Putting k=0 in the equation of electrodiffusion we neglect the electrostatic term. It leads to the following equation describing diffusion in ℜ^{n} ^{[1, }^{2]}
where D denotes a diffusion coefficient. This kind of equation represents an initial value problem. Assuming that the considered domain Ω in ℜ^{3} is of a cubic type [x_{min}, x_{max}]✗[y_{min}, y_{max}]✗[z_{min}, z_{max}] let us take u = 0 as a boundary condition. Now we seek a solution of the equation which satisfies this boundary condition and prescribed initial condition at the time t = 0. The solution of the equation is approximated by the triple sum
∞ | ∞ | ∞ | ||
u(x,y,z,t) = | ∑ | ∑ | ∑ | v_{0, kx, ky, kz} e^{-(kx2 + ky2 + kz2)Dt} sin(k_{x}x) sin(k_{y}y) sin(k_{z}z), |
k_{x} = 1 | k_{y} = 1 | k_{z} = 1 |
where v_{0, kx, ky, kz} are unknown coefficients that must be determinated from the initial condition:
∞ | ∞ | ∞ | ||
g = u(., 0) = | ∑ | ∑ | ∑ | v_{0, kx, ky, kz} sin(k_{x}x) sin(k_{y}y) sin(k_{z}z). |
k_{x} = 1 | k_{y} = 1 | k_{z} = 1 |
In the case of the domain being [0 π] ✗ [0 π] ✗ [0 π] and g = const the solution has the form
∞ | ||
u(x,y,z,t) = 64g/(π^{3}) | ∑ | e^{-(kx2 + ky2 + kz2)Dt} sin(k_{x}x) sin(k_{y}y) sin(k_{z}z)/(k_{x} k_{y} k_{z}). |
k_{x}, k_{y}, k_{z} = 1, 3, … |
In the case of cylindrical domain defined by r ∈ [0, r_{0}], θ ∈ [0, 2π) and z ∈ [0, π], and with the boundary condition of the form u = 0 the solution of diffusion equation can be expanded in an absolutely and uniformly convergent series of the form
∞ | ∞ | ||||||
u(r, θ, z, t) = | ∑ | ∑ | J_{n}(k_{n, m}r/r_{0}) | ( | a_{n, m}cos(n θ) + b_{n, m} sin(nθ) | ) | sin(k_{z} z)✗ |
n = 0 | m, k_{z} = 1 |
✗ | e^{-((kn,m/r0)2 + kz2)Dt} |
where k_{n, m} are the zeros of the Bessel functions and a_{n, m}, b_{n, m} are constants that must be found from the initial condition by making use of the orthogonality relation for the trigonometric and Bessel functions.