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Journal articles on the topic 'Fractional order heat equation'

Author: Grafiati

Published: 5 June 2025

Last updated: 16 July 2025

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1

Kosmakova, M. T., S. A. Iskakov, and L. Zh Kasymova. "To solving the fractionally loaded heat equation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 101, no. 1 (2021): 65–77. http://dx.doi.org/10.31489/2021m1/65-77.

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In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral

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2

Gundogdu, Hami, and Ömer Gozukizil. "On the approximate numerical solutions of fractional heat equation with heat source and heat loss." Thermal Science, no. 00 (2021): 321. http://dx.doi.org/10.2298/tsci210713321g.

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In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last

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3

Kochubei, Anatoly N., Yuri Kondratiev, and José Luís da Silva. "On fractional heat equation." Fractional Calculus and Applied Analysis 24, no. 1 (2021): 73–87. http://dx.doi.org/10.1515/fca-2021-0004.

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Abstract In this paper, the long-time behavior of the Cesaro mean of the fundamental solution for fractional Heat equation corresponding to random time changes in the Brownian motion is studied. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.

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4

Verma, Ninny, and Dr Anil Kumar Menaria. "FRACTIONAL ORDER DISTRIBUTION ON HEAT FLUX FOR CRYSTALLINE CONCRETE MATERIAL." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 11, no. 02 (2023): 3268–72. http://dx.doi.org/10.47191/ijmcr/v11i2.08.

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The present work devoted to analyse the impact of fractional order one dimensional heat equation on heat transfer in concrete substances, during conduction process. Throughout the evolution the thermal properties of substances are considered as constant parameters. Investigations were done through statistical phenomena with linear representation of curve.

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5

Alkhasov, A. B., R. P. Meilanov, and M. R. Shabanova. "Heat conduction equation in fractional-order derivatives." Journal of Engineering Physics and Thermophysics 84, no. 2 (2011): 332–41. http://dx.doi.org/10.1007/s10891-011-0477-9.

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6

Khan, Hassan, Rasool Shah, Poom Kumam, and Muhammad Arif. "Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method." Entropy 21, no. 6 (2019): 597. http://dx.doi.org/10.3390/e21060597.

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In the present article, fractional-order heat and wave equations are solved by using the natural transform decomposition method. The series form solutions are obtained for fractional-order heat and wave equations, using the proposed method. Some numerical examples are presented to understand the procedure of natural transform decomposition method. The natural transform decomposition method procedure has shown that less volume of calculations and a high rate of convergence can be easily applied to other nonlinear problems. Therefore, the natural transform decomposition method is considered to b

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7

Blasiak, Slawomir. "Heat Transfer Analysis for Non-Contacting Mechanical Face Seals Using the Variable-Order Derivative Approach." Energies 14, no. 17 (2021): 5512. http://dx.doi.org/10.3390/en14175512.

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This article presents a variable-order derivative (VOD) time fractional model for describing heat transfer in the rotor or stator in non-contacting mechanical face seals. Most theoretical studies so far have been based on the classical equation of heat transfer. Recently, constant-order derivative (COD) time fractional models have also been used. The VOD time fractional model considered here is able to provide adequate information on the heat transfer phenomena occurring in non-contacting face seals, especially during the startup. The model was solved analytically, but the characteristic featu

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8

Li, Shu-Nan, and Bing-Yang Cao. "Fractional-order heat conduction models from generalized Boltzmann transport equation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2172 (2020): 20190280. http://dx.doi.org/10.1098/rsta.2019.0280.

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The relationship between fractional-order heat conduction models and Boltzmann transport equations (BTEs) lacks a detailed investigation. In this paper, the continuity, constitutive and governing equations of heat conduction are derived based on fractional-order phonon BTEs. The underlying microscopic regimes of the generalized Cattaneo equation are thereafter presented. The effective thermal conductivity κ eff converges in the subdiffusive regime and diverges in the superdiffusive regime. A connection between the divergence and mean-square displacement 〈|Δ x | 2 〉 ∼ t γ is established, namely

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9

Shammakh, Wafa, A. George Maria Selvam, Vignesh Dhakshinamoorthy, and Jehad Alzabut. "Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins." Symmetry 14, no. 9 (2022): 1877. http://dx.doi.org/10.3390/sym14091877.

