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A fundamental solution, also called a heat kernel, is a solution of the heat equation corresponding 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, for an introductory treatment.
Unlike the conduction equation (a finite element solution is used), a numerical solution for the convection–diffusion equation has to deal with the convection part of the governing equation in addition to diffusion. When the Péclet number (Pe) exceeds a critical value, the spurious oscillations result in space and this problem is not unique ...
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.
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 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 Rn. Indicating by ut (x, t) the time derivative of u(x, t ...