## What is/are Heat Equation?

Heat Equation - We prove upper estimates for differences between the solution to the Cauchy problem for the heat equation with initial function u0(x) and its Stieltjes means.^{[1]}In this paper we study removable singularities for regular $(1,1/2)$-Lipschitz solutions of the heat equation in time varying domains.

^{[2]}We first study stabilization of heat equation with globally Lipschitz nonlinearity.

^{[3]}In this paper we study the local and global existence of solutions for a class of heat equation in whole \begin{document}$ \mathbb{R}^{N} $\end{document} where the nonlinearity has a critical growth for \begin{document}$ N \geq 2 $\end{document}.

^{[4]}The initial conditions for both fluid and heat equations are taken of low regularity.

^{[5]}Primarily, Laplace transformation is applied to dimensionless fractional models, and later Stehfest’s numerical algorithm is invoked to anticipate solutions of momentum and heat equations in principal coordinates.

^{[6]}In this work we benchmark this approach in the quantum realm by solving the heat equation for a square plate subject to fixed temperatures at the edges and random heat sources and sinks within the domain.

^{[7]}Our argument consists of importing information from the heat equation, through approximation and localization methods.

^{[8]}The same simulation is carried out with phonon heat transport approximated by the heat equation, and the results indicate that it is difficult for the heat equation to accurately reproduce the phonon heat transport.

^{[9]}Our model considers a DC voltage but solves in time the temperature evolution coupling the heat equation and the current continuity equation.

^{[10]}In this article, we study a certain type of boundary behaviour of positive solutions of the heat equation on the Euclidean upper half-space of R n + 1.

^{[11]}The state and adjoint equations of the heat equation are solved with integral equation techniques which avoid a discretization in the domain.

^{[12]}This paper is addressed to a study of the stability and stabilization of heat equation in non-cylindrical domain.

^{[13]}We report on the solution of the heat equation, which describes the electrocaloric effect in a ferroelectric layer.

^{[14]}In the simulation study, breast and cancerous tissue were modeled in three dimensions and the bioheat equation for active and passive modes was solved by using realistic values of the tissues.

^{[15]}We prove the existence of a sticky-reflected solution to the heat equation on the space interval [0, 1] driven by colored noise.

^{[16]}The proof of the convergence is based on a compensated compactness argument and on the derivation of sharp decay estimates for solutions to a heat equation with fast diffusion in time.

^{[17]}This paper investigates the recovery for time-dependent coefficient and free boundary for heat equation.

^{[18]}Radiation effect, Joule heating and dissipation are considered in heat equation Furthermore random and thermophoresis motion effects are scrutinized.

^{[19]}We present new extended Strichartz estimates for the solutions to the heat equation with inhomogeneous nonlinearity in the mass-energy intercritical framework in space dimension d ≥ 1.

^{[20]}Based on the equivalence of A-stability and G-stability, the energy technique of the six-step BDF method for the heat equation has been discussed in [Akrivis, Chen, Yu, Zhou, Math.

^{[21]}In this paper, this question is explored through an analytical solution of the heat equation in a representative model of a typical EV IPM heat exchanger (HX).

^{[22]}The proposed mathematical model is constructed based on the Euler–Bernoulli model, the heat equation with two-phase lag and coupled plasma wave equation that indicates the prediction of thermal, elastic and photovoltaic effects in the microbeam resonators.

^{[23]}We consider the heat equation for monolayer two-dimensional materials in the presence of heat flow into a substrate and Joule heating due to electrical current.

^{[24]}The main idea is to get an approximate solution of the temperature distribution from a simplified Pennes bioheat equation, and further fit the solution to the coagulation measurements on ex vivo porcine liver.

^{[25]}The analytical solution of the gravest mode of the heat equation compares well to the temperature profiles, and the numerical integration of the resulting forced heat equation compares favorably to the temporal evolution of the time-series data.

^{[26]}More precisely, we set logarithmic stability estimates in the determination of the two time-dependent first-order convection term and the scalar potential appearing in the heat equation.

^{[27]}The heat exchange problem has been solved through a modified version of the Pennes' bioheat equation assuming a temperature dependency of blood perfusion and metabolic heat, i.

^{[28]}The methodology is tested on the heat equation, advection equation, and the incompressible Navier-Stokes equations, to show the variety of problems the ROM can handle.