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The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vit

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10

Alharthi, Nadiyah Hussain, Abdon Atangana, and Badr S. Alkahtani. "Numerical analysis of some partial differential equations with fractal-fractional derivative." AIMS Mathematics 8, no. 1 (2022): 2240–56. http://dx.doi.org/10.3934/math.2023116.

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<abstract> <p>In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation ha

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11

Zhang, Yufeng. "Similarity solutions and the computation formulas of a nonlinear fractional-order generalized heat equation." Modern Physics Letters B 33, no. 10 (2019): 1950122. http://dx.doi.org/10.1142/s0217984919501227.

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A generalized nonlinear heat equation with the fractional derivative is proposed, whose similarity solutions are derived from a type of special scalar transformation with two parameters. With the help of separated variable method, two special series solutions of the standard heat equation are obtained. Finally, through computation of the left Riemann–Liouville fractional derivative, we obtain two approximated computation formulas of the factional-order ordinary differential equation which could be used to calculate the numerical solutions of the generalized nonlinear heat conduction equation.

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12

Kerbal, S., and N. Tatar. "Stability of a fractional heat equation with memory." Carpathian Mathematical Publications 16, no. 1 (2024): 328–45. http://dx.doi.org/10.15330/cmp.16.1.328-345.

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Of concern is a fractional differential problem of order between zero and one. The model generalizes an existing well-known problem in heat conduction theory with memory. First, we justify the replacement of the first order derivative by a fractional one. Then, we establish a Mittag-Leffler stability result for a class of heat flux relaxation functions. We will combine the energy method with some properties from fractional calculus.

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13

Singh, Brajesh K., and Vineet K. Srivastava. "Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM." Royal Society Open Science 2, no. 4 (2015): 140511. http://dx.doi.org/10.1098/rsos.140511.

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The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineeri

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14

Muratov, Kh A., and B. Kh Turmetov. "On mixed problems for a class of fractional order equations with involution." Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy) 22, no. 3 (2022): 22–31. http://dx.doi.org/10.47526/2022-3/2524-0080.02.

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In this paper, we consider new classes of differential equations of fractional order related to Hadamard derivatives. These equations generalize the well-known heat conduction equation for the fractional exponent of the time derivative. For the equations under consideration, mixed problems with Dirichlet and Neumann boundary conditions are studied. The Fourier method is used to solve these problems. Two auxiliary problems are obtained for ordinary differential equations of fractional order and ordinary differential equations with involution. The spectral properties of ordinary differential ope

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15

Gao, Zhuoyan, and JinRong Wang. "Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s Fractional Diffusion Equation." Advances in Mathematical Physics 2017 (2017): 1–6. http://dx.doi.org/10.1155/2017/9692685.

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We discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given.

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16

Korpinar, Zeliha. "On numerical solutions for the Caputo-Fabrizio fractional heat-like equation." Thermal Science 22, Suppl. 1 (2018): 87–95. http://dx.doi.org/10.2298/tsci170614274k.

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In this article, Laplace homotopy analysis method in order to solve fractional heat-like equation with variable coefficients, are introduced. Laplace homotopy analysis method, founded on combination of homotopy methods and Laplace transform is used to supply a new analytical approximated solutions of the fractional partial differential equations in case of the Caputo-Fabrizio. The solutions obtained are compared with exact solutions of these equations. Reliability of the method is given with graphical consequens and series solutions. The results show that the method is a powerfull and efficien

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17

Alshehry, Azzh Saad, Humaira Yasmin, Rasool Shah, Roman Ullah, and Asfandyar Khan. "Numerical simulation and analysis of fractional-order Phi-Four equation." AIMS Mathematics 8, no. 11 (2023): 27175–99. http://dx.doi.org/10.3934/math.20231390.

