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In the anisotropic case where the coefficient matrix A is not scalar and/or if it depends on x, then an explicit formula for the solution of the heat equation can seldom be written down, though it is usually possible to consider the associated abstract Cauchy problem and show that it is a well-posed problem and/or to show some qualitative ...
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:
As an example, number X11 denotes the Green's function that satisfies the heat equation in the domain (0 < x < L) for boundary conditions of type 1 at both boundaries x = 0 and x = L. Here X denotes the Cartesian coordinate and 11 denotes the type 1 boundary condition at both sides of the body.
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]
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.
The problem of heat transfer in the presence of liquid flowing around the body was first formulated and solved as a coupled problem by Theodore L. Perelman in 1961, [1] who also coined the term conjugate problem of heat transfer. Later T. L. Perelman, in collaboration with A.V. Luikov, [2] developed this approach further.
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]