Fractional Heat Conduction Equation Research Articles

Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition

Abstract The central symmetric time-fractional heat conduction equation with Caputo derivative of order 0 < α ≤ 2 is considered in a ball under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of values of temperature and values of its normal derivative at the boundary, and the physical condition with the prescribed linear combination of values of temperature and values of the heat flux at the boundary, which is a consequence of Newton’s law of convective heat exchange between a body and the environment. The integral transform technique is used. Numerical results are illustrated graphically.

Fractal heat conduction problem solved by local fractional variation iteration method

This paper points out a novel local fractional variational iteration method for processing the local fractional heat conduction equation arising in fractal heat transfer.

The Cattaneo-type time fractional heat conduction equation for laser heating

In this paper, the laser short-pulse heating of a solid surface is considered. The time fractional Cattaneo model is used as the heat conduction model and the corresponding fractional heat conduction equation with a volumetric heat source is built. The analytical solution for the temperature distribution is obtained using the Laplace transformation method. Finally, the numerical results are presented graphically for various values of model parameters, and their effects on the speed of heat conduction propagation are discussed.

An inverse problem to estimate relaxation parameter and order of fractionality in fractional single-phase-lag heat equation

In this paper, an inverse analysis is performed for simultaneous estimation of relaxation time and order of fractionality in fractional single-phase-lag heat equation. This fractional heat conduction equation is applied on two physical problems. In inverse procedure, solutions of a previously validated linear dual-phase-lag model on the physical problems under study have been used as the measured temperatures. The inverse fractional single-phase-lag heat conduction problem is solved using the nonlinear parameter estimation technique based on the Levenberg–Marquardt method. The results of the present study show that the Levenberg–Marquardt method can be successfully applied on the inverse fractional heat transfer problem. The solution procedures employed in the present study for direct and inverse problems have greatly increased the reliability and success of parameter estimation problem. In the present study, for the first time, relaxation time and fractionality of a non-homogeneous medium (i.e. processed meat) have been determined. Also, the results of this study show that the fractional single-phase-lag model can predict the same temperature distribution as the linear dual-phase-lag model for the problem under study. This latter result enables us to consider further generalization of the dual-phase-lag model to fractional dual-phase-lag models.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper, using the fractional Fourier law, we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system. The method of variable separation is used to solve the timefractional heat conduction equation. The Caputo fractional derivative of the order 0 < α ≤ 1 is used. The solution is presented in terms of the Mittag-Leffler functions. Numerical results are illustrated graphically for various values of fractional derivative.

The Cattaneo type space-time fractional heat conduction equation

The classical heat conduction equation is generalized using a generalized heat conduction law. In particular, we use the space-time Cattaneo heat conduction law that contains the Caputo symmetrized fractional derivative instead of gradient \({{\partial_x}}\) and fractional time derivative instead of the first order partial time derivative \({{\partial_t}}\) . The existence of the unique solution to the initial-boundary value problem corresponding to the generalized model is established in the space of distributions. We also obtain explicit form of the solution and compare it numerically with some limiting cases.

Solution of a three-dimensional boundary-value problem for a fractional differential heat conduction equation

We consider a three-dimensional axisymmetric problem for a differential heat conduction equation with fractional time derivatives. Using the method of homogeneous solutions and integral transformations, we obtain an asymptotic and a numerical solution of the problem. The results of calculations are presented. In recent years, the interest of researchers in the physicomechanical properties of materials with a fractal and multifractal structure has grown [5, 9]. Highly porous structured media, which are the result of many processes and phenomena of irreversible growth such as diffusion, aggregation, fracture, percolation, dynamic chaos, and dissolution, can serve as examples of fractals. In the literature, there exist several models for description of the mechanical, thermal, and electric properties of fractal media [5, 10]. They contain partial differential equation with fractional derivatives (with respect to space variables). The fractional differentiation index is related to the fractal dimensionality of the space. However, along with geometrical fractals, time fractals are also considered because there exist processes that have fractal nature in time. Among them are processes of charge transfer in amorphous media within the framework of the random-walk model with continuous time, impact wave processes in condensed media, anomalous diffusion, and heat conduction [9–14]. At present, in the literature, there are several approaches to the solution of differential equations with fractional derivatives. In [1], the Monte Carlo method in combination with the scheme of finite differences was used. The methods of integral transformations are efficient [10]. In [13], the problem of anomalous diffusion in an infinite medium with a cylindrical cavity was considered. Some general reasoning on the solution of fractional differential equations were presented in [11]. In the present work, we propose analytic and numerical procedures for the solution of a new threedimensional boundary-value problem of fractal heat conduction for a layer with a through cylindrical cavity. Numerical results that characterize the evolution of temperature with time for different orders of the derivative under the action of pulse load on the surface of the body are presented. Statement of the Problem In a curvilinear Cartesian system Ox 1 x 2 x 3 , we consider a layer �� < x 1 , x 2 <� , x 3 h, containing a

