Related papers: Exact solution of the Heat Equation for initial po…
We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces.…
We present a class of new explicit and stable numerical algorithms to solve the spatially discretized linear heat or diffusion equation. After discretizing the space and the time variables like conventional finite difference methods, we do…
We consider an approximate solution to the heat equation which consists of the derivatives of heat kernel. Some conditions in the initial value, under which the approximation converges to the solution of the heat equation or diverges when…
We study a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that…
We investigate exact solutions for isothermal shock problems in different one-dimensional geometries. These solutions are given as analytical expressions if possible, or are computed using standard numerical methods for solving ordinary…
In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if $\phi$ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition…
As a first approach to the study of systems coupling finite and infinite dimensional natures, this article addresses the stability of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov…
We consider the initial boundary value problem of non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random perturbation. The space boundary is Lipschitz and we impose non-zero…
We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the…
We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space $L^{1}(\mathbb{R}^{3}\times \mathbb{T}^3).$ Special…
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general…
In geophysics, wave propagation in elastic media is a crucial subject. In this context, seismology has made significant progress as a result of numerous advances, among these stands out the advancement of numerical methods such as the…
A cold, thin film of liquid impinging on an isothermal hot, horizontal surface has been investigated. An approximate solution for the velocity and temperature distributions in the flow along the horizontal surface is developed, which…
The propagation of heat and thermal signals in the form of travelling waves is investigated for a nonlinear Maxwell-Cattaneo-Vernotte equation. The exact wave solutions are derived by expressing the thermal conductivity and the relaxation…
This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility,…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
An approach for analytical description of thermal processes in harmonic lattices is presented. We cover longitudinal and transverse vibrations of chains and out-of-plane vibrations of two-dimensional lattices with interactions of an…
The subject matter of this paper concerns anisotropic diffusion equations: we consider heat equations whose diffusion matrix have disparate eigenvalues. We determine first and second order approximations, we study the well-posedness of them…
Interface dynamics in two-dimensional systems with a maximal number of conservation laws gives an accurate theoretical model for many physical processes, from the hydrodynamics of immiscible, viscous flows (zero surface-tension limit of…
The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), specifically at the critical dimension of two and near the critical temperature.…