Related papers: Solving the Heat Equation using Nonstandard Analys…

The reheating process in inflationary universe models is considered as an out-of-equilibrium mixture of two interacting and reacting fluids, and studied within the framework of causal, irreversible thermodynamics. The evolution of the…

We study anomalous transport in a one-dimensional system with two conserved quantities in presence of thermal baths. In this system we derive exact expressions of the temperature profile and the two point correlations in steady state as…

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = \Delta u + |u|^\alpha u$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < \alpha <\alpha _0$, where $\alpha _0=\frac {4} {N-4+2\sqrt{…

An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary…

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $\Omega\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = \Delta u + \lambda |u|^{p-1} u$ in $\Omega\times(0,\infty)$, $u=0$ in…

The paper considers an inverse source problem for a one-dimensional time-fractional heat equation with the generalized impedance boundary condition. The inverse problem is the time dependent source parameter identification together with the…

We establish three partial differential equation models describing the thermodynamics of the fluid, by combining the energetic variational approach, appropriate constitutive relations, and classical thermodynamics laws. What is more, by…

In this expository work we discuss the asymptotic behaviour of the solutions of the classical heat equation posed in the whole Euclidean space. After an introductory review of the main facts on the existence and properties of solutions, we…

We use the formalism of Hairer's regularity structures theory \cite{hai-14} to study a heat equation with non-linear perturbation driven by a space-time fractional noise. Different regimes are observed, depending on the global pathwise…

In this paper, we focus on the backward heat problem of finding the function $\theta(x,y)=u(x,y,0)$ such that \[ {l l l} u_t - a(t)(u_{xx} + u_{yy}) & = f(x,y,t), & \qquad (x,y,t) \in \Omega\times (0,T), u(x,y,T) & = h(x,y), & \qquad (x,y)…

In this paper, we consider the boundary stabilization and observation of the multidimensional unstable heat equation. Since we consider the heat equation in a general domain, the usual partial differential equation back-stepping method is…

For heat flux $q$ and temperature $T$ we introduce a modified Fourier--Cattaneo law $q_t+ l \frac{q}{t}= - kT_x .$ The consequence of it is a non-autonomous telegraph-type equation. % $\epsilon S_{tt} + \frac{a}{t} S_t = S_{xx}$ . This…

The paper considers the integral Volterra equations of the first kind which are related to the inverse boundary-value heat conduction problem. The algorithms have been developed to numerically solve the respective integral equations, which…

This paper delves into the Inverse Stefan problem, specifically focusing on determining the time-dependent source coefficient in the parabolic heat equation governing heat transfer in a semi-infinite rod. The problem entails the intricate…

The well known heat equation with finite speed of propagation proposed by Cattaneo is obtained by a more general fourth order PDE when a certain (small) parameters is put equal to zero. It seems that this fourth order equation has been…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^n$ and denote the regular part of the Green's function on $\Omega$ with Dirichlet boundary condition as $H(x,y)$. Assume that $q \in \Omega$ and $n\geq 5$. We prove that there exists an…

We consider a semilinear parabolic equation with flux at the boundary governed by a nonlinear memory. We give some conditions for this problem which guarantee global existence of solutions as well as blow up in finite time of all nontrivial…

We give a simple proof of the Fourier Inversion Theorem, using the methods of nonstandard analysis.

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved.…

A physics-informed neural network is developed to solve conductive heat transfer partial differential equation (PDE), along with convective heat transfer PDEs as boundary conditions (BCs), in manufacturing and engineering applications where…