Exact Fundamental Functions of the Random Walk in Onedimensional Confined System

We have found the exact discrete solutions for the one-dimensional confined system where two outer boundaries are either absorbing or reflecting. The solution can be easily generalized to the system where one boundary is absorbing and the other is reflecting. The exact solutions are proved by the mathematical induction method analytically and by the simulations numerically. We also prove that the exact discrete solution can be reduced to the well-known continuum solution.