Biological applications and numerical solution based on Monte Carlo method for a two-dimensional parabolic inverse problem

A numerical algorithm involving the combined use of the finite difference method and Monte Carlo method is proposed as a solution algorithm for a two-dimensional parabolic inverse problem with an unknown boundary condition. The algorithm is based on the discretize governing equations by finite difference method. Owing to the application of the finite difference method, some large sparse systems of linear algebraic equations are obtained. An approach of Monte Carlo method is employed to solve the linear systems. The Least squares scheme is proposed to modify unknown boundary condition. Furthermore two applications of the present problem in Biological systems are provided. Numerical test is performed in order to show the efficiency and accuracy of the present work.