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Constructing reliable approximations of the probability density function to the random heat PDE via a finite difference scheme. (English) Zbl 1435.65118

Summary: We study the random heat partial differential equation on a bounded domain assuming that the diffusion coefficient and the boundary conditions are random variables, and the initial condition is a stochastic process. Under general conditions, this stochastic system possesses a unique solution stochastic process in the almost sure and mean square senses. To quantify the uncertainty for this solution process, the computation of the probability density function is a major goal. By using a random finite difference scheme, we approximate the stochastic solution at each point by a sequence of random variables, whose probability density functions are computable, i.e., we construct a sequence of approximating density functions. We include numerical experiments to illustrate the applicability of our method.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60C05 Combinatorial probability
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