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.
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This verifies the solution.
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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[1][y] can be added to the solution, since the partial derivative of C[1][y] with respect to x is 0.
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If there are several arbitrary functions in the solution, they are labeled as
C[1],
C[2], and so on.
with c constant
where c is the speed of light
where k is the thermal diffusivity
