## On the modeling of solution of the hyperbolic equation with random initial conditions.(Ukrainian, English)Zbl 1150.60026

Teor. Jmovirn. Mat. Stat. 74, 52-67 (2006); translation in Theory Probab. Math. Stat. 74, 59-75 (2007).
The authors consider partial differential equation in the region $$Q_{T}=G\times[0,T]$$: ${\partial^2u\over\partial t^2}=\sum_{i,j=1}^{m}{\partial\over\partial x_{i}}(a_{ij}(X){\partial u\over\partial x_{j}})-a(X)u,\;\text{with initial conditions}\;u|_{t=0}=\xi(X),\;{\partial u\over\partial t}|_{t=0}=\eta(X),$ and boundary condition $$u|_{S}=0$$, $$t\in[0,T]$$, where $$\{\xi(X),\;X\in G\}$$, $$\{\eta(X),\;X\in G\}$$ - are independent strictly sub-Gaussian random fields; $$S$$ - is a boundary of set $$G=\{0\leq x_{i}\leq S_{i}, i=1,\dots,m\}$$. It is supposed that $$a(X)\geq0, a_{ij}(X)=a_{ji}(X)$$, $$\sum_{i,j=1}^{m}a_{ij}(X)\gamma_{i}\gamma_{j}\geq\alpha\sum_{i=1}^{m}\gamma_{i}^2$$, $$\alpha>0$$. The solution of the considered problem is approximated by the sum $$\hat u(x,t,N)=\sum_{k=1}^{N}(\hat A_{k}\cos(\sqrt{\lambda_{k}}t)+\hat B_{k}\sin(\sqrt{\lambda_{k}}t))V_{k}(x)$$, where $$\lambda_{k}, V_{k}(x)$$ - are eigenvalues and eigenfunctions of the corresponding Sturm-Liouville problem. It is proved that these exist such $$N$$, that $$\hat u(x,t,N)$$ approximates the solution of considered problem with given reliability and accuracy.

### MSC:

 60G60 Random fields 35R60 PDEs with randomness, stochastic partial differential equations
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