Boundary Value Problems/Introduction to BVPs

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Objective[edit]

Introduce Boundary value problems for a single independent variable.

Approach[edit]

  • 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[edit]

For a single independent variable in an interval , an initial value problem consists of an ordinary differential equation including one or more derivatives of the dependent variable, ,

and additional equations specifying conditions on the solution and the derivatives at a point


, ..., ,

Example:

The differential equation is (First order differential equation.) and the initial condition at is given as .

Solution:

.

When, and


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

and the initial condition at is given as .


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 (Second order differential equation.) and the two initial conditions at given as .

Solution:

Assume the solution has the form

The characteristic polynomial. Solve for "r".



See the Wikipedia link for more on Initial Value Problems

Two point BVPs for an ODE[edit]

Begin with second order DEs, , with conditions on the solution at and .

with and on the interval

Hello

Example[edit]

with and on the interval

See the wikipedia topic

Boundary Value Problems

References[edit]

Return to Boundary Value Problems[edit]