2.5. EXERCISES 45
With these assignments (2.243e)–(2.243g) and (2.244e)–(2.244g) reduce considerably and the
entire set of determining equations (10 C 10) can be fully integrated, eventually leading to the
compatible equations
u
t
C
c
1
x C 2c
2
t C c
3
2c
1
t C c
0
u
x
D
c
1
u .c
2
x c
4
/v
2c
1
t C c
0
;
v
t
C
c
1
x C 2c
2
t C c
3
2c
1
t C c
0
v
x
D
.c
2
x c
4
/u c
1
v
2c
1
t C c
0
;
(2.262)
where c
0
c
4
are arbitrary constants.
is chapter has considered the compatibility between PDEs: either single PDEs or sys-
tems of PDEs. It is interesting to note that several authors (see, for example, Pucci and Sac-
comandi [43], Arrigo and Beckham[44], and Niu et al. [45]) have shown that compatibility of
a given PDE and a first-order quasilinear PDE is equivalent to the nonclassical method in the
symmetry analysis of differentials (Bluman and Cole [46]). Symmetry analysis of differential
equations, first introduced by Lie [47], plays a fundamental role in the construction of exact
solutions to nonlinear partial differential equations and provides a unified explanation for the
seemingly diverse and ad hoc integration methods used to solve ordinary differential equations.
At the present time, there is extensive literature on the subject, and we refer the reader to the
books by Arrigo [48], Bluman and Kumei [49], Cherniha et al. [50], and Olver [51] to name
just a few.
2.5 EXERCISES
2.1. Solve the following PDEs by the method of characteristics:
.i/ u
2
x
3u
2
y
u D 0; u.x; 0/ D x
2
;
.i i/ u
t
C u
2
x
C u D 0; u.x; 0/ D x;
.i i i/ u
2
x
C u
2
y
D 1; u.x; 1/ D
p
x
2
C 1;
.iv/ u
2
x
u u
y
D 0; u.x; y/ D 1 along y D 1 x:
2.2. Use Charpit’s method to find compatible first-order PDEs for the following:
.i/ u
2
x
C u
2
y
D 2x;
.i i/ u
t
C uu
2
x
D 0:
Use any one of the compatible equations derived above to obtain an exact solution of
the PDEs given.
2.3. Show the PDE
u
t
D
.
uu
x
/
x
46 2. COMPATIBILITY
is compatible with
u
xxx
D 0: (2.265)
Further show that it admits solutions of the form
u D a.t/x
2
C b.t /x C c.t/:
2.4. Find the value of k such that the PDE
u
t
D uu
xx
C ku
2
x
is compatible with
.i/ u
xxxx
D 0;
.i i/ u
xxxxx
D 0:
Further show that the original PDE admits solutions of the form
.i/ u D a.t /x
3
C b.t /x
2
C c.t/x Cd.t/;
.i i/ u D a.t/x
4
C b.t /x
3
C c.t/x
2
C d.t/x C e.t/:
2.5. Show that
u
t
D u
xx
C u ln u; u
tx
D
u
t
u
x
u
;
are compatible. Further show that the first PDE admits solutions of the form
u D T .t/X.x/.
2.6. Show that
u
t
D u
xx
C u
2
x
C u
2
; u
xxx
C u
x
D 0;
are compatible. Further show that the first PDE admits solutions of the form
u D a.t/ cos x C b.t/ [52].
2.7. Consider the compatibility between
u
t
D D.u/u
xx
and u
t
D F .u/u
n
x
: (2.266)
Can you identify a relationship between D.u/ and F .u/ and for what powers n, such
that these two PDEs are compatible?
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