Fractional Heat Research Articles

Existence of solutions for time fractional semilinear parabolic equations in Besov–Morrey spaces

AbstractWe consider the Cauchy problem for a time fractional semilinear heat equation "Equation missing"where $$0<\alpha <1,\, \gamma >1,\, N\in \mathbb {Z}_{\geqslant 1}$$ 0 < α < 1 , γ > 1 , N ∈ Z ⩾ 1 and $$\mu (x)$$ μ ( x ) belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative $$^{C}\partial ^{\alpha }_{t}$$ C ∂ t α is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data.

On a class of stochastic fractional heat equations

For the fractional heat equation ∂ ∂ t u ( t , x ) = − ( − Δ ) α 2 u ( t , x ) + u ( t , x ) W ˙ ( t , x ) \frac {\partial }{\partial t} u(t,x) = -(-\Delta )^{\frac {\alpha }{2}}u(t,x)+ u(t,x)\dot W(t,x) where the covariance function of the Gaussian noise W ˙ \dot W is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that d > 2 + α d>2+\alpha is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field.

A PROBLEM IN FRACTIONAL ORDER THERMO-VISCOELASTICITY THEORY FOR A POLYMER MICRO-ROD WITH AND WITHOUT ENERGY DISSIPATION

A new study has been developed that considers the size-dependent interaction between viscoelastic deformation and thermal fields, incorporating the fractional heat conduction law with and without energy dissipation. The model is used for a particular one-dimensional problem involving a polymer micro-rod of arbitrary length experiencing three different types of thermal loading without the presence of any heat source. The study uses Laplace transforms and numerical inversion to examine how fractional order, nonlocal elasticity, and nonlocal thermal conduction impact thermal dispersion and thermo-viscoelastic response. Comparative numbers demonstrate the effects of various parameters. Findings demonstrate that nonlocal thermal and viscoelastic characteristics have a significant impact on all recorded field values, offering possible suggestions for the creation and assessment of thermal-mechanical attributes in nanoscale devices.

A semilinear problem associated to the space-time fractional heat equation in $${\mathbb {R}}^N$$

A semilinear problem associated to the space-time fractional heat equation in $${\mathbb {R}}^N$$

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0<a<1$$ 0 < a < 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

Predicting the heat release variability of Li-ion cells under thermal runaway with few or no calorimetry data

Accurate measurement of the variability of thermal runaway behavior of lithium-ion cells is critical for designing safe battery systems. However, experimentally determining such variability is challenging, expensive, and time-consuming. Here, we utilize a transfer learning approach to accurately estimate the variability of heat output during thermal runaway using only ejected mass measurements and cell metadata, leveraging 139 calorimetry measurements on commercial lithium-ion cells available from the open-access Battery Failure Databank. We show that the distribution of heat output, including outliers, can be predicted accurately and with high confidence for new cell types using just 0 to 5 calorimetry measurements by leveraging behaviors learned from the Battery Failure Databank. Fractional heat ejection from the positive vent, cell body, and negative vent are also accurately predicted. We demonstrate that by using low cost and fast measurements, we can predict the variability in thermal behaviors of cells, thus accelerating critical safety characterization efforts.

Heat and mass transfer effects on an unsteady MHD boundary layer flow of fractionalized fluid

This research investigates the transient magnetohydrodynamic (MHD) flow of a non-integer Maxwell fluid over a vertical plate, taking into account the effect of diffusion-thermo and parameter of heat absorption. The fractional derivative Caputo-Fabrizio is employed to model the problem. Using the Laplace integral transform technique, we solve the dimensionless governing equations to obtain the profiles of velocity, concentration, and temperature, which are then presented in graphical form for easy comparison and interpretation. The influence of several parameters—specifically, the fractional parameter and heat absorption is thoroughly analyzed and illustrated through a series of graphical representations. Furthermore, a comparison between traditional as well as fractionalized velocity fields is provided. The results show that chemical reactions and magnetic fields reduce the velocity profile, while diffusion-thermo and mass Grashof number increase the fluid velocity.

