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

Kozachenko, Yu. V.; Slyvka, G. I.

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.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

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

Keywords:

modeling of solution; hyperbolic equation; random initial conditions; sub-Gaussian random fields