^{[29]}Both the steady and unsteady states for each fractal bioheat equation were obtained and their implications on living cells in the presence of growth of a large tumour were analysed.

^{[30]}The beam is subject to some vibrations and it is modelled as a Timoshenko structure coupled with a heat equation of hyperbolic type.

^{[31]}Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connections between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex.

^{[32]}To determine the major affecting parameters, the dimensionless form of the heat equation is derived and solved numerically.

^{[33]}In this paper, neural solvers are implemented in TensorFlow for a variety of differential equations, namely: linear and nonlinear ordinary differential equations of the first and second order; Poisson’s equation, the heat equation, and the inviscid Burgers’ equation.

^{[34]}In the new model, Fourier’s and Fick’s laws have been improved by including the relaxation times in the Green–Naghdi theory in the framework of Moore–Gibson–Thompson (MGT) heat equation.

^{[35]}Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution.

^{[36]}We underline that instead of establishing estimates similar to the heat equation for the angular velocity $$\omega $$ in Chen and Miao (2012), we find $$\omega $$ is dominated by damping effect in the low frequency.

^{[37]}By studying and solving the direct problem for the heat equation in composite materials, we can determine the function spaces and solve the inverse problem.

^{[38]}We consider the heat equation coupled with the Maxwell system, the Ampere-Maxwell equation being coupled to the heat equation by the permittivity, which depends on the temperature due to thermal agitation, and the heat equation being coupled to the Maxwell system by the volumic heat source term.

^{[39]}Objective: Given the low bone-thermal conductivity whereby heat generated by osteotomies is not effectively dissipated and tends to remain within the surrounding tissue (peri-implant), increasing the possibility of thermal-injury; this work attempts to obtain an exact analytical solution of the heat equation under exponential thermal-stress, modeling transient heat transfer and temperature changes in Ti implants upon hot-liquid intake.

^{[40]}Applications to transport and heat equations are also provided.

^{[41]}Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties.

^{[42]}In this work, we extend the gradient discretisation method to the Navier–Stokes model coupled with the heat equation.

^{[43]}In the modern time, Bessel’s functions appear in solving many applications of engineering and natural science together with many equations such as Schrodinger equation, Laplace equation, heat equation, wave equation and Helmholtz equation in cylindrical or spherical coordinated, in this work, we introduce complex SEE integral transform of Bessel’s functions with some important applications of complex SEE transform of Bessel’s functions for evaluating the integral, which contain Bessel’s functions, are given.

^{[44]}We here propose a method that, through linear transmission lines and time-domain non linear sub-circuits, solves the heat equation for one-dimensional problems in the periodic regime, involving layered structures and boundary conditions and/or media with a non-linear behavior.

^{[45]}The construction exploits the intrinsic modularity of port-Hamiltonian systems to obtain a formulation of linear thermoelasticity as an interconnection of the elastodynamics and heat equations.

^{[46]}This study presents the mathematical model of space fractional variable-order bioheat equation and proposes a numerical method to find its solution.

^{[47]}We modeled the generated thermal wave field and the resulting PT-OCT signal as a function of concentration of this component using the bio-heat equation.

^{[48]}A more complete description of conductive heat transfer can be obtained from the heat equation by means of a two-port network.

^{[49]}This paper proposes a wavelet transformation solution to the Heat equation.

^{[50]}

## finite element method

We present a space-time least squares finite element method for the heat equation.^{[1]}We conducted our study by solving the stress and heat equations using the Finite Element Method (FEM).

^{[2]}The finite element method is used to solve a one-dimensional bioheat equation.

^{[3]}The software package includes: the two-dimensional axisymmetric heterogeneous model of a medium; the kinetic model of radiation transport and the Monte Carlo method for radiation fields modeling; the non-stationary heat equation, the finite element method and the weighted residual method for thermal fields modeling.

^{[4]}To decipher thermal properties data in this inverse heat equation problem, a 3D finite element method modelling has been used to simulate the behaviour of both the samples and the vacuum chamber.

^{[5]}The phenomenon studied is governed by Maxwell's equations coupled with the heat equation, which are solved by the finite element method.

^{[6]}The longitudinal and transversal temperature distribution in active waveguide, doped with Yb/Er ions, under conditions of laser generation was calculated as the composition of two mathematical models: computation of longitudinal pump radiation distribution by solution of rate equations for Yb/Er doped active medium and computation of transversal temperature distribution by solution of heat equation for convectively cooled fiber using finite element method.

