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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}]

enter image description here

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?

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  • $\begingroup$ …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? $\endgroup$
    – xzczd
    Aug 17, 2017 at 11:44
  • $\begingroup$ @xzczd Any suggestion on how to get a particular solution for this PDE? $\endgroup$
    – zhk
    Aug 22, 2017 at 14:53
  • $\begingroup$ Do this system have a finite b.c. at $\infty$ or $-\infty$? If so, the method mentioned here will probably work. $\endgroup$
    – xzczd
    Aug 23, 2017 at 2:40
  • $\begingroup$ @xzczd The issue is that geometry of the physical model is finite. So we can't have b.c at $\pm\infty$ $\endgroup$
    – zhk
    Aug 23, 2017 at 3:10
  • $\begingroup$ Then we need 2 more b.c.s, or we can't determine a particular solution. $\endgroup$
    – xzczd
    Aug 23, 2017 at 3:50

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