278 Boundary Conditions
An equation that relates a derivative of a function at the edge of a domain to a value of the function that is used to find a solution.
definition [d] (Boundary Conditions) From Riley, Hobson, and Bence: requirements that the general solution \(y(x)\) obeys at specified points of the domain. For homogeneous boundary conditions, in which \(y(x)\) and its derivatives are required to be zero at specified points, this may be arranged by demanding that a Green’s function \(G(x,z)\) itself obeys the boundary conditions when considered as a function of \(x\) alone. For example, if we require \(y(a) = y(b) = 0\), then we also demand \(G(a,z) = G(b,z) = 0\).
where
- \(y\) is the unknown solution function.
- \(a\) and \(b\) are endpoints of the range.
- \(G(x,z)\) is a Green’s function.
Note:
- Riley et al.: one boundary condition must be specified at each end of the range in the standard Sturm–Liouville setting they discuss.
definition [d] (Boundary Conditions) From Kreyszig: data given on the boundary of a region for a partial differential equation problem. In Neumann and mixed problems there are boundary points at which the outer normal derivative of the solution is given, while the solution \(u\) itself is not given at those points.
where
- \(u\) is the unknown solution.
- Neumann data prescribe a normal derivative on the boundary.
278.1 References
- Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — boundary conditions; homogeneous cases \(y(a)=y(b)=0\).
- Kreyszig, E. Advanced Engineering Mathematics, 10th ed. Wiley, 2011. — Neumann and mixed boundary problems.