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<abstract><p>This paper introduces a novel numerical approach for tackling the nonlinear fractional Phi-four equation by employing the Homotopy perturbation method (HPM) and the Adomian decomposition method (ADM), augmented by the Shehu transform. These established techniques are adept at addressing nonlinear differential equations. The equation's complexity is reduced by applying the Shehu Transform, rendering it amenable to solutions via HPM and ADM. The efficacy of this approach is underscored by conclusive results, attesting to its proficiency in solving the equation. With exte

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18

Ninny, Verma, and Anil Kumar Menaria Dr. "FRACTIONAL ORDER DISTRIBUTION ON HEAT FLUX FOR CRYSTALLINE CONCRETE MATERIAL." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 11, no. 02 (2023): 3268–72. https://doi.org/10.5281/zenodo.7670832.

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The present work devoted to analyse the impact of fractional order one dimensional heat equation on heat transfer in concrete substances, during conduction process. Throughout the evolution the thermal properties of substances are considered as constant parameters. Investigations were done through statistical phenomena with linear representation of curve.

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19

Mishura, Yuliya, Kostiantyn Ralchenko, Mounir Zili, and Eya Zougar. "Fractional stochastic heat equation with piecewise constant coefficients." Stochastics and Dynamics 21, no. 01 (2020): 2150002. http://dx.doi.org/10.1142/s0219493721500027.

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We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

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20

Khudair, Ayad R. "Random Fractional Laplace Transform for Solving Random Time-Fractional Heat Equation in an Infinite Medium." BASRA JOURNAL OF SCIENCE 38, no. 2 (2020): 223–47. http://dx.doi.org/10.29072/basjs.202025.

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The random Laplace and Fourier transforms are very important tools to solve random heat problems. Unfortunately, it is difficult to use these random integral transforms for solving the fractional random heat problems, where the mean square conformable fractional derivative is used to express for the time fractional derivative. Therefore, this work adopts the extension of the random Laplace transform into random fractional Laplace transform in order to solve this kind of heat problems. The stochastic process solution of the fractional random heat in an infinite medium is investigated by using r

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21

Albalawi, Wedad, Rasool Shah, Nehad Ali Shah, Jae Dong Chung, Sherif M. E. Ismaeel, and Samir A. El-Tantawy. "Analyzing Both Fractional Porous Media and Heat Transfer Equations via Some Novel Techniques." Mathematics 11, no. 6 (2023): 1350. http://dx.doi.org/10.3390/math11061350.

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It has been increasingly obvious in recent decades that fractional calculus (FC) plays a key role in many disciplines of applied sciences. Fractional partial differential equations (FPDEs) accurately model various natural physical phenomena and many engineering problems. For this reason, the analytical and numerical solutions to these issues are seriously considered, and different approaches and techniques have been presented to address them. In this work, the FC is applied to solve and analyze the time-fractional heat transfer equation as well as the nonlinear fractional porous media equation

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22

MOUS, ILHEM, and ABDELHAMID LAOUAR. "A Numerical Solution of a Coupling System of Conformable Time-Derivative Two-Dimensional Burgers’ Equations." Kragujevac Journal of Mathematics 48, no. 1 (2024): 7–23. http://dx.doi.org/10.46793/kgjmat2401.007m.

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In this paper, we deal with a numerical solution of a coupling system of fractional conformable time-derivative two-dimensional (2D) Burgers’ equations. The presence of both the fractional time derivative and the nonlinear terms in this system of equations makes solving it more diﬃcult. Firstly, we use the Cole-Hopf transformation in order to reduce the coupling system of equations to a conformable time-derivative 2D heat equation for which the numerical solution is calculated by the explicit and implicit schemes. Secondly, we calculate the numerical solution of the proposed system by using bo

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23

Kapoor, Mamta, and Simran Kour. "An analytical approach for Yang transform on fractional-order heat and wave equation." Physica Scripta 99, no. 3 (2024): 035222. http://dx.doi.org/10.1088/1402-4896/ad24ab.

