Introduction to Partial Differential Equations (PDEs)
A partial differential equation (PDE) is a relationship between an unknown function u (x1, x2, ..., xn)
and its derivatives with respect to the variables x1, x2, ..., xn
Here is an example of a PDE.
PDEs occur naturally in applications because one tries to model the rate of change of a physical quantity with respect to both space variables and time variables. At this stage of development, DSolve
typically only works with PDEs having two independent variables.
The order of a PDE is the order of the highest derivative that occurs in it. The previous equation is a first-order PDE.
A function u (x, y)
is a solution
to a given PDE if u
and its derivatives satisfy the equation.
Here is one solution to the previous equation.
This verifies the solution.
Here are some well-known examples of PDEs (clicking a link in the table will bring up the relevant examples). DSolve
gives symbolic solutions to equations of all these types, with certain restrictions, particularly for second-order PDEs.
Recall that the general solutions to PDEs involve arbitrary functions
rather than arbitrary constants
. The reason for this can be seen from the following example.
The partial derivative with respect to y
does not appear in this example, so an arbitrary function C[y]
can be added to the solution, since the partial derivative of C[y]
with respect to x
If there are several arbitrary functions in the solution, they are labeled as C
, and so on.