Finite Difference method for Numerical solution of Stochastic boundary  value Problems

Authors

Recently, the numerical solution of stochastic boundary value problems has attracted

substantial interests, owed to its broad applications in engineering, physics and

applied mathematics. These problems usually arise in differential equations with

random parameters, random forcing terms, or probabilistic boundary conditions,

where analytical solutions, if exist, are either complicated beyond belief or not

obtainable. In this paper, a powerful numerical algorithm based on the finite

difference method is proposed to solve this kind of problem. The C-methods of the

current approach approximate the spatial domain in a finite set of grid points where

difference approximations are used to reduce the initial stochastic differential equation

to a system of algebraic equations which can be numerically solved or addressed

further, e.g. within a statistical moment context. To account for the stochastic nature

of the problem, random variables as well as probabilistic distributions are introduced

in the numerical problem formulation. This enables the approach to accurately

represent the variability and uncertainty in boundary conditions and system

parameters. A recursive solution process is introduced in this way to numerically

stable and convergent solution. Furthermore, particular attention is paid to the choice

of suitable step sizes and the discretization scheme, in order to reduce these numerical

errors and consequently improve the accuracy of the solutions. The method’s accuracy

is verified with several problems possessing varying degrees of randomness and

complexity in boundaries. The results also suggest that the finite difference method

offers good approximations at a fairly low computational cost. In addition, sensitivity

analysis confirms that the approach is able to identify the effect of random parameters

upon the overall solution behaviour. The straightforward implementation and the

flexibility of the method, allow us to expect to obtain results concerning a wide range

of Stochastic Boundary Values and to be very accurate and powerful in practical

application.