Fractional Heat Research Articles

New insights on piezoelectric thermoelastic coupling and transient thermo-electromechanical responses of multi-layered piezoelectric laminated composite structure

In recent years there have been many papers that considered the pyroelectric effect in the investigations of thermo-mechanical coupling of piezoelectric ceramics serving in non-uniform thermal environment, particularly with extensive applications of ultrafast heating technologies (e.g., laser burst, laser ablation, etc.) in micro-machining of piezoelectric structures (e.g., piezoelectric sensor/actuator/generator/energy harvester, etc.). However, in such conditions, the existing piezoelectric thermoelasticity theories will not be applicable due to the lack of consideration of the memory-dependence feature in the constitutive relations and heat transport equation. To further refine the piezoelectric thermoelasticity models, present study develops a new theoretical model of generalized piezoelectric thermoelasticity which fully account for fractional order strain and heat conduction. By establishing constitutive equations within the extended thermodynamic framework, the governing equations are derived. In the aspect of application, the transient thermo-electromechanical responses of multi-layered piezoelectric laminated composite structure with non-idealized interfacial conditions is investigated. The effects of material constants ratio and fractional order parameters of each layer on structural responses are evaluated and summarized to offer some new insights and guidelines on the strength design, thermal protection, and vibration control of multi-layered piezoelectric laminated composite structure under ultrafast heating condition.

ON FUZZY PARTIAL FRACTIONAL ORDER EQUATIONS UNDER FUZZIFIED CONDITIONS

This paper is devoted to investigating or computing the solution to one-dimensional partial fuzzy fractional order heat equation. In particular, one-dimensional fuzzy partial heat equation is hybridized into two equations by hybrid techniques along with the concept of parametric fuzzy number. For this investigation, a hybrid method of decomposition due to Adomian and Laplace transform is used. The considered techniques are presented for the computation of series of solutions of partial fractional order heat equation. The applied techniques have also provided the accuracy, simplicity and efficiency as compared to other existing methods. Finally, some illustrations are solved for the justification of our theoretical solution.

On fractional and nonlocal parabolic mean field games in the whole space

We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump Lévy processes with some σ-stable like behaviour. Included are σ-stable processes and fractional Laplace diffusion operators (−Δ)σ2, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order σ∈(1,2). We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs use pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques require no moment assumptions and use a weaker topology than Wasserstein.

Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u ( t , x ) ∈ D ∞ u(t,x)\in \mathbb {D}^\infty for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.

Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise

Hölder continuity of mild solutions of space-time fractional stochastic heat equation driven by colored noise

Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model

The generalized Cattaneo model describes the heat conduction system in the perspective of time-nonlocality. This paper proposes an accurate and robust meshless technique for approximating the solution of the time fractional Cattaneo model applied to the heat flow in a porous medium. The fractional derivative is formulated in the Caputo sense with order 1<α<2. First, a finite difference technique of convergence order O(δt3−α) is adopted to achieve the temporal discretization. The unconditional stability of the method and its convergence are analysed using the discrete energy technique. Then, a local meshless method based on the radial basis function partition of unity collocation is employed to obtain a full discrete algorithm. The matrices produced using this localized scheme are sparse and, therefore, they are not subject to ill-conditioning and do not pose a large computational burden. Two examples illustrate in computational terms of the accuracy and effectiveness of the proposed method.

Analytical investigation on heat transfer performance of H2O filled with SiO2 and TiO2 in a sinusoidal wavy channel

