We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation.

The non-steady heat equation is considered in thin structures. The asymptotic expansion of the solution is constructed.The error estimates for high order asymptotic approximations are proved. The method of asymptotic partial domain decomposition is justified for the non-steady heat equation. AMS subject classifications: 35B25, 35B40, 35B27

In this paper we study some applications of the Lévy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the Lévy-Ornstein-Uhlenbeck process.

In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete noncompact Riemannian manifold M or on Kaehler-Ricci flow. We show that under a natural assumption, a new partial convexity property for smooth solutions to the heat equation is preserved.

Given a solution of the heat equation in an open strip, we state necessary and sufficient conditions for the existence of a boundary function in a given weighted Banach space. We then investigate the relationship between the smoothness of this boundary function and the growth of the solution of the heat equation.

This paper deals with the study of fractional bioheat equation for heat transfer in skin tissue with sinusoidal heat flux condition on skin surface. Numerical solution is obtained by implicit finite difference method. The effect of anomalous diffusion in skin tissue has been studied with different frequency and blood perfusion respectively, the temperature profile are obtained for different ord...

In this paper, we study the nonlinear heat equation ∂ ∂t △u(x, t) − c♦u(x, t) = f(x, t, u(x, t)), where△k is the Laplacian operator iterated k− times and is defined by (1.4)and ♦k is the Diamond operator iterated k− times and is defined by (1.2). We obtain an interesting kernel related to the nonlinear heat equation.

The following paper will first introduce the concept of a random walk and how it relates to the heat equation. The paper will look at simple random walks and the heat equation on different types of graphs, such as bipartite graphs and the integer lattice. We will also find harmonic functions on these graphs.

Three dimensional heat transfer and water flow characteristic in a set of rectangular microchannel heat sinks for advanced electronic systems investigated in this paper. The full Navier-Stoke’s approach is employed for this kind of narrow channels for the water flow assessments. The complete form of the energy equation accompanying the dissipation terms is also linked to the momentum equations....

Free Boundary Problems (FBP) motivated several studies in the past due to their relevance in applications [1 − 4]. From the mathematical point of view FBP are initial/ boundary value problems with a moving boundary [5]. The motion of the boundary is unknown (free boundary) and has to be determined together with the solution of the given partial differential equation. As a consequence the soluti...