Time-fractional Heat Conduction Research Articles

Time-Fractional Heat Conduction Effects on the Dynamic Response of a Magneto-Thermoelastic Half-Space Subjected to a Thermal Shock

Time-Fractional Heat Conduction Effects on the Dynamic Response of a Magneto-Thermoelastic Half-Space Subjected to a Thermal Shock

Fractional Heat Conduction with Heat Absorption in a Solid with a Spherical Cavity under Time-Harmonic Heat Flux

The central-symmetric time-fractional heat conduction equation with heat absorption is investigated in a solid with a spherical hole under time-harmonic heat flux at the boundary. The problem is solved using the auxiliary function, for which the Robin-type boundary condition with a prescribed value of a linear combination of a function and its normal derivative is fulfilled. The Laplace and Fourier sine–cosine integral transformations are employed. Graphical representations of numerical simulation results are given for typical values of the parameters.

Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions

In this research paper, we address the time-fractional heat conduction equation in one spatial dimension, subject to nonlocal conditions in the temporal domain. To tackle this challenging problem, we propose a novel numerical approach, the ‘Rectified Chebyshev Petrov-Galerkin Procedure,’ which extends the classical Petrov-Galerkin method to efficiently handle the fractional temporal derivatives involved. Our method is characterized by several key contributions; We introduce a set of basis functions that inherently satisfy the homogeneous boundary conditions of the problem, simplifying the numerical treatment. Through careful mathematical derivations, we provide explicit expressions for the matrices involved in the Petrov-Galerkin method. These matrices are shown to be efficiently invertible, leading to a computationally tractable scheme. A comprehensive convergence analysis is presented, ensuring the reliability and accuracy of our approach. We demonstrate that our method converges to the true solution as the spatial and temporal discretization parameters are refined. The proposed Rectified Chebyshev Petrov-Galerkin Procedure is found to be robust, and capable of handling a wide range of problems with nonlocal temporal conditions. To illustrate the effectiveness of our method, we provide a series of numerical examples, including comparisons with existing techniques. These examples showcase the superiority of our approach in terms of accuracy and computational efficiency.

Analytical solution for fractional order temperature distribution in two dimensional multilayer composite circular discs

This article deals with mathematical approach to discuss the time-fractional heat conduction in two dimensional multilayer circular discs. The fractional order heat conduction equation is generated by using caputo time derivative. All the layers of the two dimension multilayer circular are assume to be thermally isotropic and making a perfect thermal contact. The boundary and interface conditions are expressed by the Riemann-Liouville derivative and the combination of first and second kind boundary condition in angular direction of the circular disc and for the third kind boundary condition in the radial direction. This solution is also applicable for composite multiple layers problem with inner radius may be zero. Three multiple layer problem studied numerically and graphically….

Non-homogeneous impulsive time fractional heat conduction equation

This article provides a concise exposition of the integral transforms and its application to singular integral equation and fractional partial differential equations. The author implemented an analytical technique, the transform method, for solving the boundary value problems of impulsive time fractional heat conduction equation. Integral transforms method is a powerful tool for solving singular integral equations, evaluation of certain integrals involving special functions and solution of partial fractional differential equations. The proposed method is extremely concise, attractive as a mathematical tool. The obtained result reveals that the transform method is very convenient and effective.Certain new integrals involving the Airy functions are given.

A new regularization for time-fractional backward heat conduction problem

Abstract It is well known that the backward heat conduction problem of recovering the temperature u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} at a time t ≥ 0 {t\geq 0} from the knowledge of the temperature at a later time, namely g := u ⁢ ( ⋅ , τ ) {g:=u(\,\cdot\,,\tau)} for τ > t {\tau>t} , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 {t>0} and ill-posed for t = 0 {t=0} . We use this observation to obtain stable approximate solutions as solutions for t ∈ ( 0 , τ ] {t\in(0,\tau]} with t as regularization parameter for approximating the solution at t = 0 {t=0} , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.

Generalized Thermoelastic Heat Conduction Model Involving Three Different Fractional Operators

Abstract The purpose of this paper is to introduce a new time-fractional heat conduction model with three-phase-lags and three distinct fractional-order derivatives. We investigate the introduced model in the situation of an isotropic and homogeneous solid sphere. The exterior of the sphere is exposed to a thermal shock and a decaying heat generation rate. We recuperate some earlier thermoelasticity models as particular cases from the proposed model. Moreover, the effects of different fractional thermoelastic models and the effect of instant time on the physical variables of the medium are studied. We obtain the numerical solutions for the various physical fields using a numerical Laplace inversion technique. We represent the obtained results graphically and discuss them. Physical views presented in this article may be useful for the design of new materials, bio-heat transfer mechanisms between tissues and other scientific domains.

