2.2 De…nition: (i) A function F (u)is said to be homogeneous of degree
m (in u) if, for all 2 R
F ( u) = m F (u) :
(ii) A partial di¤erential equation F = 0 is called homogeneous of degree m
in u if F is. Otherwise it is called inhomogeneous.
Example 1: (e) The pde
F (u) =
@u
@x
2
2
@u
@y
+
u2 = 0
is homogeneous of degree 2. Replace u by u to obtain
F ( u)
=
=
=
@
( u)
@x
"
@u
2
@x
2
2
+
2
2
( u)
2
@u
@y
+
2
@
( u)
@y
u
2
#
F (u)
(f) The pde
F (u) =
@u
@x
2
@u
@x
2
@u
@y
+
2
u=0
is inhomogeneous since
F ( u) =
2
+
2
@u
@y
2
u = 0:
Example 2: In last lecture examples, (a) is homogeneous of degree 1 while (b),
(c), (d).are inhomogeneous.
(g) Let n = 3 then
x21
@u
@u
@u
+ x22
+ x23
+ 2u = 0
@x1
@x2
@x3
is homogeneous.
2.3 De…nition The operator L (u) is called linear if for any two functions
u, v and any constant c
(i)
L (u + v) = L (u) + L (v)
(ii)
L (cu) = cL (u) ;
otherwise it is called non-linear.
Notes:
1
(i) 2.3(ii) states that L is homogeneous of degree 1.
(ii) To see whether a pde is linear it su¢ ces to substitute cu + v for u. It is
linear , L (cu + v) = cL (u) + L (v) :
(iii) De…ne L to consist only of those terms which contain u so if the linear
pde contains a term f (x; y) independent of u then write
Lu = f:
Example 3: In operator format, let I denote the identity operator, i.e.
Iu = u
(a)
Lu
=
0
L =
2
@
@
+3
@x
@y
7I;
(b)
Lu
=
3x
@
@
L = x
+ x2 y ;
@x
@y
(c)
L = x21
@
@
@
+ x22
+ x23
+I
@x1
@x2
@x3
Lu = 0
Example 4: In last lecture examples, (a), (b) are linear, (c), (d) are nonlinear, while in example 2, (e) (f) are non-linear, (g) is linear, e.g. (a)
L (u) = 2
so
L (cu + v) = 2
c 2
@u
@u
+3
@x
@y
7u = 0
@
@
(cu + v) + 3
(cu + v)
@x
@y
@
@
+3
@x
@y
7I u + 2
@
@
+3
@x
@y
= cL (u) + L (v) :
2
7 (cu + v) = 0
7I v = 0
whereas, for (e) (put
F (cu + v)
= c to see that it is not homogeneous of degree 1)
@
(cu + v)
@x
=
= c2
2
= c
"
2
@u
@x
+ 2c
@u
@x
2
+
2
+
@
(cu + v)
@y
@u @v
+
@x @x
@u
@y
= c2 F (u) + F (v) + 2c
2
@v
@x
#
u2 +
2
2
(cu + v)
2
@u
@y
+ c2
@v
@x
@u @v
@u @v
+
@x @x @y @y
2
+
2
+ 2c
@v
@y
@u @v
+
@y @y
2
v 2 + 2c
@v
@y
uv
showing that (g) is nonlinear.
Let the function u = u (x; y).
2.4a De…nition
A linear …rst order pde for u (x; y) may be written in the form
@u
@u
+ b (x; y)
= c (x; y) u + d (x; y)
@x
@y
where the coe¢ cients a,b,c; d are known functions of the independent variables
x; y.
Note: In operator notation write
Lu
= d
L = a
@
@
+b
@x
@y
cI
Nonlinear pdes are classi…ed as follows:
2.4b De…nition A semi-linear …rst order pde for u may be written
a (x; y)
@u
@u
+ b (x; y)
= c (x; y; u)
@x
@y
where a,b, are known functions of x; y and c = c (x; y; u) is a known function of
x; y; u which is nonlinear in u.
Note: (i) c is a function of u (but not of derivatives of u).
(ii) d = d (x; y), independent of u, has been absorbed into c (x; y; u).
3
c2 u2
2cuv
@u @v
@u @v
+ 2c
@x @x
@y @y
6= cF (u) + F (v)
a (x; y)
2
v2