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Abstract A novel approach to locate the approximate analytical solutions for non-linear partial differential equations is presented in this paper: the Yang transformation method combined with the Caputo derivative. In the current work, we determine the fractional Heat and Wave equation’s approximate analytical solutions. This current work addresses the Yang transformation approach in addition with the Caputo derivative. The suggested method yields approximately analytical solutions in the form of series with a simple, straightforward mechanics and a proportionality dependent on values of the f

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24

Liu, Fengjun, Pengjie Shi, and Ying Guo. "The Thermoelastic Dynamic Response of a Rod Due to a Moving Heat Source under the Fractional-Order Thermoelasticity Theory." Symmetry 16, no. 6 (2024): 666. http://dx.doi.org/10.3390/sym16060666.

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In this paper, the thermoelastic behavior of a rod made of an isotropic material under the action of a moving heat source was investigated using a new theory of thermoelasticity related to fractional-order time with two relaxation times. A mathematical model of the one-dimensional thermoelasticity problem was established based on the new thermoelasticity theory. We considered the symmetry of the material, and the fractional-order thermoelasticity control equation was given. Subsequently, the control equations were solved and analyzed using the Laplace transform and its inverse transform. This

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25

Mohamed, Mohamed, Amjad Hamza, Tarig Elzaki, Mohamed Algolam, and Shiraz Elhussein. "Solution of Fractional Heat-Like and Fractional Wave-Like Equation by Using Modern Strategy." Acta Mechanica et Automatica 17, no. 3 (2023): 372–80. http://dx.doi.org/10.2478/ama-2023-0042.

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Abstract This paper introduces a novel form of the Adomian decomposition (ADM) method for solving fractional-order heat-like and wave-like equations with starting and boundary value problems. The derivations are provided in the sense of Caputo. In order to help understanding, the generalised formulation of the current approach is provided. Several numerical examples of fractional-order diffusion-wave equations (FDWEs) are solved using the suggested method in this context. In addition to examining the applicability of the suggested method to the solving of fractional-order heat-like and wave-li

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26

Sierociuk, Dominik, Andrzej Dzieliński, Grzegorz Sarwas, Ivo Petras, Igor Podlubny, and Tomas Skovranek. "Modelling heat transfer in heterogeneous media using fractional calculus." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1990 (2013): 20120146. http://dx.doi.org/10.1098/rsta.2012.0146.

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This paper presents the results of modelling the heat transfer process in heterogeneous media with the assumption that part of the heat flux is dispersed in the air around the beam. The heat transfer process in a solid material (beam) can be described by an integer order partial differential equation. However, in heterogeneous media, it can be described by a sub- or hyperdiffusion equation which results in a fractional order partial differential equation. Taking into consideration that part of the heat flux is dispersed into the neighbouring environment we additionally modify the main relation

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27

Oprzędkiewicz, Krzysztof, Wojciech Mitkowski, and Maciej Rosół. "Fractional Order Model of the Two Dimensional Heat Transfer Process." Energies 14, no. 19 (2021): 6371. http://dx.doi.org/10.3390/en14196371.

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In this paper, a new, state space, fractional order model of a heat transfer in two dimensional plate is addressed. The proposed model derives directly from a two dimensional heat transfer equation. It employes the Caputo operator to express the fractional order differences along time. The spectrum decomposition and stability of the model are analysed. The formulae of impluse and step responses of the model are proved. Theoretical results are verified using experimental data from thermal camera. Comparison model vs experiment shows that the proposed fractional model is more accurate in the sen

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28

Lamba, Navneet Kumar. "Thermosensitive Response of a Functionally Graded Cylinder with Fractional Order Derivative." International Journal of Applied Mechanics and Engineering 27, no. 1 (2022): 107–24. http://dx.doi.org/10.2478/ijame-2022-0008.

