Boundary Conditions

There are several types of boundary conditions commonly encountered in the solution of Partial Differential Equations.

1. Dirichlet Boundary Conditions specify the value of the function on a surface $T=f({\bf r},t)$.

2. Neumann Boundary Conditions specify the normal derivative of the function on a surface,

$${\partial T\over \partial n} = {\hat{\bf n}}\cdot \nabla T=f({\bf r},y).$$

3. Cauchy Boundary Conditions specify a weighted average of first and second kinds.

4. Robin Boundary Conditions. For an elliptic partial differential equation in a region $\Omega$, Robin boundary conditions specify the sum of $\alpha u$ and the normal derivative of $u=f$ at all points of the boundary of $\Omega$, with $\alpha$ and $f$ being prescribed.

See also Boundary Value Problem, Dirichlet Boundary Conditions, Initial Value Problem, Neumann Boundary Conditions, Partial Differential Equation, Robin Boundary Conditions

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 502-504, 1985.

Morse, P. M. and Feshbach, H. ``Boundary Conditions and Eigenfunctions.'' Ch. 6 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 495-498 and 676-790, 1953.