Time-fractional Heat Conduction Equation Research Articles

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.

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 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.

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.

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.

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'.

Time-Fractional Heat Conduction in a Plane with Two External Half-Infinite Line Slits under Heat Flux Loading

The time-fractional heat conduction equation follows from the law of conservation of energy and the corresponding time-nonlocal extension of the Fourier law with the “long-tail” power kernel. The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with two external half-infinite slits with the prescribed heat flux across their surfaces. The integral transform technique is used. The solution is obtained in the form of integrals with integrand being the Mittag–Leffler function. A graphical representation of numerical results is given.

Fractional thermoelasticity problem for an infinite solid with a cylindrical hole under harmonic heat flux boundary condition

The time-fractional heat conduction equation with the Caputo derivative results from the law of conservation of energy and time-nonlocal generalization of the Fourier law with the “long-tail” power kernel. In this paper, we consider an infinite solid with a cylindrical cavity under harmonic heat flux boundary condition. The Laplace transform with respect to time and the Weber transform with respect to the spatial coordinate are used. The solutions are obtained in terms of integrals with integrands being the Mittag-Leffler functions. The numerical results are illustrated graphically.

Fractional heat conduction in solids connected by thin intermediate layer: nonperfect thermal contact

We examine the transition region between two solids which state differs from the state of contacting media. Small thickness of the intermediate region allows us to reduce a three-dimensional problem to a two-dimensional one for a median surface endowed with equivalent physical properties. In the present paper, we consider the generalized boundary conditions of nonperfect thermal contact for the time-fractional heat conduction equation with the Caputo derivative and solve the problem for a composite medium consisting of two semi-infinite regions. Numerical results are illustrated graphically.

Time-fractional heat conduction in an infinite plane containing an external crack under heat flux loading

The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with an external half-infinite crack which surfaces are exposed to the heat flux loading. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. Numerical results are illustrated graphically.

Fractional thermoelasticity problem for a plane with a line crack under heat flux loading

Symmetric stress distribution in an infinite isotropic plane with a line crack, in which surfaces are exposed to the heat flux loading is considered in the framework of fractional thermoelasticity. The heat conduction is described by the time-fractional heat conduction equations with the Caputo derivative of fractional order α. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. The numerical results are illustrated graphically.

Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition

The time-fractional heat conduction equation with the Caputo derivative and with heat absorption term proportional to temperature is considered in a sphere in the case of central symmetry. The fundamental solution to the Dirichlet boundary value problem is found, and the solution to the problem under constant boundary value of temperature is studied. The integral transform technique is used. The solutions are obtained in terms of series containing the Mittag-Leffler functions being the generalization of the exponential function. The numerical results are illustrated graphically.

Reconstruction of the Robin boundary condition and order of derivative in time fractional heat conduction equation

This paper describes an algorithm for reconstruction the boundary condition and order of derivative for the heat conduction equation of fractional order. This fractional order derivative was applied to time variable and was defined as the Caputo derivative. The heat transfer coefficient, occurring in the boundary condition of the third kind, was reconstructed. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem was solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution an Artificial Bee Colony (ABC) algorithm and Nelder-Mead method were used. In order to stabilize the procedure the Tikhonov regularization was applied. The paper presents examples to illustrate the accuracy and stability of the presented algorithm.

Fractional heat conduction in a thin hollow circular disk and associated thermal deflection

ABSTRACTThe time nonlocal generalization of the classical 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. The solution to the fractional heat conduction equation under a Dirichlet boundary condition with zero temperature and the physical Neumann boundary condition with zero heat flux are obtained by integral transform. Thermal deflection has been investigated in the context of fractional-order heat conduction by quasi-static approach for a thin hollow circular disk. The numerical results for temperature distribution and thermal deflection using thermal moment are computed and represented graphically for copper material.

The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption

The fundamental solutions to the central symmetric time-fractional heat conduction equation with heat absorption

Hyers-Ulam Stability and Existence of Solutions for Nigmatullin’s Fractional Diffusion Equation

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.

Thermal shock fracture of a cracked thermoelastic plate based on time–fractional heat conduction

A time-fractional heat conduction equation is applied to analyze the thermal shock fracture problem of a cracked plate. For a thermoelastic plate subjected to heat shock at its surfaces, analytical temperature field and thermal stresses are obtained by using Laplace transform and finite sine transform under the assumption that crack and deformation do not alter the temperature field. With this solution, thermal stress intensity factors at the crack tips are numerically calculated through the weight function method for both an edge and a center crack, respectively. The influences of fractional order describing super-diffusion, normal diffusion, and sub-diffusion on the thermal stress intensity factors are discussed. Thermoelastic fields and the thermal stress intensity factors exhibit pronounced wave–like propagation characteristics for super-diffusion or strong diffusion, and have a similar trend to normal diffusion for sub-diffusion or weak diffusion. The most dangerous crack length and position are discussed for cold and hot shock. The classical thermal stress intensity factors can be recovered from the present results only setting the fractional order to unity.