Mathematics 
Solving PDE Problems
pdepe
Example: A Single PDE
This example illustrates the straightforward formulation, solution, and plotting of the solution of a single PDE
This equation holds on an interval for times . At , the solution satisfies the initial condition
At and , the solution satisfies the boundary conditions
Note
The demo pdex1 contains the complete code for this example. The demo uses subfunctions to place all functions it requires in a single Mfile. To run the demo type pdex1 at the command line. See PDE Solver Basic Syntax for more information.

pdepe
. See Introduction to PDE Problems for more information. For this example, the resulting equation is
pdepe
can use. The function must be of the form
c
, f
, and s
correspond to the , , and terms. The code below computes c
, f
, and s
for the example problem.
pdex1bc
.
In the function pdex1bc
, pl
and ql
correspond to the left boundary conditions (), and pr
and qr
correspond to the right boundary condition ().
pdepe
to evaluate the solution. Specify the points as vectors t
and x
.
t
and x
play different roles in the solver (see MATLAB Partial Differential Equation Solver). In particular, the cost and the accuracy of the solution depend strongly on the length of the vector x
. However, the computation is much less sensitive to the values in the vector t
.
This example requests the solution on the mesh produced by 20 equally spaced points from the spatial interval [0,1] and five values of t
from the time interval [0,2].
pdepe
with m = 0
, the functions pdex1pde, pdex1ic, and pdex1bc, and the mesh defined by x and t at which pdepe
is to evaluate the solution. The pdepe
function returns the numerical solution in a threedimensional array sol
, where sol(i,j,k)
approximates the k
th component of the solution, , evaluated at t(i)
and x(j)
.
@
to pass pdex1pde
, pdex1ic
, and pdex1bc
as function handles to pdepe
.
Note
See the function_handle (@), func2str , and str2func reference pages, and the Function Handles chapter of "Programming and Data Types" in the MATLAB documentation for information about function handles.

u = sol(:,:,1); surf(x,t,u) title('Numerical solution computed with 20 mesh points') xlabel('Distance x') ylabel('Time t')
= t = 2
. See "Evaluating the Solution at Specific Points" on
page 14119 for more information.
Evaluating the Solution at Specific Points
After obtaining and plotting the solution above, you might be interested in a solution profile for a particular value of t
, or the time changes of the solution at a particular point x
. The k
th column u(:,k)
(of the solution extracted in step 7) contains the time history of the solution at x(k)
. The j
th row u(j,:)
contains the solution profile at t(j)
.
Using the vectors x
and u(j,:)
, and the helper function pdeval
, you can evaluate the solution u
and its derivative at any set of points xout
The example pdex3
uses pdeval
to evaluate the derivative of the solution at xout = 0
. See pdeval
for details.
MATLAB Partial Differential Equation Solver  Changing PDE Integration Properties 