^{[7]}The finite element method (FEM) is used to solve the bio-heat equation at the tumor domain, consisting of radiation and convection-based boundary conditions.

^{[8]}The finite element method is used to solve the two-dimensional bio-heat equation.

^{[9]}A two-dimensional Pennes bioheat equation is solved to find the temperature variation in breast tissue with and without tumor/cyst during the menstrual cycle by using the finite element method.

^{[10]}Using the finite element method (FEM), the heat equation was solved numerically across the porous unit cell domain.

^{[11]}We use the Penne bioheat equation that accounts for the elements of conduction, perfusion, and metabolism along with the COMSOL Finite Element Method for the analysis of this thermoregulation problem.

^{[12]}The equations of hydrodynamics and heat equations were simulated by the finite element method in the FreeFem++ software.

^{[13]}Here, we create a two dimensional axisymmetric geometry of a probe embedded within a tissue material, solving the coupled electromagnetic and bioheat equation using the finite element method, utilizing hp discretisation and the NGSolve library.

^{[14]}

## boundary value problem

The projection-regularization method is used to solve the inverse boundary value problem for the heat equation and obtain error estimates of this solution correct to the order.^{[1]}On the basis of the method of generalized functions, generalized solutions of boundary value problems are constructed using the Green's function for the heat equation and the Green's tensor of the Lame equations under the action of non-stationary power and heat sources of various types.

^{[2]}In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions.

^{[3]}The first type of boundary value problem for the heat equation on a rectangle is considered.

^{[4]}The spatially and time resolved temperature field is calculated by integrating the corresponding boundary value problem of the heat equation.

^{[5]}In this paper we study the solvability of the boundary value problem for the heat equation in a domain that degenerates into a point at the initial moment of time.

^{[6]}A two layer Implicit method on hexagonal grids is proposed for approximating the solution to first type boundary value problem of heat equation on rectangle.

^{[7]}In the proposed publication, we present a geometric method for solving a boundary value problem for a nonstationary heat equation.

^{[8]}Efficient high-order integral equation methods have been developed for solving boundary value problems of the heat equation in complex geometry in two dimensions.

^{[9]}A simple idea of finding a domain that encloses an unknown discontinuity embedded in a body is introduced by considering an inverse boundary value problem for the heat equation.

^{[10]}To derive this information, substantial qualitative and quantitative analysis of a mixed boundary value problem for the heat equation and an illustrative test case for IHTP are provided.

^{[11]}The vaporized body is considered as a thin plate and the problem is formulated as a system of one-dimensional boundary-value problems of the heat equation written for the electron and lattice components.

^{[12]}

## partial differential equation

The ADMM tool wasapplied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type.^{[1]}Heat equation is a partial differential equation describing the temperature change of an object with time.

^{[2]}It is known that phenomena of nature or physical systems can be modeled using partial differential equations (PDEs) such as wave equations, heat equations, Poisson’s equation and so on.

^{[3]}The heat equation is then solved using the Method of Lines, which reduces a parabolic partial differential equation into a set of ordinary differential equations.

^{[4]}An important place in the theory of partial differential equations and its applications is occupied by the heat equation, a representative of the class of the so-called parabolic equations.

^{[5]}In the present paper we solved heat equation (Partial Differential Equation) by various methods.

^{[6]}A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition under the setting of heat equation.

^{[7]}These strategies are evaluated on two spatio-temporal evolving problems modeled by partial differential equations: the 2D propagation of acoustic waves (hyperbolic PDE) and the heat equation (parabolic PDE).

^{[8]}The results are illustrated with examples on designing controllers for linear wave and heat equations, and on studying the stability of a system of coupled partial differential equations.

^{[9]}Uncertain heat equation is a class of uncertain partial differential equations involving Liu processes.

^{[10]}

## space time white

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.^{[1]}On one hand, this work builds on the recent works on delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space-time white noise.

^{[2]}We study the Cauchy problem of the nonlinear stochastic fractional heat equation ∂ t u = − ν 2 ( − ∂ x x ) α 2 u + σ ( u ) W ˙ ( t , x ) on real line R driven by space-time white noise with bounded initial data.

^{[3]}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.

^{[4]}We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$ λ β e β u , forced by an additive space-time white noise.

^{[5]}

## initial boundary value

As examples, we consider initial boundary value problems associated with the heat equation and the wave equation.^{[1]}As a result, a new analytical method for solving initial-boundary value problems for a two-dimensional heat equation has been developed.

^{[2]}Based on the solution of the first initial-boundary value problem for an inhomogeneous two-dimensional heat equation, we state and study inverse problems, whose right-hand sides contain unknown factors depending on spatial and time variables.

^{[3]}ABSTRACTIn this paper, we study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity.

^{[4]}

## optimal control problem

We consider the integral definition of the fractional Laplacian and analyze a linearquadratic optimal control problem for the so-called fractional heat equation; control constraints are also considered.^{[1]}In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints.

^{[2]}This paper studies the time optimal control problem for systems of heat equations coupled by a pair of constant matrices.

^{[3]}As an application of the established improved regularity properties, we derive new finite element error estimates for the dG(0)-cG(1)-discretization of a purely state-constrained linear-quadratic optimal control problem governed by the heat equation.

^{[4]}

## self similar solution

It is shown that in the limiting case of a zero degree of the Riesz potential, this solution coincides with the self-similar solution of the classical heat equation.^{[1]}In this work, new properties of a radially symmetric self-similar solution of the nonlinear heat equation with source were obtained.

^{[2]}We consider positive solutions of the semilinear heat equation with supercritical power nonlinearity, and construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution.

^{[3]}

## initial value problem

As a direct application, we introduce the heat semigroup generated by the Bessel-type operators $$\Delta_{\nu}^{\mathbf{m}^{-1}}=\frac{d^{2}}{dx^{2}}+\left( \frac{2\nu +1}{x}+2i \frac{a}{b} x\right) \frac{d}{dx}-\left( \frac{a^{2}}{b^{2}}x^{2}-2i\left( \nu +1\right) \frac{a}{b}\right) $$ and use it to solve the initial value problem for the heat equation governed by $\Delta_{\nu}^{\mathbf{m}^{-1}}.^{[1]}This paper considers the initial value problem for nonlinear heat equation in the whole space R N $\mathbb{R}^{N}$.

^{[2]}By studying and solving the direct problem for the heat equation in composite materials, it is possible to determine the function spaces and solve the inverse initial value problem.

^{[3]}

## finite dimensional observer

Recently finite-dimensional observer-based controllers were introduced for the 1D heat equation, where at least one of the observation or control operators is bounded.^{[1]}In our recent paper a constructive method for finite-dimensional observer-based control of 1-D linear heat equation was suggested.

^{[2]}We study constant input delay compensation by using finite-dimensional observer-based controllers in the case of the 1D heat equation.

^{[3]}

## dimensional wave equation

These results are specified for the heat equation, the simplest transport equation and the one-dimensional wave equation.^{[1]}We demonstrate our method in the forecasting task for the two-dimensional wave equation and the Burgers-Fisher equation in multiple geometries with different boundary conditions, and the ten-dimensional heat equation.

^{[2]}In this paper, we have tried to approach the concepts of two-dimensional wave equation and one dimensional heat equation through the means of the Navier Stoke’s equation for unsteady and incompressible flow.

^{[3]}

## damped wave equation

We propose a static output-feedback controller and model the resulting closed-loop system as a disturbed (due to quantization) nonlinear heat equation (for the first-order systems) or damped wave equation (for the second-order systems) with delayed point state measurements, where the state is the relative position of the agents with respect to the desired curve.^{[1]}For the proof of the results, we use the L p - L q type estimates for the fundamental solutions of the damped wave equation and end-point maximal regularity for the inhomogeneous heat equation in that space with a detailed estimate of difference between the symbol of the heat kernel and fundamental solution of the damped wave equation.

^{[2]}

## Semilinear Heat Equation

We consider nonnegative solutions of a semilinear heat equation u t − Δ u = u p in R N ( N ≥ 3 ) with p = N / ( N − 2 ) and a nonnegative initial data u 0 ∈ L N / ( N − 2 ) ( R N ) which has a singularity at ξ 0 ∈ R N.^{[1]}We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy–Sobolev parabolic equation, in the energy space.

^{[2]}We are concerned with blow-up mechanisms in a semilinear heat equation: \begin{document}$ u_t = \Delta u + |x|^{2a} u^p , \quad x \in \textbf{R}^N , \, t>0, $\end{document} where \begin{document}$ p>1 $\end{document} and \begin{document}$ a>-1 $\end{document} are constants.