Fractional Cattaneo-Type Equations and Generalized Thermoelasticity

Following Compte and Metzler, the generalized Cattaneo-type equations with time-fractional derivatives are considered. The corresponding theory of thermal stresses is formulated. The proposed theory, on the one hand, interpolates the theory of Lord and Shulman and thermoelasticity without energy dissipation of Green and Naghdi and, on the other hand, generalizes theory of thermal stresses based on the fractional heat conduction equation. The fundamental solution to the nonhomogeneous fractional telegraph equation as well as the corresponding stresses are obtained in one-dimensional and axisymmetric cases.

The Neumann boundary problem for axisymmetric fractional heat conduction equation in a solid with cylindrical hole and associated thermal stress

The theory of thermoelasticity based on the heat conduction equation with the Caputo time-fractional derivative of order α is used to study thermal stress in an infinite medium with a cylindrical hole. Two types of Neumann boundary conditions are considered: the constant value of the normal derivative of the temperature and constant heat flux at the surface of a cavity. The solution is obtained applying Laplace and Weber integral transforms. Numerical results are illustrated graphically.

The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems

In this paper a time fractional Fourier law is obtained from fractional calculus. According to the fractional Fourier law, a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built. The fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases. In addition, we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized H -function using integral transformation methods. The fractional heat conduction equation in the case 0 < α ≤ 1 interpolates the standard heat conduction equation ( α = 1 ) and the Localized heat conduction equation ( α → 0 ) . Finally, numerical results are presented graphically for various values of order of fractional derivative.

Thermoelasticity that uses fractional heat conduction equation

A survey of nonlocal generalizations of the Fourier law and heat conduction equation is presented. More attention is focused on the heat conduction with time and space fractional derivatives and on the theory of thermal stresses based on this equation.

Fractional heat conduction equation and associated thermal stresses in an infinite solid with spherical cavity

In this work, the temperature distribution and thermal stresses in an infinite medium with a spherical cavity are studied in the framework of a quasi-static uncoupled theory of thermoelasticity based on heat conduction equation with a time fractional derivative of order 0 < α≤ 2. The Caputo fractional derivative is used. As the fractional heat conduction equation in the case 1 ≤ α ≤ 2 interpolates the standard heat conduction equation (a = 1) and the wave equation (a = 2), the proposed theory interpolates the classical thermoelasticity and the thermoelasticity without energy dissipation introduced by Green and Naghdi. The solution is obtained using the integral transform technique. Numerical results are illustrated graphically.

Fundamental Solutions to Central Symmetric Problems for Fractional Heat Conduction Equation and Associated Thermal Stresses

A quasi-static uncoupled theory of thermal stresses based on the time-fractional heat conduction equation is considered. The Caputo fractional derivative of order α is used. The fundamental solutions to the Cauchy problems as well as to the source problem are studied in the case of spherical symmetry, and corresponding thermal stresses are investigated. The solutions are obtained using the Laplace transform with respect to time and the sin-Fourier transform with respect to radial coordinate. For 1 < α < 2 the proposed theory interpolates the classical theory of thermoelasticty and that without energy dissipation introduced by Green and Naghdi. The numerical results for the temperature and stress distributions are presented graphically for various values of order of fractional derivative.

FRACTIONAL HEAT CONDUCTION EQUATION AND ASSOCIATED THERMAL STRESS

ABSTRACT A quasi-static uncoupled theory of thermoelasticity based on the heat conduction equation with a time-fractional derivative of order α is proposed. Because the heat conduction equation in the case 1≤α≤2 interpolates the parabolic equation (α = 1) and the wave equation (α = 2), the proposed theory interpolates a classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Naghdi. The Caputo fractional derivative is used. The stresses corresponding to the fundamental solutions of a Cauchy problem for the fractional heat conduction equation are found in one-dimensional and two-dimensional cases.