The diffusion problem in a rectangular container with an internal source: exact solutions obtained by the fast expansion method

Simulation of physical processes

The diffusion problem in a rectangular-shaped body with the Derichlet’s boundary conditions and an internal substance source depending on the rectangle points’ coordinates has been solved generally by the fast expansion method (FEM). The exact solution containing free parameters was obtained, and by changing them one could get many new exact solutions. Exact solutions to the problem with a constant internal source were shown as an example. From our analysis of the exact solutions it follows that the concentration and diffusion fluxes distributions should be symmetrical relative to the plane y = b/2, provided that the substance concentration in the corners of the rectangular area is equal to zero. An investigation into the difference in the diffusion fluxes along the coordinate axes showed that the constant internal source affected the difference in the nonsymmetrical fluxes, and the concentration of the substance in the area corners had no effect.