^{[3]}We consider the semilinear heat equation with Sobolev subcritical power nonlinearity in dimension $N=2$, and $u(x,t)$ a solution that blows up in finite time $T$.

^{[4]}The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.

^{[5]}We are interested in the Gevrey properties of the formal power series solution in time of the inhomogeneous semilinear heat equation with a power-law nonlinearity in $1$-dimensional time variable $t\in\mathbb{C}$ and $n$-dimensional spatial variable $x\in\mathbb{C}^n$ and with analytic initial condition and analytic coefficients at the origin $x=0$.

^{[6]}In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints.

^{[7]}In this paper, we investigate the controllability for a semilinear heat equation with memory and internal control in a bounded domain of R n.

^{[8]}These conditions generalize the ones given by the author for the linear case (Remy in J Dyn Control Syst 22(4):693–711, 2016; J Dyn Control Syst 23(4):853–878, 2017) and for the semilinear heat equation (Remy in J Math Anal Appl 494(2):124619, 2021).

^{[9]}The paper is concerned with a kind of minimal time impulse control problem for a semilinear heat equation.

^{[10]}In this paper, we study the semidiscrete approximation for the following semilinear heat equation with a singular boundary outflux ∂u ∂t = uxx + (1− u)−p, 0 < x < 1, t > 0, ux(0, t) = 0, ux(1, t) = −u(1, t)−q, t > 0, u(x, 0) = u0(x), 0 ≤ x ≤ 1.

^{[11]}We give a sufficient conditions for the existence, locally in time, of solutions to semilinear heat equations with nonlinearities of type \begin{document}$ |u|^{p-1}u $\end{document} , when the initial datas are in negative Sobolev spaces \begin{document}$ H_q^{-s}(\Omega) $\end{document} , \begin{document}$ \Omega \subset \mathbb{R}^N $\end{document} , \begin{document}$ s \in [0,2] $\end{document} , \begin{document}$ q \in (1,\infty) $\end{document}.

^{[12]}As a model problem, existence and uniqueness is proved for semilinear heat equations with polynomial growth at infinity.

^{[13]}We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term.

^{[14]}We consider positive solutions of the semilinear heat equation with supercritical power nonlinearity, and construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution.

^{[15]}ABSTRACTIn this paper, we study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity.

^{[16]}ABSTRACT In this paper, we investigate the well-posedness of the real fractional Ginzburg–Landau equation in several different function spaces, which have been used to deal with the Burgers’ equation, the semilinear heat equation, the Navier–Stokes equations, etc.

^{[17]}The ODE type behavior and its connection with the Liouville-type theorem can be considered as an analog of the well-known results of Merle and Zaag (1998) for the subcritical semilinear heat equation, with the significant difference that for the latter, the ODE behavior is in the time direction (instead of the normal spatial direction).

^{[18]}

## Dimensional Heat Equation

The governing equations for the electric potential, velocities and flow rates are solved analytically using the Debye-Huckel approximation, whereas a numerical method based on the finite difference iterative scheme is used to solve the associated non-dimensional heat equation.^{[1]}In this article, we will consider the parallel implementation of the Yanenko algorithm for the two-dimensional heat equation, and the sweep method was used to numerically solve the heat equation.

^{[2]}In this article two dimensional heat equation is solved using finite difference method for the metals such as gold, zinc, tin, marble and bronze.

^{[3]}The set of methods in solving linear system generated by high-order scheme for solving one-dimensional Poisson equation and one-dimensional heat equation with periodic boundary are presented.

^{[4]}Moreover, the efficiency of the presented method and its analysis are tested, applying the two-dimensional heat equation.

^{[5]}The ADMM tool wasapplied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type.

^{[6]}The value of the volt-watt sensitivity of the sensor is found and a data processing method based on the one-dimensional heat equation for a thin plate is presented.

^{[7]}Fixed and moving boundary problems for the one-dimensional heat equation are considered.

^{[8]}In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation.

^{[9]}When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations.

^{[10]}As a result, a new analytical method for solving initial-boundary value problems for a two-dimensional heat equation has been developed.

^{[11]}We consider the controllability of a one dimensional heat equation with nonnegative boundary controls.

^{[12]}We demonstrate our method in the forecasting task for the two-dimensional wave equation and the Burgers-Fisher equation in multiple geometries with different boundary conditions, and the ten-dimensional heat equation.