- [Instructor] Let's explore a particular type
of boundary value problem you'll frequently
come across when working with boundary value problems.
These are called
the Sturm-Liouville boundary value problems.
Let's explore what these are.
Sturm-Liouville boundary value problems
are named after the two French mathematicians,
Jacques-Charles-Francois Sturm and Joseph Liouville.
These problems are a class of eigenvalue problems
for second-order, linear differential equations.
They often provide the framework
for solving partial differential equations
using a technique called separation of variables.
This technique will be explored more
in the next chapter when you work
with partial differential equations.
Sturm-Liouville boundary value problems
are defined with the following equation
where you have d/dx multiplied by p of x
multiplied by dy/dx plus q of x multiplied by y
plus lambda multiplied by w of x
multiplied by y equals zero.
In this equation, you have your given functions,
p of x, q of x, and w of x,
your unknown function, y of x,
your eigenvalue of lambda, and your weight function w of x.
This boundary value problem
has the following boundary conditions,
so you have A1 multiplied by y of a
plus a2 multiplied by y prime of a equals zero.
Next, you have b1
multiplied by y of b plus b2
multiplied by y prime of b equals zero.
These boundary conditions are separated,
meaning each condition only involves
one of the boundary points.
These conditions also make this an eigenvalue problem
because only certain eigenvalues lambda
will allow for the eigenfunctions y of x.
Let's look at a few properties for these types of problems.
Number one, all the eigenvalues lambda
of the problem are real.
Number two, the eigenfunctions form an orthogonal set
where if ym of x and yn of x
are eigenfunctions with corresponding eigenvalues,
then you have the integral of a to b of w of x
multiplied by ym of x multiplied by yn of x dx
is equal zero for m not equal to n.
This property allows you to expand functions
in terms of eigenfunctions,
similar to how you will with Fourier series.
Number three, the eigenfunctions form a complete basis,
meaning any well-behaved function
can be expanded as a series of the eigenfunctions.
And number four, the eigenvalues are all simple,
meaning each eigenvalue corresponds
to only one linearly independent eigenfunction.
Note that there are exceptions to this last property
where it is not true.
Let's take a look at a few examples
of Sturm-Liouville boundary value problems.
The goal is to have you learn
how to identify problems
that fit this format and give you an overview
of the process to solve them,
which tends to be similar problem to problem.
Note, though, that in the next chapter,
I will give you more detailed solutions
when working with these types of problems
with partial differential equations.
First, let's take a look at the equation d/dx
multiplied by x plus one multiplied by dy/dx
plus lambda multiplied by y equals zero,
and this will be on the interval of x of zero to one,
and then you have the boundary conditions
y of zero equals zero and y of one equals one.
In this problem, you can identify p of x
equals x plus one, q of x equals zero,
and w of x equals one.
To solve this problem,
you would assume a solution
with form y of x equals capital A,
multiplied by sine of capital B,
multiplied by x plus capital C,
multiplied by cosine of capital B,
multiplied by x.
Then you would find the eigenvalues
that satisfy this equation,
along with any corresponding eigenfunctions.
Let's look at another
Sturm-Liouville boundary value problem.
This time, let's look at the function d/dx
multiplied by x squared multiplied by dy/dx
plus lambda multiplied by x squared
multiplied by y equals zero.
This will be on the interval of x
between one and two,
where you have your boundary conditions
y of one equals zero and y of two equals zero.
In this case, p of x equals x squared,
q of x equals zero, and w of x equals x squared.
To solve this problem,
you would do a change of variable
to work with Bessel's equation,
so you would have d squared y over dx squared
plus two divided by x multiplied by dy/dx
plus lambda multiplied by y equals zero
by taking the derivative,
and so this would give you p squared
multiplied by y double prime
plus p multiplied by y prime
plus P squared minus zero, multiplied by y equals zero.
This would get you the general solution
with the form of y of x equals capital A,
multiplied by J0 of the square root of lambda,
multiplied by x, plus capital B, multiplied by capital Y0
of the square root of y multiplied by x.
Finally, you would solve
for the eigenvalues
and corresponding eigenfunctions for this equation.
Like I mentioned before,
Sturm-Liouville boundary value problems frequently arise
in partial differential equations,
including the heat, wave, and Laplace's equation.
They also show up in physics
when dealing with quantum mechanics,
such as solving the Schrodinger equation.
And finally, they show in engineering problems
when you're dealing with heat, conduction,
and vibration of beams.
This should give you a general idea
of what Sturm-Liouville boundary value problems are.
That way, you can easily identify them
and know generally how you will work with them
when it comes to solving partial differential equations.
Like I mentioned before,
you will dive deeper into these in the following chapter.
Let's wrap up this chapter
by exploring Fourier series and how to use them
to solve boundary value problems.