文档介绍：Introduction to Partial Differential Equations for
Engineers
. Dodson, Department of Mathematics, UMIST
September 2001
1 What are Partial Differential Equations?
Solving ordinary differential equations involves finding a function (or a set of func-
tions) of one independent variable but partial differential equations are for functions
of two or more variables. Examples of physical models using partial differential equa-
tions are the heat equation for the evolution of the temperature distribution in a
body, the wave equation for the motion of a wavefront, the flow equation for the
flow of fluids and Laplace’s equation for an electrostatic potential or elastic strain
field. In such cases we need to have not only the initial conditions, but also bound-
ary conditions for the region in which the model applies; thus we have to solve
boundary value problems.
As with ODEs, we call a PDE linear homogeneous if a bination of deriva-
tives is equal to zero—and then a bination of solutions is another solution.
Here are typical examples of monest types of linear homogeneous PDEs, for
the simplest case—just two independent variables (x, t or x, y—it is easy to see how
they would generalize to more variables x, y, z, t)
∂u ∂u
Flow Equation c + = 0, given initial or boundary values for()u
∂x ∂t
∂2u ∂u
Heat Equation c2 −= 0, given initial or boundary values for()u
∂x2 ∂t
∂2u ∂2u
Wave Equation c2 −= 0, given initial or boundary values for()u
∂x2 ∂t2
∂2u ∂2u
Laplace0s Equation + = 0, given boundary values for u. ()
∂x2 ∂y2
An example of a linear but non homogeneous PDE—Poisson’s equation:
∂2u ∂2u
+ = f(x, y). ()
∂x2 ∂y2
An example of a nonlinear PDE—a nonlinear heat equation:
∂2u ∂u
c2 − u = 0. ()
∂x2 ∂t
1
2 2M1: Introduction to Partial Differential Equations
Figure 1: A snapshot of a particular solution of the wave equation, for a vibrating
drum.
Notation
∂u
Sometimes we denote partial