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The heat equation governs heat diffusion, as well as other diffusive processes, such as particle diffusion or the propagation of action potential in nerve cells. Although they are not diffusive in nature, some quantum mechanics problems are also governed by a mathematical analog of the heat equation (see below).
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 ...
v. t. e. In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. [1] It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable.
The heat kernel represents the evolution of temperature in a region whose boundary is held fixed at a particular temperature (typically zero), such that an initial unit of heat energy is placed at a point at time t = 0. Fundamental solution of the one-dimensional heat equation. Red: time course of . Blue: time courses of for two selected points.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
FTCS scheme. In numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations. [1] It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation.
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 Rn. Indicating by ut (x, t) the time derivative of u(x, t ...
Reduced specific heat for KCl, TiO2, and graphite, compared with the Debye theory based on elastic measurements (solid lines). [1]In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 to estimate phonon contribution to the specific heat (heat capacity) in a solid. [2]