- [Instructor] So far, you have focused
on exploring ordinary differential equations.
In this chapter,
you will explore partial differential equations.
These types of differential equations tend
to be more difficult to work with
than ordinary differential equations.
Let's begin by learning exactly
what partial differential equations are.
Partial differential equations contain
at least one partial derivative in them.
These types of equations are sometimes abbreviated as PDE.
This means at least one of the derivatives in the equation
are represented with two or more independent variables.
The typical notation used for a partial derivative
is denoted by partial N of Y divided by partial XN.
So note that the curly D looking symbol
is known as partial,
which represents a partial derivative.
Let's look at a few examples
of partial differential equations.
With the first equation,
you have partial U over partial T
plus U multiplied by partial U over partial DX
equals V multiplied by partial squared U
over partial X squared.
So you'll note that you have your partial derivatives
throughout this equation,
along with multiple different variables
you are working with.
In the next equation,
you have partial squared X
over partial Y squared plus three
multiplied by partial x partial T
equals X squared plus five.
So as you can tell,
there are still multiple different variables
you are working with,
and both of these derivatives have those partial symbols.
Finally, you have partial cubed over X over partial T cubed
minus partial squared X over partial Y squared
plus partial X over partial T
equals E to the X plus sign of X.
Like before, these three derivatives are partial derivatives
due to working with multiple different variables
and having those partial symbols.
When it comes to partial differential equations,
you can have an equation that contains one
or more ordinary derivatives in it.
For example,
you can have partial squared X over partial Y squared
plus partial X over partial T
plus X multiplied by DXDY equals Y plus E to the X.
So you'll notice here you have two partial derivatives
and one regular derivative.
When it comes to classifying a differential equation
as ordinary or partial,
you need to determine if the differential equation has
at least one partial derivative in it.
If it does have at least one partial derivative,
then the equation is automatically classified
as a partial differential equation.
Otherwise, if all the derivatives
in the equation are ordinary,
then it is just an ordinary differential equation.
The following is the general form used
for partial differential equations.
So you have capital F of X1, X2 to XN, and then U,
and then partial U over partial X1,
partial U over partial X2,
all the way to partial squared U
over partial X squared one equals zero.
So in this case,
you have your independent variables X1, X2 to XN,
your dependent variable U,
and however many partial derivatives you need,
starting with partial U over partial X1.
Note, a simpler form of this equation is denoted
by partial U, partial X of X, Y equals zero.
The partial differential equations
you will work with in this chapter can be classified
with a variety of orders,
with the first order tending to be the easiest to work with.
You'll be working with both linear
and non-linear differential equations in this chapter.
Note that for second-order partial differential equations,
you can also classify these as elliptic, parabolic,
or hyperbolic depending on the discriminant
of the principle part when it comes to working
with partial differential equations.
When it comes to solving partial differential equations,
you will learn how to use the separation of variables
for your transforms
and method of characteristics techniques.
You will also explore
and learn how to solve the diffusion equation,
wave equation, string equation, Schrodinger equation,
and Laplace's equation.
You will also explore various Fourier series,
including Fourier sine and cosine series,
and just regular Fourier series
with partial differential equations.
Note that you can also use numerical methods
to solve partial differential equations,
but I will not explore that in this particular course.
There are various practical applications
for partial differential equations.
Many of them you'll be exploring throughout this chapter,
such as various physics equations,
such as the diffusion and wave equations.
In biology, these types of equations are used
to model the development of biological tissues.
And finally, in engineering,
they are used to model the behavior
of electromagnetic fields for products
such as microwaves and antennas.
Partial differential equations can be more complex
to work with than ordinary differential equations,
but they could be extremely helpful
in a variety of practical scenarios.
The goal of this chapter is
to help you understand how to identify
and work with various partial differential equations,
but note that this is a subject
you could dive much deeper into on your own.
Let's begin
by exploring the separation of variables technique
to solve partial differential equations.