Abstract
This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes. By using a random Fourier transform, an inf- nite integral form of the solution stochastic process is firstly obtained. Afterwards, explicit expressions for the expectation and standard deviation of the solution are obtained. Since these expressions depend upon random improper integrals, which are not computable in an exact manner, random Gauss-Hermite quadrature formulae are introduced throughout an illustrative example.
Highlights
Random partial differential initial value problems (IVP) was considered an emergent mathematical subject since the celebrated surveys [3] edited by Albert T
The aim of this paper is just to progress in this direction and we propose the construction of analytic-numerical solutions for random parabolic-type models
For the sake of clarity, since the order of magnitud of relative errors associated to N ∈ {3, 5, 8, 10, 12} are very different, the latter two figures (Figure 5b and Figure 6b) have been plotted on a shorted spatial domain, −2 ≤ x ≤ 2, for N ∈ {8, 10, 12}
Summary
Random partial differential initial value problems (IVP) was considered an emergent mathematical subject since the celebrated surveys [3] edited by Albert T. One of the main difficulties in dealing with random partial differential equations is the fact the search for solutions and the analysis has to be carried out for every realization of the random parameters of the model equation In this respect one usually faces arduous problems when trying to apply the usual and well known numerical techniques of the deterministic case. The aim of this paper is just to progress in this direction and we propose the construction of analytic-numerical solutions for random parabolic-type models To achieve this goal, we use a mean square Fourier transform approach.
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