The peristaltic transport of hybrid nanofluid (SiO2–TiO2/H2O) flow in a sinusoidal wavy channel under the heat generation effect is reported in present analysis. We have simplified two dimensional equations of hybrid nanofluid for peristaltic transport by using long wavelength (δ≪1) and small Reynolds (Re) number assumptions. We also modeled temperature equation for hybrid nanofluid flow with the effect of heat generation. The two dimensional peristaltic flow equations of (SiO2–TiO2/H2O) are analytically solved by using perturbation method. The numerical integration and solutions of governing two dimensional equations are obtained in Mathematica software. The graphically behavior of solutions for temperature, pressure rise, streamlines and pressure gradient are elucidated with Matlab software. It is noticed that the pressure rise for hybrid nanofluid (SiO2–TiO2/H2O) increases for both amplitudes (a and b), solid fractional volume (ξ1) and heat generation (β). We observe that higher pressure gradient requires to retain same flux in narrow part of sinusoidal channel, also due to increase of amplitudes (a and b) pressure gradient increases. Moreover size of trapping bolus decreases by increase of channel width d and increases due to increase of heat generation parameter β.

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.

Identification of magnetic interactions and high-field quantum spin liquid in \u03b1-RuCl3

The frustrated magnet α-RuCl3 constitutes a fascinating quantum material platform that harbors the intriguing Kitaev physics. However, a consensus on its intricate spin interactions and field-induced quantum phases has not been reached yet. Here we exploit multiple state-of-the-art many-body methods and determine the microscopic spin model that quantitatively explains major observations in α-RuCl3, including the zigzag order, double-peak specific heat, magnetic anisotropy, and the characteristic M-star dynamical spin structure, etc. According to our model simulations, the in-plane field drives the system into the polarized phase at about 7 T and a thermal fractionalization occurs at finite temperature, reconciling observations in different experiments. Under out-of-plane fields, the zigzag order is suppressed at 35 T, above which, and below a polarization field of 100 T level, there emerges a field-induced quantum spin liquid. The fractional entropy and algebraic low-temperature specific heat unveil the nature of a gapless spin liquid, which can be explored in high-field measurements on α-RuCl3.

The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique

This research considers an inverse source problem for fractional diffusion equation that containing fractional derivative with non-singular and non-local kernel, namely, Atangana-Baleanu-Caputo fractional derivative. In our study, an explicit solution set is acquired via the expansion method and the overdetermination condition at a final time. The problem is ill-posed in the meaning of Hadamard and thus the solution does not continuously depend on the input data. We have applied the Tikhonov regularization method to regularize the unstable solution. For the estimation of convergence between the exact and the regularized solutions, we focus on two parameter choice rules, a-priori and a-posteriori parameter. In the end, a simulation example is utilized and discussed to affirm the presented theoretical results.

Quantitative normal approximations for the stochastic fractional heat equation

In this article we present a quantitative central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space–time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius R converges, as R tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.

Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.

Direct and inverse Cauchy problems for generalized space-time fractional differential equations

In this paper, explicit solutions of a class of generalized space-time fractional Cauchy problems with time-variable coefficients are given. The representation of a solution involves kernels given by convergent infinite series of fractional integro-differential operators, which can be extensively and efficiently applied for analytic and computational goals. Time-fractional operators of complex orders with respect to a given function are used. Further, we study inverse Cauchy problems of finding time dependent coefficients for fractional wave and heat type equations, which involve the explicit representation of the solution of the direct Cauchy problem and a recent method to recover variable coefficients for the considered inverse problems. Concrete examples and particular cases of the obtained results are discussed.

Electro‐osmotic flow of fractional second‐grade fluid with fractional Cattaneo heat flux through a vertical microchannel

AbstractThis study investigates the unsteady electro‐osmotic flow (EOF) of a fractional second‐grade fluid through a vertical microchannel with convection heat transfer. The fractional Cattaneo heat flux model will be used to modify the heat equation. The solutions for the velocity and the temperature have been derived by employing the Laplace and finite Fourier sine transforms and their numerical inverses. The results show that at the beginning of the time period, the fractional parameter postpones the movement of the fluid. Furthermore, the results show that at the high values of retardation time (non‐Newtonian case), the required time for the velocity and the flow rate to reach the steady state increases. Moreover, the heat relaxation time reduces the heat transfer until a critical time, and then the effect reverses.

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.

Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

Transient thermoelastic response of a size‐dependent nanobeam under the fractional order thermoelasticity

AbstractWith the miniaturization of structures, such as MEMS/NEMS, the size‐dependent effect has become an issue and attracted much attention. The well‐known theories describing the size‐dependent effect mainly include the nonlocal elasticity theory, the strain gradient theory and the modified coupled stress theory. Based on these theories, a number of works have been conducted to explore the size‐dependent behaviors of structures or devices in micro/nano‐scale, among them, majorities are on elastic performances, while, minorities are on thermoelastic performances. It is inevitable for structures suffering changeable temperature, as a consequence, thermal‐induced stress and deformation occur in structures and they are worth being fully concerned. For thermoelastic behaviors limited to small scale problems, the classical Fourier's heat conduction law may fail, meanwhile, new models, for example, fractional order heat conduction model, have been developed to modify Fourier's law. In present paper, the transient thermoelastic response of a nanobeam subjected to a ramp heating is investigated by combining the nonlocal elasticity theory and the fractional order heat conduction model. The governing equations are formulated and then solved by Laplace transform and its numerical inversion. The non‐dimensional temperature, displacement, stress, and deflection in the nanobeam are obtained and illustrated graphically. In calculation, the effects of the ramp‐heating time parameter, the nonlocal parameter and the fractional order parameter on the considered physical quantities are examined and discussed in detail.

The regular free boundary in the thin obstacle problem for degenerate parabolic equations

This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t − Δ x ) s (\partial _t - \Delta _x)^s for s ∈ ( 0 , 1 ) s \in (0,1) . The regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. The approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. By using several results from the elliptic theory, including the epiperimetric inequality, the optimal regularity is established for solutions as well as the H 1 + γ , 1 + γ 2 H^{1+\gamma ,\frac {1+\gamma }{2}} regularity of the free boundary near such regular points.

Thermal Performance of a Convective Functionally Graded Fin Using Fractional Non-Fourier Heat Conduction

To analyze the thermal performance of longitudinal fins made of functionally graded materials, this article studies the transient heat transfer in a longitudinally graded fin with power-law thermal conductivity in a convective ambiance using a fractional heat conduction model. This model has an advantage over the Fourier heat conduction in describing the anomalous heat diffusion and non-Fourier effect, and in avoiding the infinite velocity of heat waves and abrupt change of temperature. For heat shock at the base, instantaneous temperature response in an axially graded fin surrounded by convective fluid is obtained through numerical inversion of the Laplace transform. The effects of non-Fourier parameters such as the fractional order and phase lag of heat flux (relaxation time), and the gradient index of the thermal conductivity on the temperature distribution and fin efficiencies are analyzed and shown graphically. Two definitions of instantaneous fin efficiency are discussed for homogeneous and longitudinally graded fins. The occurrence of heat convection leads to a subdiffusion term, and the temperature change exhibits a significant difference when adopting the fractional-order model and the integer-order model. The obtained results are of great benefit to the design of enhancing the thermal performance of extended surfaces.

FRACTIONAL ORDER THERMOELASTICITY FOR PIEZOELECTRIC MATERIALS

This work is aimed at establishing a unified fractional thermoelastic model for piezoelectric structures, and shedding light on the influence of different definitions on the transient responses. Theoretically, based upon Cattaneo-type equation, a unified form of fractional heat conduction law is proposed by adopting Caputo Fabrizio fractional derivative, Atangana Baleanu fractional derivative and Tempered Caputo fractional derivative. Then, thermoelastic model of fractional order is formulated for piezoelastic materials by combining the unified heat conduction law and the governing equations of elastic and electric fields. Numerically, the present theoretical model is applied to study the transient responses of piezoelectric medium that is subjected to a thermal shock. The governing equations are analytically derived and numerically solved with the aids of Laplace transform method. The obtained results are graphically illustrated, and the influences of different definitions of fractional derivative and different fractional order are revealed. This work may be helpful for understanding the multi-coupling effect of elastic, thermal and electric fields, and for inspiring further developments of fractional calculus.