Fractional Heat Conduction Equation Research Articles

Dynamic behaviors of the non-linear local fractional heat conduction equation on the cantor sets

Dynamic behaviors of the non-linear local fractional heat conduction equation on the cantor sets

Investigating Thermal Deflection in a Finite Hollow Cylinder Using Quasi-Static Approach and Space-Time Fractional Heat Conduction Equation

Investigating Thermal Deflection in a Finite Hollow Cylinder Using Quasi-Static Approach and Space-Time Fractional Heat Conduction Equation

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.

An Application of the Homotopy Analysis Method for the Time- or Space-Fractional Heat Equation

This paper focuses on the usage of the homotopy analysis method (HAM) to solve the fractional heat conduction equation. In the presented mathematical model, Caputo-type fractional derivatives over time or space are considered. In the HAM, it is not necessary to discretize the considered domain, which is its great advantage. As a result of the method, a continuous function is obtained, which can be used for further analysis. For the first time, for the considered equations, we proved that if the series created in the method converges, then the sum of the series is a solution of the equation. A sufficient condition for this convergence is provided, as well as an estimation of the error of the approximate solution. This paper also presents examples illustrating the accuracy and stability of the proposed algorithm.

An Efficient Approach for Solving One-Dimensional Fractional Heat Conduction Equation

An Efficient Approach for Solving One-Dimensional Fractional Heat Conduction Equation

Fast method and convergence analysis for the magnetohydrodynamic flow and heat transfer of fractional Maxwell fluid

This work investigates the unsteady magnetohydrodynamic (MHD) flow and heat transfer of fractional Maxwell fluids in a square cavity, which is under the influence of the Hall effect and radiation heat. The coupled model is formed from the momentum equation based on the fractional constitutive relationship and the fractional heat-conduction equation derived from the Fourier law. The fractional coupled model is solved numerically by combining the weighted and shifted Grünwald difference method in the temporal direction with the spectral method based on Lagrange-basis polynomials in the spatial direction. In addition, we propose a fast method to reduce the computational time and the memory requirements of the actual calculation. We also prove the stability and convergence of the numerical scheme with the fast method. Furthermore, a numerical example is given to verify the efficiency of the numerical method and of the theoretical analysis. An example of non-smooth solutions is dealt with by adding correction terms. Finally, an example is considered to discuss the effects of the Hartmann number, the Hall parameter, and the thermal radiation parameter on the MHD flow and heat transfer of a fractional Maxwell fluid in a square cavity.

Non-differentiable solutions for non-linear local fractional heat conduction equation

Fractional calculus has many advantages. Under consideration of this paper is a (2+1)-dimensional non-linear local fractional heat conduction equation with arbitrary degree non-linearity. Backlund transformation of a reduced form of the local fractional heat conduction equation is constructed by Painleve analysis. Based on the Backlund transformation, some exact non-differentiable solutions of the local fractional heat conduction equation are obtained. To gain more insights of the obtained solutions, two solutions are constrained to a Cantor set and then two spatio-temporal fractal structures with profiles of these two solutions are shown. This paper further reveals by local fractional heat conduction equation that fractional calculus plays important role in dealing with non-differentiable problems.

Thermoelastic guided wave in fractional order functionally graded plates: An analytical integration Legendre polynomial approach

In this paper, an analytical integration Legendre polynomial approach (AILPA) is proposed to investigate the guided thermoelastic wave in functionally graded material (FGM) plates in the context of the fractional order Lord-Shulman (LS) thermoelastic theory. Coupled wave equations and fractional order heat conduction equation are solved by the presented approach, which proposes the analytical integral instead of numerical integration in the available conventional Legendre polynomial approach (CLPA). Comparison of the CPU time between two approaches indicates the higher efficiency of the presented approach. Furthermore, a new treatment of the adiabatic boundary condition for the Legendre polynomial is developed, other than the CLPA can only deal with the isothermal boundary condition. Finally, the phase velocity dispersion curves, attenuation curves, the displacement and temperature distributions for functionally graded plates with different fractional orders are analysed. Both the fractional order and relaxation time have weak influence on the elastic mode velocity, but they have considerable influence on the elastic mode attenuation.

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 <i>0 <</i> α ≤ <i>2</i>, 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.

Neural network method for fractional-order partial differential equations

In this paper, neural network method is first proposed to solve the fractional-order partial differential equations. The neural network based on the sine and the cosine functions is established on the sample points which are evenly distributed in the solution area. In view of the fractional heat conduction equation and the fractional wave equation, the training error function is established, and the related algorithm for the initial boundary value problem of the above equations is given. Then, the learning rate of the neural network is analyzed. Finally, several numerical examples are given to verify the effectiveness of solving the fractional-order partial differential equations by the present method.

Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs.

Numerical simulation of fractional non-Fourier heat transfer in thin metal films under short-pulse laser

In this paper, heat transfer is investigated in thin metal films subjected to a short-pulse laser. Due to the short temporal scales and high intensity of laser pulses during the heating process, the non-Fourier phenomenon can be observed in the temperature response of thin metal films. Therefore, the fractional non-Fourier model has been applied to investigate the heat transport in thin metal films to capture the thermal behaviors of nano-scale metal films. The fractional non-Fourier model is presented to investigate the heat transfer in single-layer gold film and gold‑chromium double-layer films under short-pulse laser. The governing fractional heat conduction equation is solved numerically using the finite difference method based on implicit scheme, and the obtained results are compared to the experimental data. The numerical results obtained from the fractional model represent its accuracy and suitability to predict the thermal behavior of the thin metal films irradiated by short-pulse laser. In addition, upon comparing the results of the fractional model with the results of other models which have been employed in this field, it can be seen that the fractional model has a higher capability than others to examine the non-Fourier effects during laser heating of thin metal films.

