6.2. REFERENCES 145
6.5. Find conditions on D.u/ and Q.u/ (in terms of differential equations) such that the
following admit functional separable solutions (see [81] and [82]):
.i/ u
t
D
.
D.u/u
x
/
x
C Q.u/;
.i i/ u
tt
D
.
D.u/u
x
/
x
C Q.u/:
Can any of these be solved explicitly?
6.6. Find separable solutions to
u
t
D
.
D.u/u
x
/
x
C
D.u/u
y
y
(6.155)
of the form
f .u/ D A.t/ C B.x/ C C.y/: (6.156)
6.7. e stream function formulation of the boundary layer equations is
y
xy
x
yy
D
yyy
C U.x/: (6.157)
Find separable solutions of the form D A.y/x C B.y/ (Polyanin [84]).
6.2 REFERENCES
[78] A. M. Grundland and E. Infeld, A family of nonlinear Klein–Gordon equations J. Math.
Phys., 33(7), pp. 2498–2503, 1992. DOI: 10.1063/1.529620. 144
[79] W. Miller and L. A. Rubel, Functional separation of variables for Laplace equations in
two dimensions, J. Phys. A: Math. Gen., 26, pp. 1901–1913, 1993. DOI: 10.1088/0305-
4470/26/8/017. 144
[80] R. Z. Zhdanov, Separation of variables in the nonlinear wave equation, J. Phys. A: Math.
Gen., 27, pp. L291–297, 1994. DOI: 10.1088/0305-4470/27/9/009. 144
[81] C. Qu, S. Zhang, and R. Liu, Separation of variables and exact solutions to quasilin-
ear diffusion equations with nonlinear source, Physica D, 144, pp. 97–123, 2000. DOI:
10.1016/s0167-2789(00)00069-5. 145
[82] P. G. Estevez and C. Z. Qu, Separation of variables in nonlinear wave equation
with a variable wave speed, eor. Math. Phys., 133(2), pp. 1490–1497, 2002. DOI:
10.1023/A:1021190509331. 145
[83] C. Z. Qu, W. He, and J. Dou, Separation of variables and exact solutions of generalized
nonlinear Klein–Gordon equations, Prog. eor. Phys., 105, pp. 379–398, 2001. 144
[84] A. D. Polyanin, Exact solutions and transformations of the equations of a stationary lami-
nar boundary layer, eoretical Foundations of Chemical Engineering, 35, pp. 319–328, 2001.
DOI: 10.1023/A:1010462116343. 145
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