Introduction 15
In practice, the things that make ODEs nonlinear also make PDEs nonlin-
ear, for example, powers ofuand its derivatives (
√
u,u^1 x,u^3 yy,...), products
involvinguand its derivatives (uxuy,uuxxy,...), various functions ofuand
its derivatives (eu,cosux,...) and the like.
As with linear ODEs, we distinguish between homogeneous and nonhomo-
geneous equations.
Definition 1.2Given thelinearPDEL[u]=f,iff ≡ 0 on some region
(that is,f is the zero-functionon some region), we say that the PDE is
homogeneouson that region. Otherwise, the PDE isnonhomogeneous.
Example 6The PDExuxx− 5 uxy+y^2 ux= 0 is homogeneous (on thex-y
plane).
Example 7The PDEux+5u=x^2 yis nonhomogeneous (on thex-yplane).
Example 8ux=
{
1 ,ifx<0ory< 0
0 ,otherwise is nonhomogeneous on thex-yplane, but
it ishomogeneouson the first quadrant.
Example 9u^2 x+u^2 y= 0 cannot be said to be homogeneous or nonhomoge-
neous, because it is not alinearPDE to start with.
Exercises 1.4
In Exercises 1–7, determine whether the PDE is linear or nonlinear, and prove
your result. If it is linear, decide if it is homogeneous or nonhomogeneous. If
it is nonlinear, point out the term or terms which make it nonlinear.
- Burger’s equation,ut+uux=0
2.uxxy−(sinx)uyy+x−y=0- 2uy− 5 u^3 =x
4.uxx=sinu- Thethree-dimensional heat equation,ut=α^2 (uxx+uyy+uzz), where
α^2 is a constant. - Poisson’s equation is two dimensions (in polar coordinates),
urr+1
rur+1
r^2uθθ=f(r, θ).7.
√
1+x^2 y^2 uxyy−cos(xy^3 )uxxy+e−y3
ux−(5x^2 − 2 xy+3y^2 )u=0