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Abstract The present paper deals with thermal behaviour analysis of an axisymmetric functionally graded thermosensitive hollow cylinder. The system of coordinates are expressed in cylindrical-polar form. The heat conduction equation is of time-fractional order 0 < α ≤ 2, subjected to the effect of internal heat generation. Convective boundary conditions are applied to inner and outer curved surfaces whereas heat dissipates following Newton’s law of cooling. The lower surface is subjected to heat flux, whereas the upper surface is thermally insulated. Kirchhoff’s transformation is used to re

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29

Sitthiwirattham, Thanin, Muhammad Arfan, Kamal Shah, Anwar Zeb, Salih Djilali, and Saowaluck Chasreechai. "Semi-Analytical Solutions for Fuzzy Caputo–Fabrizio Fractional-Order Two-Dimensional Heat Equation." Fractal and Fractional 5, no. 4 (2021): 139. http://dx.doi.org/10.3390/fractalfract5040139.

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In the analysis in this article, we developed a scheme for the computation of a semi-analytical solution to a fuzzy fractional-order heat equation of two dimensions having some external diffusion source term. For this, we applied the Laplace transform along with decomposition techniques and the Adomian polynomial under the Caputo–Fabrizio fractional differential operator. Furthermore, for obtaining a semi-analytical series-type solution, the decomposition of the unknown quantity and its addition established the said solution. The obtained series solution was calculated and approached the appro

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30

Riaz, Muhammad Bilal, Aziz Ur Rehman, Jan Martinovic, and Muhammad Abbas. "Special function form solutions of multi-parameter generalized Mittag-Leffler kernel based bio-heat fractional order model subject to thermal memory shocks." PLOS ONE 19, no. 3 (2024): e0299106. http://dx.doi.org/10.1371/journal.pone.0299106.

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The primary objective of this research is to develop a mathematical model, analyze the dynamic occurrence of thermal shock and exploration of how thermal memory with moving line impact of heat transfer within biological tissues. An extended version of the Pennes equation as its foundational framework, a new fractional modelling approach called the Prabhakar fractional operator to investigate and a novel time-fractional interpretation of Fourier’s law that incorporates its historical behaviour. This fractional operator has multi parameter generalized Mittag-Leffler kernel. The fractional formul

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31

Jagdish, Sonawane, Sonatakke Bhausaheb, and Takale Kalyanrao. "Numerical Solution of Subdiffusion Bioheat equation with Single Phase lag." Indian Journal of Science and Technology 17, no. 11 (2024): 955–66. https://doi.org/10.17485/IJST/v17i11.3220.

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Abstract <strong>Objectives:</strong> The aim of this study is to investigate thermal beheviour of living tissues, for this we use subdiffusion bioheat equation with single phase lag model. <strong>Methods:</strong> The Crank-Nicolson finite difference scheme is used for subdiffusion bioheat equation with single phase lag by using Caputo fractional derivative. A Python program is used to calculate a numerical solution, which is then visually depicted through graphical representation.<strong> Finding:</strong> We discuss significance of heat flux time relaxation paramet

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32

Pskhu, A. V., M. T. Kosmakova, D. M. Akhmanova, L. Zh Kassymova, and A. A. Assetov. "Boundary value problem for the heat equation with a load as the Riemann-Liouville fractional derivative." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 105, no. 1 (2022): 74–82. http://dx.doi.org/10.31489/2022m1/74-82.

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A boundary value problem for a fractionally loaded heat equation is considered in the first quadrant. The loaded term has the form of the Riemann-Liouville’s fractional derivative with respect to the time variable, and the order of the derivative in the loaded term is less than the order of the differential part. The study is based on reducing the boundary value problem to a Volterra integral equation. The kernel of the obtained integral equation contains a special function, namely, the Wright function. The kernel is estimated, and the conditions for the unique solvability of the integral equa

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Cesarano, Clemente. "Generalized special functions in the description of fractional diffusive equations." Communications in Applied and Industrial Mathematics 10, no. 1 (2019): 31–40. http://dx.doi.org/10.1515/caim-2019-0010.

