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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).
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.
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.
In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium. [1] Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary ∂V (also called S) of any smooth region V is zero, provided there is no source or sink within V: =, where n is the outward ...
Thermal conduction (power) is the heat per unit time transferred some distance ℓ between the two temperatures. κ is the thermal conductivity of the material. A is the cross-sectional area of the object. ΔT is the difference in temperature from one side to the other. ℓ is the length of the path the heat has to be transferred.
Quantity (common name/s) (Common) symbol/s Defining equation SI unit Dimension Temperature gradient: No standard symbol K⋅m −1: ΘL −1: Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer
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 ...
The von Neumann method is based on the decomposition of the errors into Fourier series.To illustrate the procedure, consider the one-dimensional heat equation = defined on the spatial interval , with the notation = (,) where are the specific x values, and are the sequence of t values.