Time-fractional Heat Conduction Equation Research Articles

Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses

ABSTRACTTime-nonlocal generalization of the classical Fourier law with the “long-tail” power kernel can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and leads to the time-fractional heat conduction equation with the Caputo derivative. Fractional heat conduction equation with the harmonic source term under zero initial conditions is studied. Different formulations of the problem for the standard parabolic heat conduction equation and for the hyperbolic wave equation appearing in thermoelasticity without energy dissipation are discussed. The integral transform technique is used. The corresponding thermal stresses are found using the displacement potential.

Application of fractional order theory of thermoelasticity to a bilayered structure with interfacial conditions

ABSTRACTIn this work, fractional order theory of thermoelasticity is applied to a bilayered structure being in imperfect thermal and mechanical contact. The model is subjected to a sudden heating at the traction-free end, assumed to be undisturbed at infinity. The heat conduction in each medium is described by the time-fractional heat conduction equations with two fractional order parameters, respectively. An analytical technique based on Laplace transform is adopted. Numerical results are computed and represented graphically, from which the effects of fractional derivative parameters of both media, thermal contact resistance, elastic wave impedance ratio on the responses are discussed.

Time-Fractional Heat Conduction in a Half-Line Domain due to Boundary Value of Temperature Varying Harmonically in Time

The Dirichlet problem for the time-fractional heat conduction equation in a half-line domain is studied with the boundary value of temperature varying harmonically in time. The Caputo fractional derivative is employed. The Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate are used. Different formulations of the considered problem for the classical heat conduction equation and for the wave equation describing ballistic heat conduction are discussed.

Implicit finite difference approximation for time fractional heat conduction under boundary condition of second kind

The time fractional heat conduction in an infinite plate of finite thickness, when both faces are subjected to boundary conditions of second kind, has been studied. The time fractional heat conduction equation is used, when attempting to describe transport process with long memory, where the rate of heat conduction is inconsistent with the classical Brownian motion. The stability and convergence of this numerical scheme has been discussed and observed that the solution is unconditionally stable. The whole analysis is presented in dimensionless form. A numerical example of particular interest has been studied and discussed in details.

Similarity solutions for phase change problems with fractional governing equations

In this paper, the scale-invariant solution of the fractional generalized Lame–Clapeyron–Stefan melting problem with a mushy region is considered. The time fractional heat conduction equations of orders α 1 and α 2 are used for the governing equations of the heat transfer in the liquid and solid phases. The Lie group method is used to determine the scale-invariant variables of the problem. To study the existence and characteristics of the similarity solutions, two cases α 1 ≠ α 2 and α 1 = α 2 are discussed. The solutions are obtained in forms of generalized Wright function and the computations of these solutions are introduced in detail.

Reconstruction of the boundary condition for the heat conduction equation of fractional order

This paper describes reconstruction of the heat transfer coefficient occurring in the boundary condition of the third kind for the time fractional heat conduction equation. Fractional derivative with respect to time, occurring in considered equation, is defined as the Caputo derivative. Additional information for the considered inverse problem is given by the temperature measurements at selected points of the domain. The direct problem is solved by using the implicit finite difference method. To minimize functional defining the error of approximate solution the Nelder-Mead algorithm is used. The paper presents results of computational examples to illustrate the accuracy and stability of the presented algorithm.

Analytical solutions to a fractional generalized two phase Lame-Clapeyron-Stefan problem

Purpose – The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the analytical solutions of the models and show the performance of several parameters. Design/methodology/approach – To describe the heat conduction, the Caputo type time fractional heat conduction equation is used and a convective term is included since the changes in density give rise to motion of the liquid phase. The similarity variables are used to simplify the models. Findings – The analytical solutions describing the changes of temperature in both liquid and solid phases are obtained. For the solid phase, the solution is given in the Wright function form. While for the liquid phase, since the appearance of the advection term, an approximate solution in series form is given. Based on the solutions, the performance of the parameters is discussed in detail. Originality/value – From the point of view of mathematics, the moving boundary problems are nonlinear, so barely any analytical solutions for these problems can be obtained. Furthermore, there are many applications in which a material undergoes phase change, such as in melting, freezing, casting and cryosurgery.

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.

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.

Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

Abstract Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.

Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion

The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R) and a matrix (R < r < ∞) being in perfect thermal contact at r = R is considered. The heat conduction in each region is described by the time-fractional heat conduction equation with the Caputo derivative of fractional order 0 < a ≤ 2 and 0 < β ≤ 2, respectively. The Laplace transform with respect to time is used. The approximate solution valid for small values of time is obtained in terms of the Mittag-Leffler, Wright, and Mainardi functions.

Fundamental Solutions to Robin Boundary-Value Problems for the Time-Fractional Heat-Conduction Equation in a Half Line

The time-fractional heat-conduction equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a half line. Two types of Robin boundary condition are examined: the mathematical condition with prescribed linear combination of the values of temperature and the values of its normal derivative and the physical condition with prescribed linear combination of the values of temperature and the values of heat flux on the boundary of the domain. These two types of Robin boundary condition coincide only for the classical heat-conduction equation.

Fractional Heat Conduction in Infinite One-Dimensional Composite Medium

The problem of fractional heat conduction in a composite medium consisting of two semi-infinite regions being in perfect thermal contact is considered. The heat conduction in each region is described by the time-fractional heat conduction equations with the Caputo derivative of fractional order α and β, respectively. The solution is obtained using the Laplace transform with respect to time and is expressed in terms of the Mittag–Leffler function and Mainardi function. Numerical results are illustrated graphically.

Time-fractional heat conduction in an infinite medium with a spherical hole under robin boundary condition

Abstract The time-fractional heat conduction equation with the Caputo derivative of the order 0 &lt; α ≤ 2 is considered in an infinite medium with a spherical hole in the central symmetric case under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of the values of temperature and the values of its normal derivative at the boundary and the physical condition with the prescribed linear combination of the values of temperature and the values of the heat flux at the boundary. The integral transforms techniques are used. Several particular cases of the obtained solutions are analyzed. The numerical results are illustrated graphically.

Thermoelasticity of thin shells based on the time-fractional heat conduction equation

AbstractThe time-nonlocal generalizations of Fourier’s law are analyzed and the equations of the generalized thermoelasticity based on the time-fractional heat conduction equation with the Caputo fractional derivative of order 0 &lt; α ≤ 2 are presented. The equations of thermoelasticity of thin shells are obtained under the assumption of linear dependence of temperature on the coordinate normal to the median surface of a shell. The conditions of Newton’s convective heat exchange between a shell and the environment have been assumed. In the particular case of classical heat conduction (α = 1) the obtained equations coincide with those known in the literature.

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.

Nonaxisymmetric solutions of the time-fractional heat conduction equation in a half-space in cylindrical coordinates

Nonaxisymmetric solutions of the time-fractional heat conduction equation with source term in cylindrical coordinates are obtained for a half-space. The solutions are found using the Laplace transform with respect to time, the Hankel transform with respect to the radial coordinate, the finite Fourier transform with respect to the angular coordinate, and the sine or cosine Fourier transform with respect to the bulk coordinate. Numerical results are illustrated graphically.

Axisymmetric Solutions to Time-Fractional Heat Conduction Equation in a Half-Space under Robin Boundary Conditions

The time-fractional heat conduction equation with the Caputo derivative of the order <svg style="vertical-align:-0.30096pt;width:63.349998px;" id="M1" height="11.0625" version="1.1" viewBox="0 0 63.349998 11.0625" width="63.349998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.0625)"> <g transform="translate(72,-63.15)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">0</tspan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0">&lt;</tspan> <tspan style="font-size: 12.50px; " x="21.75522" y="0">𝛼</tspan> <tspan style="font-size: 12.50px; " x="32.282745" y="0">&lt;</tspan> <tspan style="font-size: 12.50px; " x="44.323135" y="0">2</tspan> </text> </g> </g> </svg> is considered in a half-space in axisymmetric case under two types of Robin boundary condition: the mathematical one with the prescribed linear combination of the values of temperature and the values of its normal derivative and the physical condition with the prescribed linear combination of the values of temperature and the values of the heat flux at the boundary.

Exact solutions of time-fractional heat conduction equation by the fractional complex transform

The Fractional Complex Transform is extended to solve exactly time-fractional differential equations with the modified Riemann-Liouville derivative. How to incorporate suitable boundary/initial conditions is also discussed.

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.