Thermodynamic properties and performance improvements of fractional Otto heat engine with repulsive bosons

This study presents calculations of a multiparticle system within the framework of fractional quantum mechanics. We specifically explore the energy levels of a bosonic system with repulsive interactions confined in a hard-wall box. The impacts of fractional parameters on the system’s thermodynamic properties are meticulously analyzed. Furthermore, utilizing this model, we construct a quantum Otto cycle and discover that the system exhibits Bose–Fermi duality under varying fractional parameters. Intriguingly, the introduction of fractional parameters enables to optimize the performance of the quantum heat engine, edging it closer to the Carnot efficiency.

On the fractional heat semigroup and product estimates in Besov spaces and applications in theoretical analysis of the fractional Keller–Segel system

This paper is concerned with the fractional Keller–Segel system in temporal and spatial variables. We consider fractional dissipation for the physical variables including a fractional dissipation mechanism for the chemotactic diffusion, as well as a time fractional variation assumed in the Caputo sense. We analyze the fractional heat semigroup obtaining time decay and integral estimates of the Mittag–Leffler operators in critical Besov spaces, and prove a bilinear estimate derived from the nonlinearity of the Keller–Segel system, without using auxiliary norms. We use these results in order to prove the existence of global solutions in critical homogeneous Besov spaces employing only the norm of the natural persistence space, including the existence of self-similar solutions, which constitutes a persistence result in this framework. In addition, we prove a uniqueness result without assuming any smallness condition on the initial data.

Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity

Abstract In this article, we consider the global and local well-posedness of the mild solutions to the Cauchy problem of fractional drift diffusion system with higher-order nonlinearity. The main difficulty comes from the higher-order nonlinearity. Instead of the convention that people always focus on the properties of the solution in critical spaces, here we are interested in non-critical spaces such as supercritical Sobolev spaces and subcritical Lebesgue spaces. For the initial data in these non-critical spaces, using the properties of fractional heat semigroup and the classical Hardy-Littlewood-Sobolev inequality, we obtain the existence and uniqueness of the mild solution, together with the decaying rate estimates in terms of time variable.

Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets

AbstractWe give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic $$\alpha $$ α -stable Lévy process. To derive the representation, we first establish the existence of the parabolic Martin kernel. This involves proving new boundary regularity results for both the fractional heat equation and the fractional Poisson equation with Dirichlet exterior conditions. Specifically, we demonstrate that the ratio of the solution and the Green function is Hölder continuous up to the boundary.

Effects of external magnetic field on the reflection and transmission of thermoelastic coupled waves with consideration of fractional order thermoelasticity

Effects of external magnetic field on the dispersion and attenuation feature as well as the reflection and transmission behavior of thermoelastic coupled waves are studied in the paper. The heat conduction behavior at instant heating is modeled by non-Fourier law with the fractional order derivatives. The effects of external magnetic field are incorporated into the motion equation by consideration of the Lorentz force. The dispersion and attenuation feature of the two longitudinal waves and one transversal wave and their reflection and transmission behaviors at interface are studied. The effects of external magnetic field on the dispersion and attenuation and on the reflection and transmission behavior are discussed based on the theoretic formulation and the numerical examples. It is found that the external magnetic field have influences on three kinds of coupled wave while the fractional order heat conduction only has evident influences on the thermal wave.

A sparse spectral method for fractional differential equations in one-spatial dimension

We develop a sparse spectral method for a class of fractional differential equations, posed on R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}$$\\end{document}, in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document} whereas the latter have global support. The global approximation space may contain different affine transformations of the basis, mapping [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document} to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving K independent sparse linear systems of size O(n)×O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)\ imes \\mathcal {O}(n)$$\\end{document}, with O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)$$\\end{document} nonzero entries, where K is the number of different intervals and n is the highest polynomial degree contained in the sum space. This results in an O(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(n)$$\\end{document} complexity solve. Applications to fractional heat and wave equations are considered.

