Monte Carlo Method For Solving Numerical Problems For Partial Differential Equations

Monte Carlo method comprise that branch of experimental mathematics which is concerned with experiments on random numbers. They have found extensive use in the fields of operational research and nuclear physics, where there are a variety of problems beyond the available resources of theoretical mathematics. They have been employed sporadically in numerous other fields of science, including chemistry, biology, and medicine.Many computational tasks arising in modern science and engineering are so complex that they cannot be solved exactly, but rather have to be treated in an approximate manner through an appropriate stochastic model. Standard example are the calculation of high-dimensional of complicated multivariate functions, and several classical problems of statistical mechanics and computational physics. Monte Carlo methods provide a tool for analyzing such stochastic models and for setting up numerical schemes for actual computations. Quasi-Monte Carlo methods can be described as deterministic versions of Monte Carlo method. Instead of random samples as in a Monte Carlo method, one employs judiciously chosen deterministic samples, i.e., quasi-random points. Monte Carlo and Quasi-Monte Carlo methods are entering a fascinating stage: on the one hand, the availability of more computer power masks many classical problems computationally feasible, but on the other hand it also leads to even more challenging questions. For instance, supercomputers and modern computer architectures such as parallel computers call for new types of algorithms that use the hardware more efficiently. Furthermore, the large-scale Monte Carlo and Quasi-Monte Carlo calculations that can be run on present-day computers require large and better ensembles of random samples and quasi-random points.In this paper we mainly use the Monte Carlo methods to solve numerical problems for partial differential equations. We considered the heat equations:1．Homogenenous equation: satisfies the starting value and the boundary conditions: 2: Inhomogenenous equation: satisfies the starting value and the boundary conditions: . A probability model was first established for the starting value of stochastic differential equation. Then a series of pseudo-random number was generated using Monte Carlo method. Based on the finite difference format, we established a random wander ,and get the results. At last, we give numerical examples for a set of problems, and estimated the errors in solving the equations.Especially for homogenenous equation, let (3.2.4) use the six points difference format . (3.2.5)let , get . (3.2.6)when, we can obtain whereAssume a random wander, the particle on move to the neighbor points each according the probility . Stop at the point who satisfied or on the boundary. Let the function value is, let it be one sample of . Repeat itself times. We obtain . Unbiased estimate for : . A multivariate integration method based on the quasi-Monte Carlo method which alleviates the problem of domain discretization has been implemented for computing particular solutions of Poisson's equation: where are given function.Coupled with the method of fundamental solutions, a simpler and versatile technique, we have demonstrated a simple computational method which provides means of shifting the effort from analyst to computer.