Generalized Heat Equation Research Articles

Harmonic analysis associated with the linear canonical Fourier–Jacobi transform

The main objective of this paper is to develop a new harmonic analysis related to a Fourier–Jacobi type operator Δ α , β m of the real line. We define and study the linear canonical Fourier–Jacobi transform F α , β m . We study some important properties, inversion formula, Plancherel theorem, Paley–Wiener theorem and Riemann–Lebsgue lemma. An application in solving a generalized heat equation is given.

Mathematical modeling of the thermal process of arc erosion with current carrying heating effect in a temperature gradient

This paper presents a mathematical model aimed at comprehensively understanding the thermal processes associated with arc erosion of closed electrical contacts triggered by the instantaneous explosion of micro-asperities including Thomson effect and Joule heat source. The phenomenon involves vaporization and liquid zones, and the temperature distribution is governed by a generalized heat equation, accounting for the heating effect due to current flow in temperature gradients and effect of heat source. The proposed model allows for a nuanced analysis of the thermal dynamics, shedding light on the complex phenomena involved in arc erosion. The proposed method in this paper employs similarity transformation techniques to effectively reduce the complexity of the problem, transforming it into a set of manageable ordinary differential equations. The study establishes the existence and uniqueness of the solution through rigorous analysis. The behavior of the solution of the problem is successfully considered for special cases of Thomson and thermal coefficients. By leveraging similarity transformations, the paper offers a powerful approach for unraveling the intricacies of the thermal processes in arc erosion of closed electrical contacts, providing valuable insights into the phenomena associated with current-carrying heating in the presence of temperature gradients.

Special affine Fourier transform of tempered distributions and pseudo-differential operators

The primary aim is to develop a new class of pseudo-differential operators by incorporating the Special Affine Fourier Transform (SAFT) and some of its fundamental properties in Schwartz and tempered distribution space. The secondary aim is to explore the utility of the SAFT in constructing a new generalized heat equation and derive its analytical solution. Further, we have investigated particular cases of the proposed generalized heat equation. Furthermore, we have visually illustrated solutions of this equation via MATLAB R2023b.

Characterization of Pseudo-Differential Operators Associated with the Coupled Fractional Fourier Transform

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation.

Solar Disc Concentrator: Material Selection for the Receiver

Solar concentration is the ability to harness solar radiation in order to increase the temperature of a receiver. The receiver is a component into which a heat transfer fluid can flow in an ORC system, which produces electricity, or it can be used for high-temperature thermal storage or even to implement thermochemical cycles. The choice of material is critical to ensure optimal performance and long-lasting operation. It is also essential that such material can operate at high temperatures and high thermal gradients. In short, material identification involves high thermal stresses that result in structural deformation. Different metal alloys were used to verify that the yield strength limit was not exceeded due to thermal stress induced by concentrated solar radiation. Starting with the general heat equation, the problem was implemented in Matlab. The purpose was to test whether thermal stress exceeds the yield strength, which is the condition in which elastic bonds in the material are changed, causing deformation. This condition, if exceeded, is sufficient to discard the material; otherwise, it is a necessary but not sufficient condition to resist over time. The best material identified was Inconel 740H, which had a high yield strength value and the lowest temperature difference. Under extreme working conditions, it withstood induced thermal shocks.

A coupled model of transport-reaction-mechanics with trapping, Part II: Large strain analysis

A coupled finite strain chemo-transport-mechanical formulation with trapping is here proposed to extend a previous work set in the realm of small strain theory in continuum mechanics. The theory is rooted in non-equilibrium rational thermodynamics. The kinematics is based on a multiplicative decomposition of the deformation gradient to account for swelling and shrinking, thermal, elastic and inelastic contributions. Mass balance laws and balance of linear and angular momentum, as well as the laws of thermodynamics for a convecting body, are directly formulated in their material description, after specifications of some standard transformation rules between current and reference configuration. Thermodynamic restrictions are identified based on the functional dependence of the referential Helmholtz free energy density, which is chosen as the thermodynamic potential, and further subjected to a constitutive additive decomposition. Constitutive prescriptions for the chemical potentials, referential heat and mass fluxes, chemical kinetics and the generalized heat equation lead to the establishment of the governing equations. The theoretical framework is complemented by numerical simulations, highlighting the potential of the proposed formulation in multi-physics applications.