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Abstract Starting from the heat equation, we discuss some fractional generalizations of various forms. We propose a method useful for analytic or numerical solutions. By using Hermite polynomials of higher and fractional order, we present some operational techniques to find general solutions of extended form to d'Alembert and Fourier equations. We also show that the solutions of the generalized equations discussed here can be expressed in terms of Hermite-based functions.

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Shior, M. M., B. C. Agbata, U. Karim, Marcos Salvatierra, D. J. Yahaya, and S. Abraham. "Approximate Solution of Fractional Order of Partial Differential Equations Using Laplace-Adomian Decomposition Method in MATLAB." International Journal of Mathematics and Statistics Studies 12, no. 3 (2024): 61–70. http://dx.doi.org/10.37745/ijmss.13/vol12n36170.

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This article presents the application of the Laplace-Adomian Decomposition Method (LADM) for solving partial differential equations (PDEs) in the context of heat conduction and wave propagation. The LADM combines Laplace transform and Adomian decomposition to approximate solutions to PDEs efficiently in MATLAB. The procedure involves transforming the PDE into simpler differential equations, which are then solved iteratively using the Adomian decomposition method. The advantages of LADM include simplicity, flexibility, and applicability to a wide range of PDEs. We demonstrate the effectiveness

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Zaki, Golamreza, Aliasghar Akbarfam, and Safar Irandoust-Pakchin. "On the inverse problem of time fractional heat equation using Crank-Nicholson type method and genetic algorithm." Filomat 38, no. 19 (2024): 6829–49. https://doi.org/10.2298/fil2419829z.

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In this paper, the time-fractional heat equation with the Caputo derivative of order ? where 0 < ? ? 1 is considered. The parametric Crank-Nicholson type method for direct problems is used. But for the inverse problem, for finding the best conduction parameter c and the best order of fractional derivative ?, we use genetic algorithm (GA) for minimizing fitting error such that the numerical solution obtained for direct problem at the final time be fitted by the final conditions. Several examples are carried out to describe the method and to support the theoretical claims. Finally, we conclud

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Aguilar, J. F. Gómez, T. Córdova-Fraga, J. Tórres-Jiménez, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, and G. V. Guerrero-Ramírez. "Nonlocal Transport Processes and the Fractional Cattaneo-Vernotte Equation." Mathematical Problems in Engineering 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/7845874.

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The Cattaneo-Vernotte equation is a generalization of the heat and particle diffusion equations; this mathematical model combines waves and diffusion with a finite velocity of propagation. In disordered systems the diffusion can be anomalous. In these kinds of systems, the mean-square displacement is proportional to a fractional power of time not equal to one. The anomalous diffusion concept is naturally obtained from diffusion equations using the fractional calculus approach. In this paper we present an alternative representation of the Cattaneo-Vernotte equation using the fractional calculus

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Felipe, Gabriel Antonio, Carlos Alberto Valentim, and Sergio Adriani David. "A Combined Separation of Variables and Fractional Power Series Approach for Selected Boundary Value Problems." Dynamics 5, no. 3 (2025): 24. https://doi.org/10.3390/dynamics5030024.

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Fractional modeling has emerged as an important resource for describing complex phenomena and systems exhibiting non-local behavior or memory effects, finding increasing application in several areas in physics and engineering. This study presents the analytical derivation of equations pertinent to the modeling of different systems, with a focus on heat conduction. Two specific boundary value problems are addressed: a Helmholtz equation modified with a fractional derivative term, and a fractional formulation of the Laplace equation applied to steady-state heat conduction in circular geometry. T

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Tiwari, Rakhi, and Santwana Mukhopadhyay. "On harmonic plane wave propagation under fractional order thermoelasticity: an analysis of fractional order heat conduction equation." Mathematics and Mechanics of Solids 22, no. 4 (2015): 782–97. http://dx.doi.org/10.1177/1081286515612528.

