Abstract
A numerical method to solve a general random linear parabolic equationwhere the diffusion coefficient, source term, boundary and initial condi-tions include uncertainty, is developed. Diffusion equations arise in manyfields of science and engineering, and, in many cases, there are uncertaintiesdue to data that cannot be known, or due to errors in measurements andintrinsic variability. In order to model these uncertainties the correspon-ding parameters, diffusion coefficient, source term, boundary and initialconditions, are assumed to be random variables with certainprobabilitydistributions functions. The proposed method includes finite differenceschemes on the space variable and the differential transformation methodfor the time. In addition, the Monte Carlo method is used to deal withthe random variables. The accuracy of the hybrid method is investigatednumerically using the closed form solution of the deterministic associated equation. Based on the numerical results, confidence intervals and ex-pected mean values for the solution are obtained. Furthermore, with theproposed hybrid method numerical-analytical solutions are obtained.
Highlights
Differential equations, in general, describe the rate of change in the physical property of matter with respect to time and/or space
In order to obtain accurate numerical solutions it is necessary to consider three issues: the first is the time step size used in the differential transformation method
The second one is the step size in the space for the finite difference scheme and the last is the order of the differential method
Summary
Differential equations, in general, describe the rate of change in the physical property of matter with respect to time and/or space. Random effects and the variations produced by using different probability distributions can be studied using Monte Carlo simulations. The Monte Carlo method is used here with the aim of obtaining qualitative and quantitative behavior of the numerical solutions of the random diffusion P DE. The Monte Carlo simulation differs from traditional simulation in that the model input parameters are treated as random variables, rather than as fixed values. These parameters must be identified with their uncertainty ranges and shapes of their probability density functions prescribed. A useful criteria on to select the optimal number of realizations is to stop when the increasing them does not significantly change expected value and variance of the physical process solution.
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