Mathematical modeling of the heat process in closure electrical contacts with a heat source

This study presents a mathematical model that describes the heat process during the vaporization of metal at electrical contacts. The arc erosion model for the micro-asperity in the liquid state is established using a one-phase Stefan problem. Two problems involving Joule heating effects on the generalized heat equation are considered. The solution of these two problems is based on the similarity principle, which allows for the reduction of the problem to ordinary differential equations. The existence and uniqueness of the solutions are proven using fixed point theory in a Banach space. Additionally, a solution for the particular case of constant thermal coefficients is provided.

Ground Surface‐Soil Thermal Inversion From In Situ Measurement

AbstractSoil heat flux at the ground surface (G) is an essential component of energy for the recirculation of water and other masses on the land surface. However, direct measurement is difficult and inaccurate, and available data are inadequate owing to the inconvenience of installing the heat sensor plate immediately beneath the true surface. Additionally, thermal properties such as the diffusivity, conductivity, and volumetric heat capacity are intricately influenced by water, vegetation, labile weather, and environmental conditions. Errors of estimation for them from indirect measurement of water content can be substantial and almost impossible without adequately and accurately determination of the soil parameters in situ. In this study, a more general parabolic heat equation is analytically solved for thermal diffusion of gradually varying state properties, and an inverse method is used to determine the parameters so that the properties and G can be simultaneously obtained at a site of measurement. Comparison and validation against data from a numerical model and experimental‐site measurements indicate well agreement and consistency. The retrieved G reveals relatively larger in scale that indicates a non‐negligible disparity from the under‐surface plate measured and the estimations by previous heat‐correction methods, suggesting that G should be regarded more significant in roles for the studies of near‐surface processes, particularly in rather dry conditions and in sub‐diurnal time scale. The present inversion provides some insights into the soil thermal physics of the land surface. Finally, a formula is suggested for the completion of the plate‐based heat flux measurements.

Two-phase Stefan problem for generalized heat equation with nonlinear thermal coefficients

In this article we study a mathematical model of the heat transfer in semi infinite material with a variable cross section, when the radial component of the temperature gradient can be neglected in comparison with the axial component. In particular, the temperature distribution in liquid and solid phases of such kind of body can be modeled by Stefan problem for the generalized heat equation. The method of solution is based on similarity principle, which enables us to reduce generalized heat equation to nonlinear ordinary differential equation. Moreover, we determine temperature solution for two phases and free boundaries which describe the position of boiling and melting interfaces. Existence and uniqueness of the similarity type solution is provided by using the fixed point Banach theorem.

Generalized Heat Equation with the Caputo–Fabrizio Fractional Derivative for a Nonsimple Thermoelastic Cylinder with Temperature-Dependent Properties

Generalized Heat Equation with the Caputo–Fabrizio Fractional Derivative for a Nonsimple Thermoelastic Cylinder with Temperature-Dependent Properties

Reconstructing the Potential of the Generalized Heat Equation

We reconstruct the potential q(x) for the generalized heat equation of the form u_{t}-b(x)u_{xx}-a(x)u_{x}-q(x)u=0 under general conditions, and by means of a Carleman’s estimate.

The Reflection Principle in the Control Problem of the Heat Equation

We consider the control problem for the generalized heat equation for a Schrödinger operator on a domain with a reflection symmetry with respect to a hyperplane. We show that if this system is null-controllable, then so is the system on its respective parts and the corresponding control cost does not exceed the one on the whole domain. As an application, we obtain null-controllability results for the heat equation on half-spaces, orthants, and sectors of angle π/2n. As a byproduct, we also obtain explicit control cost bounds for the heat equation on certain triangles and corresponding prisms in terms of geometric parameters of the control set.

Investigation of the impact of pipe geometric form on earth-to-air heat exchanger performance using Complex Finite Fourier Transform analysis. Part I: Operation in cooling mode

