On the other hand, assuming that the time derivative in the diffusion equation equals 0 and putting D = 1 we end up with the boundary value problem of the Laplace type^{[1, }^{2]}
Let us consider Ω being a cubic domain i.e. [0 π] ✗ [0 π] ✗ [0 π]. And for the g function equals 0 everything on ∂Ω apart from g(x = π, y,z) = φ_{0} one can approximate the exact solution by
∞ | ||||
φ(x,y,z) = 16φ_{0}/π^{2} | ∑ | ( | sinh((n^{2} + m^{2})^{1/2}x) sin(ny) sin(mz) | )/ |
n, m = 1, 3, … |
/( | nm sinh((n^{2} + m^{2})^{1/2}π) | ) |
For φ(x,y,z) = v(r) where r = (x^{2} + y^{2} + z^{2})^{1/2} the Laplace equation has the solution defined in ℜ^{3} for r ≠ 0
where α(3) denotes volume of B(0,1) in ℜ^{3} and equals π^{3/2}/Γ(3/2 + 1).