The Solution Comparison of Fractional Heat Transfer and Porous Media Equations Using Analytical Techniques

The Solution Comparison of Fractional Heat Transfer and Porous Media Equations Using Analytical Techniques

Stability of a fractional heat equation with memory

Of concern is a fractional differential problem of order between zero and one. The model generalizes an existing well-known problem in heat conduction theory with memory. First, we justify the replacement of the first order derivative by a fractional one. Then, we establish a Mittag-Leffler stability result for a class of heat flux relaxation functions. We will combine the energy method with some properties from fractional calculus.

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter α∈(0,1). This highlights the distinct feature of the alignment operator.

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable.

This study suggests a strategy for calculating the fuzzy analytical solutions to a two-dimensional fuzzy fractional-order heat problem including a diffusion variable connected externally. We propose Sawi residual power series scheme (SRPSS) which is the amalgamation of Sawi transform and residual power series scheme under the Caputo fractional differential operator. We demonstrate three different examples to derive the fuzzy fractional series solution which is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity. The most significant aspect of this scheme is that it derives the results without time effort compared with the traditional residual power series approach. Our findings confirm that the SRPSS is a robust and valuable method for approximating the solution of fuzzy fractional problems. Furthermore, we provide 2D and 3D symbolic representations to present the physical behavior of fuzzy fractional problems under the lower and upper bounded solutions.

Highly stable mesoporous-controlled corncob biochar confined paraffinic material with exceptional latent heat retention performance

Bio-based composites fabricated from bio-sourced waste feedstocks have attracted increasing interest and have shown potential applications in sustainable energy infrastructure development. Thermal management using phase-change materials (PCMs) has appeared as an innovative approach for energy storage and cooling, where pure PCMs that lack thermal stability are encapsulated in solid matrices to ensure the charge/discharge of the latent heat of the composite. However, their energy storage density, thermal stability, and durability are reduced after multiple thermal cycles, besides the complicated and expensive nonrenewable fossil feedstock utilization. Biochar, mostly used for soil remediation, can be redirected and repurposed for the encapsulation of PCMs in renewable energy storage infrastructures. In this study, a “greener” biochar-based composite was fabricated from agricultural corncob-derived biochar (88.9 % mesopore proportion) and a commercially available paraffinic PCM (n-hexadecane) following vacuum impregnation strategies. Corncob biochar combined with hexadecane exhibited an efficient energy per unit mass of 250.6 kJ kg−1 and a latent heat retention of 110.3 % after 500 consecutive thermal cycles. In addition, the corncob biochar/hexadecane composite demonstrated high fractional crystallinity and relative latent heat efficiency (98.9 %). The composite exhibited promising chemical compatibility with composite components, stability, and competitive energy density even compared with a composite made of nonrenewable fossil-based materials, like exfoliated graphene nanoplatelets, which demonstrated reduced latent heat retention after multiple heating-cooling cycles. Intermolecular interactions, such as hydrogen bonding, between the guest molecular chains and host materials played a significant role in the thermal performance of the as-synthesized composite materials. This approach will contribute to overcoming the complexities and costs of composite PCM development in thermal management and thermal energy storage applications and support and advance the UN's sustainable development goal (affordable and clean energy).

Carleson Measure Associated with the Fractional Heat Semigroup of Schrödinger Operator

Carleson Measure Associated with the Fractional Heat Semigroup of Schrödinger Operator

Thermomagnetic Transient Analysis of an Infinitely Long Transverse Isotropic Annular Cylinder Using the MGT Fractional Heat Conduction Model with a Non-Singular Kernel

Thermomagnetic Transient Analysis of an Infinitely Long Transverse Isotropic Annular Cylinder Using the MGT Fractional Heat Conduction Model with a Non-Singular Kernel