This paper deals with the impact of the geometric form of pipe section upon the performance of earth-to-air heat exchangers operating in cooling mode. For this purpose, three different configurations, namely circular, rectangular and elliptical were selected. Based on the heat balance analysis, a transient two-dimensional model was developed for the disturbed soil around the buried pipe. The discretization of the partial differential equation by applying the Finite Control-Volume Method yields a new generalized heat equation valid for all kinds of cylinders. Then, the resulting set of ordinary differential time-equations was linearized by using the technique of Complex Finite Fourier Transform. After successfully validating the model against available data in the literature, a thorough sensitivity analysis was carried out for assessing the impact of geometric and operating parameters including pipe length and cross-section and air speed. Further, the effect of pipe flatness relating to the rectangular and elliptical configurations was also considered through the geometric aspect ratio.The results show that the rectangular and elliptical buried pipes are more effective because they possess large exchange surfaces compared to the circular (conventional) one. For instance, at a pipe cross-section of 0.07 m2 and a geometric aspect of 9, the exchange surface between the flowing air and pipe wall in the rectangular and elliptical exchangers are larger of 88.3% and 93.6% than that of the conventional system, respectively. Moreover, at a geometric aspect of 4 and a pipe length of 60 m, the outlet temperatures supplied by the rectangular and elliptical systems are of 1.5 and 1.3 °C less than that obtained with the conventional one. Regarding the influence of pipe dimensions, it is shown that the effect of pipe length is dominant compared to that of pipe section and the performance of the system is proportional to the pipe length. Likewise, the EAHE performance increases with the increasing of the pipe geometric aspect, but a value of 4 is found to be a good compromise. Though, the performance of EAHE is found to drop dramatically at high speeds of air.

Generalized heat equation with the Caputo-Fabrizio fractional derivative for a nonsimple thermoelastic cylinder with temperature-dependent properties

Generalized heat equation with the Caputo-Fabrizio fractional derivative for a nonsimple thermoelastic cylinder with temperature-dependent properties

Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

The fractional energy balance equation

AbstractClassical Energy Balance Equations (EBEs) are differential equations of integer order (h = 1), here we generalize this to fractional orders: the Fractional EBE (FEBE, 0 < h ≤ 1). In the FEBE, when the Earth is perturbed by a forcing, the temperature relaxes to equilibrium via a slow power‐law process: h = 1 is the exceptional (but standard) exponential case. Our FEBE derivation is phenomenological, it complements derivations based on the classical continuum mechanics heat equation (that imply h = 1/2 for the surface temperature) and of the more general Fractional Heat Equation which allows for 0 < h < 2. Unlike some of the earlier “scale free” models based purely on scaling, the FEBE has an extra blackbody radiation term that allows for energy balance. It therefore has two scaling regimes (not one), it has the advantage of being stable to infinitesimal step‐function perturbations and it has a finite Equilibrium Climate Sensitivity. We solve the FEBE using Green's functions, whose high‐ and low‐frequency limits are power laws with a relaxation scale transition (several years). When stochastically forced, the high‐frequency parts of the internal variability are fractional Gaussian noises that can be used for monthly and seasonal forecasts; when deterministically forced, the low‐frequency response describes the consequences of anthropogenic forcing, it has been used for climate projections. The FEBE introduces complex climate sensitivities that are convenient for handling periodic (especially annual) forcing. The FEBE obeys Newton's law of cooling, but the heat flux crossing a surface nonetheless depends on the fractional time derivative of the temperature. The FEBE's ratio of transient to equilibrium climate sensitivity is compatible with GCM estimates. A simple ramp forcing model of the industrial‐epoch warming combining deterministic (external) with stochastic (internal) forcing is statistically validated against centennial‐scale temperature series.

Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

New perspectives for modelling ballistic-diffusive heat conduction

Among the three heat conduction modes (diffusive, second sound, and ballistic propagation), the ballistic one is the most difficult to model. In the present paper, we discuss its physical interpretations and show different alternatives to its modelling. We highlight two of them: a thermo-mechanical model in which the generalized heat equation—the so-called Maxwell–Cattaneo–Vernotte equation—is coupled to thermal expansion. The other one uses internal variables in the framework of non-equilibrium thermodynamics. For the first one, we developed a numerical solution and tested it on the heat pulse experiment performed by McNelly et al. on NaF samples. For the second one, we found an analytical solution that emphasizes the role of boundary conditions. This analytical method is used to validate an earlier developed numerical code.

Asymptotic Distributions for Power Variations of the Solutions to Linearized Kuramoto–Sivashinsky SPDEs in One-to-Three Dimensions

We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in d=1,2,3. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient.

HEAT EQUATION OBTAINED BY l -DIFFERENCE OPERATOR CHANGE OF TIME WITH TWO VARIABLES

We investigate the generalized partial difference operator and propose a model of it in discrete heat equation. For example, the temperature in an object changes with time and the position within the object. In this paper Generalized heat equation obtained by two dimensional l -difference operator change of time with two variables