Heat equation: in B = Q ∪ Ω ∂ t (c(x)u(x, t)) − ∇ · (κ(x)∇u(x, t)) = 0 u(x, t): temperature c(x): heat capacity κ(x): heat conductivity Special case: c(x) = 1 in Ω 0 in Q κ(x) = κ 1 > 1 in Ω 1 in Q Ω Q parabolic equation in Ω (" ∂ t u − ∆u = 0 ") elliptic equation in Q (" ∆u = 0 ") Motivation:

Consider the heat equation driven by a smooth, Gaussian random potential: ∂tuε = 1 2 ∆uε + uε(ξε − cε), t > 0, x ∈ R, where ξε converges to a spacetime white noise, and cε is a diverging constant chosen properly. For any n > 1, we prove that uε converges in Ln to the solution of the stochastic heat equation. Our proof is probabilistic, hence provides another perspective of the general result of...

<p style='text-indent:20px;'>In this paper, we investigate the pointwise time analyticity of three differential equations. They are biharmonic heat equation, equation with potentials and some nonlinear equations power nonlinearity order <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>. The include all nonnegative one...

In this article we study the large-time behavior of the solution to a general Rosenau type approximation to the heat equation [16], by showing that the solution to this approximation approaches the fundamental solution of the heat equation at a sub-optimal rate. The result is valid in particular for the central differences scheme approximation of the heat equation, a property which to our knowl...

In this paper the heat signaling in microtubules (MT) is investigated. It is argued that for the description of the heat signaling phenomena in MT, the hyperbolic heat transport (HHT) equation must be used. It is shown that HHT is the Klein-Gordon (K-G) equation. The general solution for the K-G equation for MT is obtained. For the undistorted signal propagation in MT the Heisenberg uncertainty...

In this paper the heat signaling in microtubules (MT) is investigated. It is argued that for the description of the heat signaling phenomena in MT, the hyperbolic heat transport (HHT) equation must be used. It is shown that HHT is the Klein-Gordon (K-G) equation. The general solution for the K-G equation for MT is obtained. For the undistorted signal propagation in MT the Heisenberg uncertainty...

Heat and mass convection ............................................................................................................................. 1 Heat convection: what it is ........................................................................................................................ 1 Types of heat convection .......................................................................

In this work, we derive different systems of mesoscopic moment equations for the heat-conduction problem and analyze the basic features that they must hold. We discuss twoand three-equation systems, showing that the resulting mesoscopic equation from two-equation systems is of the telegraphist’s type and complies with the Cattaneo equation in the Extended Irreversible Thermodynamics Framework. ...

In this work, we derive different systems of mesoscopic moment equations for the heat-conduction problem and analyze the basic features that they must hold. We discuss twoand three-equation systems, showing that the resulting mesoscopic equation from two-equation systems is of the telegraphist’s type and complies with the Cattaneo equation in the Extended Irreversible Thermodynamics Framework. ...

Conjugate heat transfer problems are of importance in many engineering applications, since flows are usually confined by some material with heat transfer properties. Whenever there is a temperature difference between the fluid and solid, heat will be transfered and change the flow properties in a non-trivial way (Giles 1997; Henshaw & Chand 2009; Lindström & Nordström 2010; Roe et al. 2008). Th...