Boundary Value Problems/Introduction to BVPs

Objective
Introduce Boundary value problems for a single independent variable.

Approach

 * What is a Boundary Value problem?


 * Solution of a Boundary Value Problem is directly related to solution of an Initial Value Problem. So let's review the material on IVPs first and then make the connection to BVPs.


 * Details of solving a two point BVP.

Initial Value Problems
For a single independent variable $$ x $$ in an interval $$ I : a < x < b $$, an initial value problem consists of an ordinary differential equation including one or more derivatives of the dependent variable, $$ y$$,

$$  y^{(n)}+p_{n-1}(x) y^{(n-1)}  +...+p_1(x)y'(x)+p_0 y(x)=f(x) $$ and $$ n$$ additional equations specifying conditions on the solution and the derivatives  at a point $$ x_0 \in I$$

$$ y^{(n-1)}(x_0) = b_{n-1} $$, ..., $$ y'(x_0) = b_1 $$, $$  y(x_0) = b_0 $$

Example:

The differential equation is $$  y' = x $$ (First order differential equation.) and the initial condition at $$ x=0 $$ is given as  $$  y(0)=1 $$.

Solution:

$$ \int y' dx = \int x dx  $$

$$ y = \frac{x^2}{2} + C $$.

When,$$ x=0 $$   $$  1 = C $$ and $$   y = \frac{x^2}{2} + 1 $$

Get out a piece of paper and try to solve the following IVP in a manner similar to the preceding example:

$$ y' = xy $$ and the initial condition at $$ x=0 $$ is given as  $$  y(0)=3 $$.

Once you have an answer (or are stuck) check your solution here. Click here for the solution: /IVP-student-1/ A second order ODE example:

The differential equation is $$  y+ 5y'+4y = 0 $$ (Second order differential equation.) and the two'' initial conditions at $$ x=0 $$ given as  $$  y(0)=1,y'(0)=-2 $$.

Solution:

Assume the solution has the form $$ y = e^{rx} $$

$$ y' = re^{rx}, y'' = r^2 e^{rx}  $$

$$ y'' + 5y' + 4y = r^2 e^{rx} + 5 re^{rx} + 4e^{rx} $$

$$ 0 = r^2 e^{rx} + 5 re^{rx} + 4e^{rx} $$

$$ 0 = r^2 + 5 r + 4 $$  The characteristic polynomial. Solve for "r".

$$ \begin{array}{c} r_1 = 4 \\ r_2 = 1 \end{array}

$$

See the Wikipedia link for more on Initial Value Problems

Two point BVPs for an ODE
Begin with second order DEs, $$ x'' = f(t,x,x')$$, with conditions on the solution at $$ t=a $$ and $$ t=b $$.

$$\scriptstyle \frac {d^2 x}{dt^2} + p(t) \frac {d x}{dt} + q(t) x(t) = f(t)$$ with $$\scriptstyle a_0x(a) +a_1x'(a)= g$$ and  $$\scriptstyle b_0x(b) +b_1x'(b)= h$$ on the interval $$\scriptstyle I_{ab} = \{x | a \leq t \leq b \} $$



Example
$$\scriptstyle \frac {d^2 x}{dt^2} + 4 \frac {d x}{dt} + 2 x(t) = f(t)$$ with $$\scriptstyle x(0) = 0$$ and  $$\scriptstyle x(1)=0$$ on the interval $$\scriptstyle I_{ab} = \{x | 0 \leq t \leq 1 \} $$

See the wikipedia topic

Boundary Value Problems