Stochastic Boundary Value Problem Research Articles

On the generalized sample solutions of stochastic boundary value problems

In this paper, we investigate the stochastic two-point boundary value problems: Under some less restricted conditions, by considering the related initial value problem for every sample path, we prove the existence, uniqueness, measurability and vibro-stability of the solution processes.

Existence of solutions and stability of a class of parabolic systems perturbed by generalized white noise on the boundary∗

This paper is concerned with a class of stochastic boundary value problems and their stability questions. The system, we consider, is governed by a parabolic partial differential equation perturbed by generalized white noise on the boundary. Existence of weak solutions and their regularity properties are established. It is also shown that the solution of the autonomous system generates a Feller process in a Hilbert space, in case the spatial operator is time invariant. The questions of Lyapunov type stability of this class of systems are also examined. The system is shown to be almost surely globally asymptotically stable with respect to a ball centered at the origin. Further, it is shown that there exists a measure, supported on the attractor, which is invariant with respect to the adjoint Feller semigroup. An explicit expression for the generator of the semigroup is also given

Stochastic Analysis of a Compressible Gas Lubrication Slider Bearing Problem

By employing the theory of stochastic differential inequalities and a comparison result for the stochastic boundary value problem, the effects of roughness for a nonlinear gas lubricated problem are analyzed. In particular, by summarizing analytic techniques, estimates on the absolute mean and root-mean-square deviation of a normalized load carrying capacity from the smooth case are obtained for a general nonlinear one-dimensional problem.

Iteration method of solving stochastic boundary-value problems of elasticity theory

Iteration method of solving stochastic boundary-value problems of elasticity theory

On the fundamental theory of nonlinear second order stochastic boundary value problems

A study of nonlinear second order stochastic boundary value problems (SBVP for short) is initiated through sample calculus approach. A basic existence result for bounded nonlinearities is established. The method of upper and lower solution processes and a general existence theorem are established. After proving the stochastic version of the needed maximum principle , the monotone iterative technique is developed which yields existence of muptiple solution processes of SBVP. Finally , by developing a stochastic comparison result, the important problem of finding error estimates between the sample solutions of SBVP and the solutions of corresponding deterministic BVP, is considered

Propagation in a randomly perturbed multimode matched waveguide

A stochastic two-point boundary value problem, modeling propagation in a randomly perturbed two-mode waveguide, is studied in the diffusion limit. A lowpass spectral approximation is introduced, from which the forward scattering approximation (e.g. coupled power equations) emerges as the leading order description. First order corrections are explicitly computed. The results are compared with two-mode radiative transfer theory and computer simulations.

Variational formulation of stochastic linear boundary-value problems of the theory of elasticity

Variational formulation of stochastic linear boundary-value problems of the theory of elasticity

Statistical analysis of linear stochastic boundary-value problems by the method of reduction to a Cauchy problem

Statistical analysis of linear stochastic boundary-value problems by the method of reduction to a Cauchy problem

Gaussiann-Markovian processes and stochastic boundary value problems

In this paper a class of Gaussian processes on the interval is considered. The processes arise naturally as formal solutions of generalized stochastic boundary value problems with Gaussian white noise input. This is briefly explained in preliminary Sect. 1. Such processes were also studied in connection with interpolation by generalized spline functions [8, 9]. We give the characterization of those boundary value problems for which the solutions are n-Markovian (Sect. 2). The proof is based on the characterization of Markov property in terms of reproducing kernel Hilbert space associated to a Gaussian process similar to the one given in e.g. [7] or [5].

A note on stochastic boundary-value problems

A note on stochastic boundary-value problems

Random vibrations of orthotropic plates clamped or simply supported all round

An approximate method is presented for determining the probabilistic response of rectangular orthotropic plates clamped all round. For solving the stochastic boundary value problem, the probabilistically given and sought functions are expressed in terms of series of approximate modes of vibration, which satisfy the boundary conditions but not the field equation. Galerkin's procedure then yields a set of linear equations for the cross-spectral densities of the displacements. The cross-spectral density of the external pressure is taken to be a product of longitudinal and transverse correlation coefficients which depend on frequency and separation distance.

