This tag is for questions regarding to the boundary conditions (b.c.) which expresses the behaviour of a function on the boundary (border) of its area of definition. The choice of the b.c. is fundamental for the resolution of the computational problem: a bad imposition of b.c. may lead to the divergence of the solution or to the convergence to a wrong solution.
Boundary conditions (B.C.) are constraints necessary for the solution of a boundary value problem.
The introduction of the boundary conditions into its general solution has accomplished for us the following three note- worthy things:
$(1)~~$ specified the type of mathematical function to represent the physical disturbance;
$(2)~~$ evaluated the arbitrary constants appearing in the expression for this disturbance;
$(3)~~$ introduced a certain characteristic discreteness into the resulting motion.
It is just these things which are necessary in the answer to the physicist's desire for concrete knowledge, as expressed in the question stated.
Boundary conditions play a more significant role in physical modeling and explanation than the student’s division would suggest. Not all boundary conditions allow for solutions, but usually the physics suggests what makes sense.
Types of Boundary Conditions:
- A boundary condition which specifies the value of the function itself is a Dirichlet boundary condition (or first-type boundary condition).
For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.
- A boundary condition which specifies the value of the normal derivative of the function is a Neumann boundary condition (or second-type boundary condition).
For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.
- If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition.