Solution of counter diffusion problem with position dependent diffusion coefficent by using variational methods

Unsteady state counter diffusion problem with position dependent diffusion coefficient can be modeled using Fick’s second law. A mathematical model was constructed and solved to quantitatively describe the dynamic behavior of solute diffusion through non-homogeneous materials where diffusion coefficient is a function of position. The eigenfunction expansion approach was utilized to solve the model. The eigenvalues and eigenfunction of the system were obtained using a variational method. It has been shown that position dependency of the material can be neglected if the thickness of the material is relatively small. Mathematical models were solved for different thicknesses and different diffusion coefficient functions.