transform of the temperature field and utilized in the algorithm. An implicit relation is derived for the order of fractional differentiation. In the algorithm, this relation is resolved numerically. An example illustrates the proposed technique.

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Section: Of 14mentioning

confidence: 99%

A Semi-explicit Algorithm for Parameters Estimation in a Time-Fractional Dual-Phase-Lag Heat Conduction Model

Lukashchuk

2024

Preprint

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“…In this approach, the considered anomalous diffusion equation describes the phenomenon of heat flow in porous medium [ 11 , 21 , 22 ]. In Equation ( 1 ) we assume the following notations:

—temperature,

—spatial variable,

—time,

—specific heat,

—density,

—order of derivative and

is scaled heat conduction coefficient, where

is scale parameter.…”

Section: Anomalous Diffusion Modelmentioning

confidence: 99%

Comparison of Heuristic Algorithms in Identification of Parameters of Anomalous Diffusion Model Based on Measurements from Sensors

Brociek

Wajda²,

Słota

2023

Sensors

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In recent times, fractional calculus has gained popularity in various types of engineering applications. Very often, the mathematical model describing a given phenomenon consists of a differential equation with a fractional derivative. As numerous studies present, the use of the fractional derivative instead of the classical derivative allows for more accurate modeling of some processes. A numerical solution of anomalous heat conduction equation with Riemann-Liouville fractional derivative over space is presented in this paper. First, a differential scheme is provided to solve the direct problem. Then, the inverse problem is considered, which consists in identifying model parameters such as: thermal conductivity, order of derivative and heat transfer. Data on the basis of which the inverse problem is solved are the temperature values on the right boundary of the considered space. To solve the problem a functional describing the error of the solution is created. By determining the minimum of this functional, unknown parameters of the model are identified. In order to find a solution, selected heuristic algorithms are presented and compared. The following meta-heuristic algorithms are described and used in the paper: Ant Colony Optimization (ACO) for continous function, Butterfly Optimization Algorithm (BOA), Dynamic Butterfly Optimization Algorithm (DBOA) and Aquila Optimize (AO). The accuracy of the presented algorithms is illustrated by examples.

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“…Using the derivative of the Legendre function [15] and the boundary condition (15), we obtain the following equation are determined numerically, and some of them are given in Table 1. The functions  , m P   0, 1, 2, ... , m  create an orthogonal set of functions [16],…”

Section: Solution Of the Problem Under Considerationsmentioning

confidence: 99%

Modelling of the solar heating of a multi-layered spherical cone

Siedlecka

2023

J. Appl. Math. Comput. Mech.

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In this paper, the solar heating of a multi-layered spherical body with azimuthal symmetry is considered. The mathematical model is related to the determination of the steady state of the temperature distribution in the spherical cone consisting of concentric spherical layers. The solar heating is composed of two parts of the heat flux: direct and diffusion. Also, the simultaneous cooling of the cone by its outer surface (as convective heat flow to the environment) is taken into account. The proposed system of the partial differential equations supplemented by the adequate boundary conditions is solved in the analytical way by using, among others, the Legendre functions of the first kind. The sample results of temperature distribution in the cross-section of the cone with different polar angles are also presented.

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Fractional Order Dual-Phase-Lag Model of Heat Conduction in a Composite Spherical Medium

Cited by 12 publications

References 37 publications

A Semi-explicit Algorithm for Parameters Estimation in a Time-Fractional Dual-Phase-Lag Heat Conduction Model

A Semi-explicit Algorithm for Parameters Estimation in a Time-Fractional Dual-Phase-Lag Heat Conduction Model

Comparison of Heuristic Algorithms in Identification of Parameters of Anomalous Diffusion Model Based on Measurements from Sensors

Modelling of the solar heating of a multi-layered spherical cone

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