Fourier and time-phase-lag heat conduction analysis of the functionally graded porosity media

We performed numerical heat conduction analysis for the new proposed functionally graded porosity (FGP) media. Fourier, Cattaneo-Vernotte (CV), dual-phase lag (DPL) and time-fractional heat conduction responses for a porous solid-gas media are compared with the different flux pulse duration experimental results. Four configurations of the FGP domain in which the porosity gradations are controlled by the power-law volume fractions, are introduced. Accordingly, the porosity-dependent effective thermal properties of the domain are estimated and the temperature responses are discussed for different heat conduction models. Even a slight change in the porosity, leads to a different temperature-time history and its gradation affects the temperature distributions through the medium; therefore, by tuning the porosity gradations the desired temperature distributions can be achievable. Finally, the diffusion-like DPL and the classical Fourier models in descending configuration predict about 35% and 41% higher peak temperatures than the constant porosity ones, respectively. Furthermore, these models demonstrate the same temperature distributions in constant porosity; that is, in the case of graded porosity the maximum difference is less than 4% due to not taking into account the time laggings as the function of porosity gradations.

A meshless collocation method for solving the inverse Cauchy problem associated with the variable-order fractional heat conduction model under functionally graded materials

A localized meshless collocation method, namely the generalized finite difference method (GFDM), is introduced to cope with the inverse Cauchy problem associated with the fractional heat conduction model under functionally graded materials (FGMs). The variable-order time-fractional heat conduction equation under the Caputo definition is employed to describe anomalous heat conduction problems. In the present numerical framework, temporal-discretization is implemented by using the standard implicit finite difference method with the spatial-discretization through the GFDM. On the basis of moving least squares and Taylor series expansion, the GFDM is capable of avoiding the ill-posedness in inverse Cauchy problems and solving the time fractional heat conduction equations (TFHCE) under FGMs. To give evidence of the efficiency and accuracy of the proposed approach for solving inverse heat conduction of FGMs, three numerical experiments are considered in the results and discussions section.

Thermoelastic Modeling of Time Fractional Heat conduction in Circular Disk with Internal Heat Generation

In this paper, thermoelastic modeling of a circular sector disk is done under the influence of fractional time theory of thermoelasticity to evaluate the changes in temperature flow, displacement and stress behavior subjected to internal heat generation within it. At the fixed edge a time dependent heat flux is applied while thermally insulation type boundary are maintained at upper part of surface as well as the lower part surface is kept at zero temperature. Also certain boundary surfaces at 0 are fixed as a function of temperatures field , 1 f 2 f which is function of respectively. Furthermore, method of Integral transformations is employed to determine the required solution of equation of heat conduction. The obtained expressions areof Bessel’s type and in form of series. For numerical analysis the pure aluminum circular sector disk considered and obtained results are plotted as shown in figures.

An External Circular Crack in an Infinite Solid under Axisymmetric Heat Flux Loading in the Framework of Fractional Thermoelasticity.

In a real solid there are different types of defects. During sudden cooling, near cracks, there can appear high thermal stresses. In this paper, the time-fractional heat conduction equation is studied in an infinite space with an external circular crack with the interior radius R in the case of axial symmetry. The surfaces of a crack are exposed to the constant heat flux loading in a circular ring . The stress intensity factor is calculated as a function of the order of time-derivative, time, and the size of a circular ring and is presented graphically.

A localized collocation scheme with fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials

This paper makes the first attempt to develop a novel localized collocation scheme based on fundamental solutions for long-time anomalous heat conduction analysis in functionally graded materials. In the present numerical framework, the Laplace transformation and Fixed Talbot numerical inverse Laplace transformation are employed to deal with the time fractional derivative in anomalous time fractional heat conduction model. The localized collocation scheme based on fundamental solutions is applied to solve Laplace-transformed time-independent partial differential equations (PDEs), and the generalized reciprocity method (GRM) coupled with the derived problem-dependent generalized fundamental solution is proposed to deal with the inhomogeneous term in Laplace-transformed time-independent PDEs. Numerical tests demonstrate that the developed computational strategy can be considered as one of the competitive alternatives in the simulation of anomalous heat conduction behavior of functionally graded materials. Furthermore, the effect of time fractional derivative order on the behavior of anomalous heat conduction is numerically investigated.

Numerical simulation and parameters estimation of the time fractional dual-phase-lag heat conduction in femtosecond laser heating

In this paper, considering the heat flux and the temperature gradient approximated by the fractional Taylor's series expansions with different orders in dual-phase-lag (DPL) model, we propose the time fractional DPL heat conduction model for thin gold films heating by femtosecond laser pulses. The solutions of the model are investigated analytically and numerically. Firstly, the semi-analytical solution is presented by the Laplace transform method and the Fourier cosine transform method. With the L1 approximation of the Caputo fractional derivative, the finite difference scheme is derived to describe the effects of femtosecond laser pulses on the temperature distributions of the gold films. The effectiveness of the numerical algorithm is examined by the comparison with the semi-analytical solution. And then based on the numerical algorithm and the experimental datasets of femtosecond laser pulses heating on gold films of various thicknesses, the estimations of the model parameters are considered by the modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO). Using the optimal values of the parameters estimation, we verify the consistencies between the semi-analytical solutions of the model and the datasets. It proves the accuracy of the time fractional DPL heat conduction model in characterizing the temperature distributions of thin gold films. Finally, the varieties of temperature distribution with time and position are considered and then the effects of the model parameters on the temperature distribution are also discussed.

