Examples of boundary value problems

We will use k to denote the square root of the absolute value of $lambda$.

If $lambda\; =\; 0$ then

:$y(x)\; =\; Ax\; +\; B,$

solves the ODE.

Substituted boundary conditions give that both "A" and "B" are equal to zero.

For positive $lambda$ we obtain that

:$y(x)\; =\; A\; e^\{kx\}\; +\; B\; e^\{-kx\},$

solves the ODE.

Substitution of boundary conditions again yields "A" = "B" = 0.

For negative $lambda$ it is easy to show that

:$y(x)\; =\; A\; sin(kx)\; +\; B\; cos(kx),$

solves the ODE.

From the first boundary condition,

:$0\; =\; y(0)\; =\; A\; sin(0)\; +\; B\; cos(0)\; =\; B,$.

Now, after the cosine is gone, we will substitute the second boundary condition:

:$y(pi)\; =\; A\; sin(k\; pi)\; =\; 0,$.

So either "A" = 0 or "k" is an integer.

Thus we get that the eigenfunctions which solve the "boundary value problem" are:$y\_n(x)\; =\; A\; sin(nx)\; quad\; n\; =\; 0,1,2,3,...$.

One may easily check that they satisfy the boundary conditions.

Example (partial)

Consider the elliptic eigenvalue problem (boundary value problem):

with boundary conditions

:

:

Using the separation of variables, we suppose the solution is of the form

:$v\; =\; X(x)Y(y),$

substituting,

:$\{partial^2overpartial\; x^2\}(X(x)Y(y))+\{partial^2overpartial\; y\}(X(x)Y(y))+lambda\; X(x)Y(y),$

:$=\; X"(x)Y(y)+X(x)Y"(y)+lambda\; X(x)Y(y)=\; 0,$.

Divide throughout by "X"("x"):

:$=\; \{X"(x)Y(y)\; over\; X(x)\}+\{X(x)Y"(y)over\; X(x)\}+\{lambda\; X(x)Y(y)over\; X(x)\}$

:$=\; \{X"(x)Y(y)\; over\; X(x)\}+Y"(y)+lambda\; Y(y)\; =\; 0$

and then by "Y"("y"):

:$=\; \{X"(x)over\; X(x)\}+\{Y"(y)+lambda\; Y(y)over\; Y(y)\}\; =\; 0$.

Now "X"′′("x")/"X"("x") is a function of "x" only, as is ("Y"′′("y") + λ"Y"("y"))/"Y"("y"), so there are separation constants so

:$\{X"(x)over\; X(x)\}\; =\; k\; =\; \{Y"(y)+lambda\; Y(y)over\; Y(y)\}.$

which splits up into ordinary differential equations

:$\{X"(x)over\; X(x)\}\; =\; k,$

:$X"(x)\; -\; k\; X(x)\; =\; 0,$

and

:$\{Y"(y)+lambda\; Y(y)over\; Y(y)\}\; =\; k,$

: $Y"(y)+(lambda-k)\; Y(y)\; =\; 0,$From our boundary conditions we have

:$v(x,0)\; =\; X(x)Y(0)\; =\; 0,\; v(x,1)\; =\; X(x)Y(1)\; =\; 0$,

:$\{partialoverpartial\; x\}v(0,y)\; =\; X\text{'}(0)Y(y)\; =\; 0,\; \{partialoverpartial\; x\}v(1,y)=X\text{'}(1)Y(y)=0$

we want

:$Y(0)\; =\; 0,\; Y(1)\; =\; 0,\; X\text{'}(0)\; =\; 0,\; X\text{'}(1)\; =\; 0$

for which we can evaluate the boundary conditions and solutions accordingly.

ee also

* differential equation
* boundary value problem
* initial value problem
* Sturm-Liouville theory