# How to solve in Mathematica gradient transport model for point source? [closed]

Problem is to get numerical solution for equation

equ = D[c[x, y, z, t], t] - Kx * D[c[x, y, z, t], x, x] - Ky * D[c[x, y, z, t], y, y] - Kz * D[c[x, y, z, t], z, z] == 0;
c[x, y, z, t] == 0, t-> Infinity,
c[x, y, z, 0] == 0 // Assumptions -> {x != 0, y != 0, z != 0},
Integrate[Integrate[Integrate[c[x,y,z,t], {x, -Infinity, +Infinity}], {y, -Infinity, +Infinity}], {z, 0, Infinity}] == Qip


with

NDSolve[{equ,c[x, y, z, t] == 0, t-> Infinity,c[x, y, z, 0] == 0 // Assumptions -> {x != 0, y != 0, z != 0}, Integrate[Integrate[Integrate[c[x,y,z,t], {x, -Infinity, +Infinity}], {y, -Infinity, +Infinity}], {z, 0, Infinity}] == Qip}, x,y,z,t, {x,0,10}, {y,0,10}, {z,0,10}, {t, 0, 50}]


where the solution is given by

c[x_,y_,z_,t_]=Qip (4 Pi t)^1/2 exp[-1/4t (x^2/kx + y^2/ky + z^2/kz)]


and to compare with numerical

• Okay, it seems to be about the fundamental solution of the heat equation on $\mathbb{R}^3$ (heat kernel). This is also the fundamental solution of the advection-diffusion equation without advection (= wind?). What's not clear to me: What would you accept as numerical solution? Do you have a specific method in mind? Jun 3, 2018 at 21:09
• Yes, you are right, it is related to the advection diffusion, the diffusivity is assumed constant in any given direction but can be different for different directions Jun 3, 2018 at 21:25
• What is your sugestion for numerical solution? Jun 3, 2018 at 21:27
• It will be good to plot generalised solution and to compare with numerica l solution Jun 3, 2018 at 21:31
• I try to make question clear, please every coment is welcome Jun 11, 2018 at 7:14