# Find the arbitrary functions of integration of a PDE

The PDE in question is,

sol = DSolve[{epsilon^2*D[u[eta, zeta], eta, eta] +
D[u[eta, zeta], zeta, zeta] == -1}, u, {eta, zeta}]


DSolve is unable to find C[1] and C[2] with the following boundary conditions,

bc1 = 2*Kn*epsilon*(D[u[eta, zeta], eta] /. eta -> 1) == -u[eta, zeta];
bc2 = 2*Kn*epsilon*(D[u[eta, zeta], zeta] /. zeta -> 1) == -u[eta, zeta];
bc3 = D[u[eta, zeta], eta] == 0 /. eta -> 0
bc4 = D[u[eta, zeta], zeta] == 0 /. zeta -> 0


Is there an alternate way to find C[1] and C[2] subject to the above conditions?

• …I think the /. eta -> 1 should be at the end of the equation i.e. 2*Kn*epsilon*(D[u[eta, zeta], eta] ) == -u[eta, zeta]/. eta -> 1? Commented Aug 17, 2017 at 11:44
• @xzczd Any suggestion on how to get a particular solution for this PDE?
– zhk
Commented Aug 22, 2017 at 14:53
• Do this system have a finite b.c. at $\infty$ or $-\infty$? If so, the method mentioned here will probably work. Commented Aug 23, 2017 at 2:40
• @xzczd The issue is that geometry of the physical model is finite. So we can't have b.c at $\pm\infty$
– zhk
Commented Aug 23, 2017 at 3:10
• Then we need 2 more b.c.s, or we can't determine a particular solution. Commented Aug 23, 2017 at 3:50