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:

$\int y'dx=\int xdx$

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

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 $y=e^{rx}$

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

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

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

$0=r^{2}+5r+4$ 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 $t=b$.

$\scriptstyle {\frac {d^{2}x}{dt^{2}}}+p(t){\frac {dx}{dt}}+q(t)x(t)=f(t)$ with $\scriptstyle a_{0}x(a)+a_{1}x'(a)=g$ and $\scriptstyle b_{0}x(b)+b_{1}x'(b)=h$ on the interval $\scriptstyle I_{ab}=\{x|a\leq t\leq b\}$

Example

$\scriptstyle {\frac {d^{2}x}{dt^{2}}}+4{\frac {dx}{dt}}+2x(t)=f(t)$ 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