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In the present paper, we investigate the propagation of an harmonic plane wave propagating with assigned frequency by implementing the thermoelasticity theory based on a fractional order heat conduction law where the fractional order parameter [Formula: see text] satisfies [Formula: see text]. After formulating the problem, the exact dispersion relation solutions for the plane wave are determined analytically and asymptotic expressions of different characterization of the wave are analyzed in two special cases, namely for a high-frequency field and low-frequency field. We consider the case of

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Dong, Guanqiang, and Mingcong Deng. "Operator & Fractional Order Based Nonlinear Robust Control for a Spiral Counter-Flow Heat Exchanger with Uncertainties and Disturbances." Machines 10, no. 5 (2022): 335. http://dx.doi.org/10.3390/machines10050335.

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This paper studies operator and fractional order nonlinear robust control for a spiral counter-flow heat exchanger with uncertainties and disturbances. First, preliminary concepts are presented concerning fractional order derivative and calculus, fractional order operator theory. Then, the problem statement about nonlinear fractional order derivative equation with uncertainties is described. Third, the design of an operator fractional order controller and fractional order PID controller and determination of several related parameters is described. Simulations were performed to verify tracking

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Sehra, Haleema Sadia, Nadia Gul, Anwar Zeb, and Zareen A. Khan. "Convection heat–mass transfer of generalized Maxwell fluid with radiation effect, exponential heating, and chemical reaction using fractional Caputo–Fabrizio derivatives." Open Physics 20, no. 1 (2022): 1250–66. http://dx.doi.org/10.1515/phys-2022-0215.

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Abstract This article is directed to analyze the transfer of mass and heat in a generalized Maxwell fluid flow unsteadily on a vertical flat plate oscillating in its respective plane and heated exponentially. It explains the transfer of mass and heat using a non-integer order derivative usually called a fractional derivative. It is a generalization of the classical derivatives of the famous Maxwell’s equation to fractional non-integer order derivatives used for one-dimensional flow of fluids. The definition given by Caputo–Fabrizio for the fractional derivative is used for solving the problem

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Thakare, S., and M. Warbhe. "Analysis of Time-Fractional Heat Transfer and its Thermal Deflection in a Circular Plate by a Moving Heat Source." International Journal of Applied Mechanics and Engineering 25, no. 3 (2020): 158–68. http://dx.doi.org/10.2478/ijame-2020-0040.

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AbstractMathematical modeling of a thin circular plate has been made by considering a nonlocal Caputo type time fractional heat conduction equation of order 0 < α ≤ 2, by the action of a moving heat source. Physically convective heat exchange boundary conditions are applied at lower, upper and outer curved surface of the plate. Temperature distribution and thermal deflection has been investigated by a quasi-static approach in the context of fractional order heat conduction. The integral transformation technique is used to analyze the analytical solution to the problem. Numerical computation

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Žecová, Monika, and Ján Terpák. "Fractional Heat Conduction Models and Thermal Diffusivity Determination." Mathematical Problems in Engineering 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/753936.

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The contribution deals with the fractional heat conduction models and their use for determining thermal diffusivity. A brief historical overview of the authors who have dealt with the heat conduction equation is described in the introduction of the paper. The one-dimensional heat conduction models with using integer- and fractional-order derivatives are listed. Analytical and numerical methods of solution of the heat conduction models with using integer- and fractional-order derivatives are described. Individual methods have been implemented in MATLAB and the examples of simulations are listed

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Elamin, Mawahib, Khadeeja Helalb, Sayed Abdel-Khalek, Ali Mubaraki, Tahani Albogami, and Emad Abdel-Salam. "Analyzing fractional order variable coefficients heat model." Thermal Science 28, no. 6 Part B (2024): 5143–52. https://doi.org/10.2298/tsci2406143e.

