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Random heat equation: Solutions by the stochastic adaptive interpolation method. (English) Zbl 0661.60082
The authors consider a procedure providing a basis, under reasonable conditions, for some practical problems of stochastic continuum mechanics using a generalization to the stochastic case of Bellman’s differential quadrature method. As an application, the authors consider the nonlinear and stochastic heat equation in one space dimension to obtain approximate solutions for long-time intervals both in a bounded space region and in a semi-infinite half-space. The equation considered is $\partial u/\partial t=(\partial /\partial x)[h(u,r(\omega,x))(\partial u/\partial x)],\quad t\in R_+,\quad x\in [0,1],\quad u=u(\omega,t,x),$ given consistent conditions $$u(\omega,x,t=0)$$ and $$u(\omega,t,x=0)$$ and $$u(\omega,t,x=1)$$, or u($$\omega$$,t,x$$\to \infty)$$ for the half-space problem. The solution is compared with results from standard numerical techniques.