Finite Difference method for Numerical solution of Stochastic boundary value Problems

الملخص

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.