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The manuscript?s primary goal is to utilize the decomposition Adomian approach to approximate solutions for a specific class of space-time fractional order heat model characterized by variable coefficients and appropriate initial values. This method allows for the computation of a power series representation of the solution without the need for linearization, assumptions about weak non-linearity, or reliance on perturbation theory. By employing mathematical software like MATHEMATICA or Maple, the Adomian formulas are employed to evaluate the resulting series solution. Furthermore, this approac

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Yuan, Yirang, Qing Yang, Changfeng Li, and Tongjun Sun. "A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis." Numerical Mathematics: Theory, Methods and Applications 10, no. 3 (2017): 541–61. http://dx.doi.org/10.4208/nmtma.2017.y15013.

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AbstractA coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two

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Wang, Wensheng. "Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient." Symmetry 13, no. 3 (2021): 380. http://dx.doi.org/10.3390/sym13030380.

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High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and sto

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Abdel Aal, Mohammad. "New Perturbation–Iteration Algorithm for Nonlinear Heat Transfer of Fractional Order." Fractal and Fractional 8, no. 6 (2024): 313. http://dx.doi.org/10.3390/fractalfract8060313.

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Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on extending the nonlinear heat equations to a fractional order in a Caputo order. A new perturbation iteration algorithm (PIA) of the fractional order is applied to solve the nonlinear heat equations. Solving numerical problems that involve fractional differential equations can b

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Tuan, Nguyen Huy. "Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation." Discrete & Continuous Dynamical Systems - S 14, no. 12 (2021): 4551. http://dx.doi.org/10.3934/dcdss.2021113.

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<p style='text-indent:20px;'>In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source

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Li, Pengtao, Tao Qian, Zhiyong Wang, and Chao Zhang. "Regularity of Fractional Heat Semigroup Associated with Schrödinger Operators." Fractal and Fractional 6, no. 2 (2022): 112. http://dx.doi.org/10.3390/fractalfract6020112.

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Let L=−Δ+V be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup {e−tLα}t>0, 0<α<1, associated with L. By the aid of the fundamental solution of the heat equation: ∂tu+Lu=∂tu−Δu+Vu=0, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel Kα,tL(·,·), respectively. This method is independent of the Fourier transform, and can be applied to the second-order differential operators whose heat kernels satisfy the Gaussian upper bounds. As an applicatio

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Beshtokov, M. Kh, and M. Z. Khudalov. "The Third Boundary Value Problem for a Loaded Thermal Conductivity Equation with a Fractional Caputo Derivative." Mathematics and Mathematical Modeling, no. 3 (September 20, 2020): 52–64. http://dx.doi.org/10.24108/mathm.0320.0000222.

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Recently, to describe various mathematical models of physical processes, fractional differential calculus has been widely used. In this regard, much attention is paid to partial differential equations of fractional order, which are a generalization of partial differential equations of integer order. In this case, various settings are possible.Loaded differential equations in the literature are called equations containing values of a solution or its derivatives on manifolds of lower dimension than the dimension of the definitional domain of the desired function. Currently, numerical methods for

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Povstenko, Y., and T. Kyrylych. "Fractional thermoelasticity problem for an infinite solid with a penny-shaped crack under prescribed heat flux across its surfaces." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2172 (2020): 20190289. http://dx.doi.org/10.1098/rsta.2019.0289.

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The time-nonlocal generalization of the Fourier law with the ‘long-tail’ power kernel can be interpreted in terms of fractional calculus and leads to the time-fractional heat conduction equation with the Caputo derivative. The theory of thermal stresses based on this equation was proposed by the first author ( J. Therm. Stresses 28 , 83–102, 2005 ( doi:10.1080/014957390523741 )). In the present paper, the fractional heat conduction equation is solved for an infinite solid with a penny-shaped crack in the case of axial symmetry under the prescribed heat flux loading at its surfaces. The Laplace

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