Generalized solutions of a hyperbolic equation with a \(\varphi\)-sub-Gaussian right hand side. (Ukrainian, English) Zbl 1224.60158

Dovgaj, B. V.

Teor. Jmovirn. Mat. Stat. 81, 25-30 (2009); translation in Theory Probab. Math. Stat. 81, 27-33 (2010).

The author deals with the boundary value problem for the nonhomogeneous hyperbolic equation \[ \frac{\partial}{\partial x} \left(p(x) \frac{\partial u}{\partial x}\right) -q(x)u-\rho(x)\frac{\partial^2 u}{\partial t^2}=-\rho(x)\xi(x,t),\quad x\in[0,\pi],\quad t\in[0,T], \]
\[ u(x,0)=0,\quad u(0,t)=0,\quad u(\pi,t)=0,\quad \frac{\partial u}{\partial t}|_{t=0}=0, \] where \(p(x)>0,\;x\in[0,\pi]\), and \(\rho(x)>0,\;x\in[0,\pi]\), are twice continuously differentiable functions, \( q(x)\geq0,\;x\in[0,\pi],\) is a continuously differentiable function, and \(\xi(x, t),\;x\in[0,\pi],\;t\in[0,T],\) is a strictly \(\varphi\)-sub-Gaussian almost surely continuous random field. For definitions and related results see the book by V. V. Buldygin and Yu. V. Kozachenko [Metric characterization of random variables and random processes. Providence, RI: AMS (2000; Zbl 0998.60503)]. The corresponding Sturm-Liouville problem is of the form \[ \frac{d}{dx}\left(p\frac{d X}{dx} \right)-qX+\lambda\rho X=0,\quad X(0)=X(\pi)=0. \] Let \(X_n(x)\) be the orthonormal eigenfunctions with respect to the weight \(\rho\), and let \(\lambda_n\) be the corresponding eigenvalues of the Sturm-Liouville problem, and let \(\mu_n=\sqrt{\lambda_n}\).
The author proposes the sufficient conditions under which the random field \(u(x, t),\;x\in[0,\pi],\;t\in[0,T],\) represented as the series \[ u(x,t)=\sum_{n=1}^{\infty}X_n(x)\frac{1}{\mu_n}\int_0^t\sin(\mu_n(t-u))\zeta_n(u)\,du, \quad \zeta_n(t)=\int_0^{\pi}\xi(x,t)X_n(x)\rho(x)\,dx, \] is a generalized solution to the described boundary value problem.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35L20	Initial-boundary value problems for second-order hyperbolic equations
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

generalized solution; hyperbolic equation; \(\varphi\)-sub-Gaussian stochastic process

Citations:

Zbl 0998.60503