Fundamental Solutions
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, (Evans 1998) for an introductory treatment.
In one variable, the Green's function is a solution of the initial value problem
where δ is the Dirac delta function. The solution to this problem is the fundamental solution
One can obtain the general solution of the one variable heat equation with initial condition u(x, 0) = g(x) for -∞ < x < ∞ and 0 < t < ∞ by applying a convolution:
In several spatial variables, the fundamental solution solves the analogous problem
in -∞ < x i < ∞, i = 1,...,n, and 0 < t < ∞. The n-variable fundamental solution is the product of the fundamental solutions in each variable; i.e.,
The general solution of the heat equation on Rn is then obtained by a convolution, so that to solve the initial value problem with u(x, t = 0) = g(x), one has
The general problem on a domain Ω in Rn is
with either Dirichlet or Neumann boundary data. A Green's function always exists, but unless the domain Ω can be readily decomposed into one-variable problems (see below), it may not be possible to write it down explicitly. The method of images provides one additional technique for obtaining Green's functions for non-trivial domains.
Read more about this topic: Heat Equation
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