General Solution
For a domain having a sufficiently smooth boundary, the general solution to the Dirichlet problem is given by
where is the Green's function for the partial differential equation, and
is the derivative of the Green's function along the inward-pointing unit normal vector . The integration is performed on the boundary, with measure . The function is given by the unique solution to the Fredholm integral equation of the second kind,
The Green's function to be used in the above integral is one which vanishes on the boundary:
for and . Such a Green's function is usually a sum of the free-field Green's function and a harmonic solution to the differential equation.
Read more about this topic: Dirichlet Problem
Famous quotes containing the words general and/or solution:
“Every writer is necessarily a critic—that is, each sentence is a skeleton accompanied by enormous activity of rejection; and each selection is governed by general principles concerning truth, force, beauty, and so on.... The critic that is in every fabulist is like the iceberg—nine-tenths of him is under water.”
—Thornton Wilder (1897–1975)
“Any solution to a problem changes the problem.”
—R.W. (Richard William)