Analysis of Solutions of 2nd Order Stochastic Parabolic Equations

Schmidt, Erik
Using a computational method to solve parabolic differential equations is a feat that can be done with relative simplicity. However, when the stochasticity, i.e. the randomness of coefficients, is introduced into the equation, things become more complicated. The goal in solving a stochastic differential equation is different from the goal in solving a deterministic equation. In solving a stochastic equation we try to determine the expectation or the variance of the set of "possible" solutions of the considered equation. A method already exists which yields reasonably good results that can be used as a benchmark for checking the accuracy of new computational methods: Monte Carlo trials. In the Monte Carlo trials, the same problem is solved a sufficiently large number of times using randomly chosen coefficients employing some sort of a random number generator to simulate randomness. Then the expectation is computed as the average of all obtained solutions, and the variance is computed as the average of the square of difference between the average solution and the set of all solutions. The higher the number of trials, the higher is the accuracy. The algorithm suggested by Zhang and Lu (Zhang & Lu 2004) and implemented in the project uses the Karhunen-Loève decomposition of the random coefficient and the series expansions of this coefficient and of the solution itself to divide the original problem into a sequence of much simpler deterministic problems. Then the numerical scheme called the Crank-Nicolson method is used to solve those deterministic parabolic differential equations.
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