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I am new to Mathematica and am trying to figure out how to solve the following second order partial differential equation using DSolve.

$\qquad U C_{x}+V C_y= C_{xx}+ C_{yy}+ C_{zz} + q' \delta(x)\,\delta(y)\,\delta(z)$

where $U=uK_x^{-1/2},\ V=vK_y^{-1/2},\ q'=q(k_x k_y k_z )^{-1/2}$ with two boundary conditions given by

  1. $\qquad C\rightarrow 0$ as $|x|,|y|,z\rightarrow \infty$

  2. $\qquad -K_zC_z=0 $ at $z=0$

So far I have written code as

pde = 
  u Sqrt[kx] D[c[x, y, z], x] + v Sqrt[ky] D[c[x, y, z], y]  == 
    D[c[x, y, z], {x, 2}] + D[c[x, y, z], {y, 2}] + D[c[x, y, z], {z, 2}] + 
      q Sqrt[kx ky kz] DiracDelta[x] DiracDelta[y] DiracDelta[z]

but I don't know how to write the boundary conditions.

Also,

DSolve[pde, c[x, y, z], {x, y, z}]

returns

$\qquad \mathsf{DSolve}\left[\sqrt{\text{kx}}\,u\,c^{(1,0,0)}(x,y,z) + \sqrt{\text{ky}}\,v\,c^{(0,1,0)}(x,y,z) = c^{(0,0,2)}(x,y,z) + c^{(0,2,0)}(x,y,z)+c^{(2,0,0)}(x,y,z) + q\,\delta(x)\,\delta(y)\,\delta(z) \sqrt{\text{kx}\,\text{ky}\,\text{kz}}\,c(x,y,z),\{x,y,z\}\right]$

but does not solve the problem.

Please help me with code for the solution of this partial differential equation.

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  • $\begingroup$ Katherina, DSolve cannot do that. You need to go for NDSolve , that is, to solve it numerically, then see Menu/Help/WolframDocumentation/NDSolve/Scope/PartialDifferentialEquations. Another way is to go to the k-space and solve it it analytically. This latter way I recommend. Have fun! $\endgroup$ – Alexei Boulbitch Sep 24 '15 at 8:46
  • $\begingroup$ I wish to solve this equation analytically. Any Suggestions about how to go about doing it. $\endgroup$ – Katherine Sep 24 '15 at 9:06
  • $\begingroup$ But I already have suggested. I am not quite sure about the second boundary condition, since the equation itself already contains the boundary condition in the origine, but do it! Go to Fourier transform and solve it, not a great deal. $\endgroup$ – Alexei Boulbitch Sep 24 '15 at 15:41