A mollified approach to reconstruct an unknown boundary condition for the heat conduction equation of fractional order

We consider an inverse problem of time fractional heat conduction problem. It is shown that the problem is ill-posed. A method is investigated based on the finite difference to find heat distribution and boundary values. The discrete mollification regularisation is applied to obtain a stable numerical solution. Finally, some test problems are investigated to show the ability of the proposed scheme.

Wavelet regularization strategy for the fractional inverse diffusion problem

This manuscript deals with an inverse fractional-diffusing problem, the time-fractional heat conduction equation, which is a physical model of a problem, where one needs to identify the temperature distribution of a semi-conductor, but one transient temperature data is unreachable to measurement. Mathematically, it is designed as a time-fractional diffusion problem in a semi-infinite region, with polluted data measured at x = 1, where the solution is wanted for 0 ≤ x < 1. In view of Hadamard, the problem extremely suffers from an intrinsic ill-posedness, i.e., the true solution of the problem is computationally impossible to measure since any measurement or numerical computation is polluted by inevitable errors. In order to capture the solution, a regularization scheme based on the Meyer wavelet is therefore applied to treat the underlying problem in the presence of polluted data. The regularized solution is restored by the Meyer wavelet projection on elements of the Meyer multiresolution analysis (MRA). Furthermore, the concepts of convergence rate and stability of the proposed scheme are investigated and some new order-optimal stable estimates of the so-called Holder-Logarithmic type are rigorously derived by carrying out both an a priori and a posteriori choice approaches in Sobolev scales. It turns out that both approaches yield the same convergence rate, except for some different constants. Finally, the computational performance of the proposed method effectively verifies the applicability and validity of our strategy. Meanwhile, the thrust of the present paper is compared with other sophisticated methods in the literature.

Analysis of Time-Fractional Heat Transfer and its Thermal Deflection in a Circular Plate by a Moving Heat Source

Mathematical modeling of a thin circular plate has been made by considering a nonlocal Caputo type time fractional heat conduction equation of order &lt;i&gt;0 &lt;&lt;/i&gt; α ≤ &lt;i&gt;2&lt;/i&gt;, 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 including the effect of the fractional order parameter has been done for temperature and deflection and illustrated graphically for an aluminum material.

Numerical analysis of time-fractional non-Fourier heat conduction in porous media based on Caputo fractional derivative under short heating pulses

In this research, the heat transport in porous media (sand particles and interstitial gas) under short heating pulses is examined, taking into account the non-Fourier effects. The time-fractional heat conduction equation based on the Caputo definition is proposed to examine the thermal response of porous media in short-time scales. The results of the time-fractional model are compared to experimental data, which proves the accuracy of this model as well as its capability to describe the fast transient thermal process, the local thermal nonequilibrium condition, and the energy exchange between the solid and gaseous phases. Additionally, the influence of variation in the different parameters, including the order of fractionality and heating pulse widths are investigated, and it has been proved that the gas-solid interactions in the fast transient process play a vital role in heat transfer mechanism of porous media, especially in the near-field locations to the heater with short pulse width.

Time-fractional heat conduction in a finite composite cylinder with heat source

Time-fractional heat conduction in a finite composite cylinder with heat source

Fractional thermoelasticity problem for an infinite solid with a penny-shaped crack under prescribed heat flux across its surfaces.

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, Hankel and cos-Fourier integral transforms are used. The solution for temperature is obtained in the form of integral with integrands being the generalized Mittag-Leffler function in two parameters. The associated thermoelasticity problem is solved using the displacement potential and Love's biharmonic function. To calculate the additional stress field which allows satisfying the boundary conditions at the crack surfaces, the dual integral equation is solved. The thermal stress field is calculated, and the stress intensity factor is presented for different values of the order of the Caputo time-fractional derivative. A graphical representation of numerical results is given. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Parameter estimation for the time fractional heat conduction model based on experimental heat flux data

In this paper, a time fractional heat conduction model is built to describe the heat transfer process in coalbed methane adsorption. A spectral collocation scheme with time discretized by the finite difference method and space discretized by Legendre collocation method is introduced to obtain the numerical solution of the fractional model. In addition, a new optimization algorithm, called the cuckoo search (CS) algorithm, is first proposed to estimate the parameters of fractional heat conduction model with the adsorption experimental heat flux data and is proved to be effective. Sensitivity analysis of model parameters and stability analysis of the CS algorithm for the fractional model are carried out at last.