LOCAL SOLUTIONS OF LAPLACE, HEAT, AND OTHER EQUATIONS BY ITO PROCESSES

A method is presented for finding the solution of deterministic partia l differential equations at an arbitrary point of the domain of defini tion of these equations referred to as the local solution. The method is based on the Ito calculus, properties of diffusion processes, and M onte Carlo simulation. The theoretical background of the proposed meth od is relatively difficult. However, the method has attractive feature s for applications. For example, the numerical algorithms based on the proposed method are simple, stable, accurate, local, and ideal for pa rallel computation. Numerical examples from mechanics are presented to demonstrate the use and the accuracy of the proposed method.