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 subdiffusion model for thin metal films under femtosecond laser pulses based on Caputo fractional derivative to examine anomalous diffusion process

In this paper, the non-Fourier effects are examined in thin metal films exposed to femtosecond laser pulses. For the first time, the time-fractional subdiffusion model is presented based on the Caputo fractional derivative to examine the anomalous diffusion process in nano-scale metal films. The fractional heat conduction equation is solved numerically, using the finite difference method based on implicit scheme and the numerical results are compared to experimental data. The comparison implies that the numerical results well coincide with experimental data, proving the accuracy and suitability of fractional model to capture thermal behaviors of thin metal films heated by short-pulse laser. Furthermore, the results of time-fractional model are compared to other models including parabolic and hyperbolic two-step models. Having compared, one can be found is that the fractional model is more reliable than other models, because its results are well matched with experimental data. The use of Fourier's law in parabolic and hyperbolic two-step model leads to deviation from experimental data which can be addressed upon applying the proposed time-fractional model in this work. Moreover, the presented time-fractional model can be effectively utilized for studying the heat transport in thin metal films under laser pulses as well as the laser heating process and its applications in engineering. Ultimately, the effect of changes in various parameters such as order of fractional derivative, laser pulse duration, metal materials (gold and chromium), gold film thickness, relaxation time and absorption depth of laser pulses are investigated on temperature response of thin metal films.

Analytical and numerical analysis of time fractional dual-phase-lag heat conduction during short-pulse laser heating

In this study, we analytically and numerically investigate the non-Fourier heat conduction behavior within a finite medium based on the time fractional dual-phase-lag model. Firstly, the time fractional dual-phase-lag model and the corresponding fractional heat conduction equation for short-pulse laser heating is built. Laplace and Fourier cosine transforms are performed to derive the semi-analytical expression of temperature distribution in the Laplace domain. Then, by the L1 approximation for the Caputo derivative, the finite difference algorithm is developed for the short-pulse laser heating problem. The solvability, stability, and convergence of this algorithm are also examined. Meanwhile, the efficiency and accuracy of this method have been verified by using three numerical examples. Finally, based on numerical analysis, we study the non-Fourier heat conduction behavior and discuss the effect of variability of parameters, such as fractional parameter and the ratio between the relaxation and retardation times, on the temperature distribution graphically. We believe that this analysis, besides benefiting the laser heating applications, will also provide a deep theoretical insight to interpret the anomalous heat transport process.

Investigations on the thermal behavior and associated thermal stresses for short pulse laser heating

The fractional type Cattaneo heat conduction equation and the associated thermal stress equation are established for studying the thermal and related stress behavior of the short pulse laser heating. With the Laplace transform method, the analytical solution of the temperature distribution and thermal stress field are derived. Compared to the classical Fourier heat transfer model and the standard C-V one, the results of the present model show the thermal diffusion feature as well as the thermal wave behaviour, which is more realistic. Furthermore, the influences of the fractional order parameter on the thermal and related stress behavior are discussed. The thermal velocity, thermal variation rate and the peak value of the temperature and the thermal stress depend on the fractional order parameter. The fractional type Cattaneo heat model and associated thermal stress exhibit the non-local nature.

On integrability of the time fractional nonlinear heat conduction equation

Under investigation in this letter is a time fractional nonlinear heat conduction equation which usually appears in mathematics physics, integrable system, fluid mechanics and nonlinear areas, by means of applying the fractional symmetry group method with the sense of Riemann-Liouville (R-L)fractional derivative. First of all, we use the fractional symmetry group method to obtain symmetries of the time fractional nonlinear heat conduction equation. Second, according to the above find symmetries, this equation can be reduced to a fractional ordinary differential equation. Moreover, invariant solutions of the time fractional nonlinear heat conduction equation are yielded. Finally, with the aid of the Ibragimov theorem, the conservation laws are also find to the time fractional nonlinear heat conduction equation. These new results are an effective complement to existing knowledge.

Effects of fractional and two-temperature parameters on stress distributions for an unbounded generalized thermoelastic medium with spherical cavity

Effects of fractional and two-temperature parameters on the distribution of stresses of an unbounded isotropic thermoelastic medium with spherical cavity are studied in the context of the theory of two-temperature generalized thermoelasticity based on the Green-Naghdi model III using fractional order heat conduction equation. The surface of the cavity is considered to be free from traction and is subjected to a smooth and time-dependent-heating effect. A spherical polar coordinate system has been used to describe the problem and the resulting governing equations are solved in Laplace transform domain. Numerical Laplace transform inversion method has been then applied to get the stresses in time domain. The numerical estimates of the distributions of stresses are obtained and are presented graphically to study the effects of fractional and two-temperature parameters.

The infrared thermal wave imaging detection of micro-crack defects in Ti-Al alloy plate

The Ti-Al alloy has good physical, chemical and mechanical properties, and it is the preferred material in aerospace and other special fields. The laser spots array is used for thermal excitation of the Ti-Al alloy specimen. Based on the fractional differential equation and Fourier heat conduction equation, the fractional heat transfer model of Ti-Al alloy plate specimen ex-cited by short pulse laser spots array is established, and the infrared thermal imaging simulation analysis is carried out by finite element method. The effects of crack width, crack depth and the distance between the crack and its nearest laser spot center on temperature abrupt jump is analyzed. With the increase of crack width and depth, the temperature abrupt jump increases, but the trend gradually slows down. The distance between the crack and its nearest laser spot center has a significant effect on the temperature abrupt jump. With the increase of the distance, the temperature abrupt jump first increases and then decreases. When the distance equals the laser spot radius, that is, the crack is tangent to the spot, the temperature abrupt jump reaches its maximum.