How to | Solve a Partial Differential Equation
Mathematica's differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms without the need for preprocessing by the user. One such class is partial differential equations (PDEs).
Using
D to take derivatives, this sets up the
transport equation,
, and stores it as
:
| Out[14]= |  |
Use
DSolve to solve the equation and store the solution as
. The first argument to
DSolve is an equation, the second argument is the function to solve for, and the third argument is a list of the independent variables:
| Out[15]= |  |
The answer is given as a rule and
C
is an arbitrary function.
To use the solution as a function, say
, use
(the short form of
ReplaceAll) and
(the short form of
Part):
| Out[16]= |  |
You can then evaluate
like any other function:
| Out[17]= |  |
You can also add an initial condition like
by making the first argument to
DSolve a list. The solution is stored as
:
| Out[18]= |  |
Use
Plot3D to plot the solution:
| Out[20]= |  |
Use
DSolve with the inhomogeneous PDE
with the initial condition
:
| Out[1]= |  |
Get just the solution from the nested list:
| Out[2]= |  |
Evaluate the solution for given values of the parameters:
| Out[3]= |  |
Now, use
Plot3D to plot the solution:
| Out[5]= |  |
You can also work with PDEs that have non-numeric coefficients.
Use
DSolve to solve a inhomogeneous PDE, for example,
with the initial condition
. The solution is stored as
:
| Out[6]= |  |
Define a function
, corresponding to the solution
:
| Out[7]= |  |
Evaluate the solution function for given values of the parameters:
| Out[8]= |  |
Substitute values for the parameters:
| Out[9]= |  |
Plot the solution
for a given set of values of parameters:
| Out[10]= |  |
Use
Manipulate to show how the solution
changes with respect to the parameters
,
, and
:
The examples so far use
DSolve to obtain symbolic solutions to PDEs. When a given PDE does not contain parameters,
NDSolve can be used to obtain numerical solutions. The results of
NDSolve are given as
InterpolatingFunction objects.
Here, the solution produced by
NDSolve is stored as
:
| Out[12]= |  |
Plot the solution with
Plot3D:
| Out[15]= |  |
The
InterpolatingFunction object can be evaluated, plotted, and used in other operations.
Get just the
InterpolatingFunction solution from
and assign it to the new symbol
:
| Out[13]= |  |
Evaluate the solution with values specified for
and
:
| Out[14]= |  |
Plot the solution
with
Plot3D:
| Out[24]= |  |
When the PDE contains parameters,
NDSolve can be used for each specific value of the parameters. In addition, you can set up a function that uses
NDSolve and takes parameter values.
Use
NumericQ to prevent the function
from evaluating for non-numeric values of the parameter:
Find the solution corresponding to a specific value of the parameter, 5 in this case:
| Out[9]= |  |
Plot the solution using
Plot3D.
Evaluate is necessary so that the evaluations occur in the correct order:
| Out[10]= |  |
NDSolve can also be used with other
Mathematica functions, like
Manipulate.
Solve the PDE corresponding to a given value of the parameter
, and then plot the resulting solution:
