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A fundamental solution of the heat equation is a solution that corresponds to the initial condition of an initial point source of heat at a known position. These can be used to find a general solution of the heat equation over certain domains (see, for instance, ( Evans 2010 )).
The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases ...
The philosophy underlying Duhamel's principle is that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example of the heat equation modeling the distribution of heat energy u in R n.
In the case of a moving heat source applied to a plate that is so thin that temperature does not vary in the through-thickness dimension, the third term becomes zero, and the problem is two-dimensional conduction. [2] [3] The factors that determine whether temperature varies through the thickness include:
It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation. When used as a method for advection equations, or more generally hyperbolic partial differential equations, it is unstable unless artificial viscosity is included. The abbreviation FTCS was first used by Patrick Roache. [2] [3]
Fundamental solution of the one-dimensional heat equation. Red: time course of (,).Blue: time courses of (,) for two selected points. Interactive version. The most well-known heat kernel is the heat kernel of d-dimensional Euclidean space R d, which has the form of a time-varying Gaussian function, (,,) = / (| |), which is defined for all , and >. [1]
This article describes how to use a computer to calculate an approximate numerical solution of the discretized equation, in a time-dependent situation. In order to be concrete, this article focuses on heat flow, an important example where the convection–diffusion equation applies. However, the same mathematical analysis works equally well to ...
The physical processes and solutions of the governing equations are considered separately for each object in two subdomains. Matching conditions for these solutions at the interface provide the distributions of temperature and heat flux along the body–flow interface, eliminating the need for a heat transfer coefficient.