Editing Superposition principle (section)

==Boundary-value problems==
{{further|Boundary-value problem}}

A common type of boundary value problem is (to put it abstractly) finding a function ''y'' that satisfies some equation
<math display="block">F(y) = 0</math>
with some boundary specification
<math display="block">G(y) = z.</math>
For example, in [[Laplace's equation]] with [[Dirichlet problem|Dirichlet boundary conditions]], ''F'' would be the [[Laplacian]] operator in a region ''R'', ''G'' would be an operator that restricts ''y'' to the boundary of ''R'', and ''z'' would be the function that ''y'' is required to equal on the boundary of ''R''.

In the case that ''F'' and ''G'' are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation:
<math display="block">F(y_1) = F(y_2) = \cdots = 0 \quad \Rightarrow \quad F(y_1 + y_2 + \cdots) = 0,</math>
while the boundary values superpose:
<math display="block">G(y_1) + G(y_2) = G(y_1 + y_2).</math>
Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary-value problems.