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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 )).
This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance ...
Any solution function will both solve the heat equation, and fulfill the boundary conditions of a temperature of 0 K on the left boundary and a temperature of 273.15 K on the right boundary. A boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if ...
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectrum of the Laplace operator , and is thus of some auxiliary importance throughout mathematical physics .
Lions's generalization is an important one since it allows one to tackle boundary value problems beyond the Hilbert space setting of the original Lax–Milgram theory. To illustrate the power of Lions's theorem, consider the heat equation in n spatial dimensions (x) and one time dimension (t):
The most classical example is the melting of ice: Given a block of ice, one can solve the heat equation given appropriate initial and boundary conditions to determine its temperature. But, if in any region the temperature is greater than the melting point of ice, this domain will be occupied by liquid water instead.
The particles undergo a characteristic cooling process, with the heat profile at = for initial temperature as the maximum at = and = at = and =, and the heat profile at = for as the boundary conditions. Splat cooling rapidly ends in a steady state temperature, and is similar in form to the Gaussian diffusion equation.
The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.