Random vibrations of orthotropic plates clamped or simply supported all round

An approximate method is presented for determining the probabilistic response of rectangular orthotropic plates clamped all round. For solving the stochastic boundary value problem, the probabilistically given and sought functions are expressed in terms of series of approximate modes of vibration, which satisfy the boundary conditions but not the field equation. Galerkin's procedure then yields a set of linear equations for the cross-spectral densities of the displacements. The cross-spectral density of the external pressure is taken to be a product of longitudinal and transverse correlation coefficients which depend on frequency and separation distance. When the approximate method presented here is applied to cases capable of closed solutions (i.e. plates having a pair of opposite edges simply supported), the result coincides with that obtained by the classical normal-mode approach.

One Dimensional Random Wave Propagation

We study wave propagation in a one-dimensional random medium whose index of refraction is characterized randomly and is assumed to have small fluctuations about the mean. The appropriate stochastic boundary value problem for the scattering region is transformed into a Cauchy type initial value problem for the boundary values of the random Green’s function. The stochastic differential equation derived is a first order, nonlinear equation of the Riccati type. The initial value problem is solved in two ways: (i) by conventional power series perturbation expansion, and (ii) by quasi-linearization. In both cases we consider the refracting medium to be characterized by a general stationary process in the broad sense, and for such a process, general expressions for the statistical properties of the reflected and transmitted amplitude waves are derived. In the former case the solutions obtained are uniformly valid for $\lambda < O( 1/\varepsilon )$, $\lambda $ being the size of the scattering region and $\varepsilon \ll 1$, while in the second case the solutions are uniformly valid for all $\lambda $. As an example, we consider an Ornstein–Uhlenbeck process and its limiting white-Gaussian process, for which expressions for the mean power reflected and transmitted are obtained.

On the Relationship Between the Solution of a Stochastic Boundary Value Problem and Parameters in the Boundary Conditions

Previous article Next article On the Relationship Between the Solution of a Stochastic Boundary Value Problem and Parameters in the Boundary ConditionsWilliam B. Day and William E. BoyceWilliam B. Day and William E. Boycehttps://doi.org/10.1137/0119021PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. T. Bharucha-Reid, On the theory of random equationsProc. Sympos. Appl. Math., Vol. XVI, Amer. Math. Soc., Providence, R.I., 1964, 40–69 MR0189071 0142.13603 CrossrefGoogle Scholar[2] William E. Boyce, Stochastic nonhomogeneous Sturm-Liouville problems, J. Franklin Inst., 282 (1966), 206–215 10.1016/0016-0032(66)90563-1 MR0204797 0221.34023 CrossrefISIGoogle Scholar[3] William E. Boyce, A. T. Bharucha-Reid, Random eigenvalue problemsProbabilistic Methods in Applied Mathematics, Vol. 1, Academic Press, New York, 1968, 1–73 MR0263171 Google Scholar[4] William B. Day, A monotone property of the solution of a stochastic boundary value problem, Quart. Appl. Math., 28 (1970), 411–425 MR0268979 0254.60036 CrossrefISIGoogle Scholar[5] C. W. Haines, Hierarchy methods for random vibrations of elastic strings and beams, J. Engrg. Math., 1 (1967), 293–305 10.1007/BF01540512 0153.28102 CrossrefGoogle Scholar[6] E. L. Ince, Ordinary Differential Equations, Longmans, Green, London, 1927 Google Scholar[7] Athanasios Papoulis, Probability, random variables, and stochastic processes, McGraw-Hill Book Co., New York, 1965xi+583 MR0176501 0191.46704 Google Scholar[8] Rheinhoud H. van der Linde, Eigenfunctions of random eigenvalue problems and their statistical properties, SIAM J. Appl. Math., 17 (1969), 1298–1304 10.1137/0117121 MR0256480 0188.51902 LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Bibliography Cross Ref Thermoelastic wave propagation in a random medium and some related problemsInternational Journal of Engineering Science, Vol. 11, No. 9 Cross Ref Volume 19, Issue 1| 1970SIAM Journal on Applied Mathematics History Submitted:03 June 1969Published online:12 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0119021Article page range:pp. 221-234ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

A monotone property of the solution of a stochastic boundary value problem

A monotone property of the solution of a stochastic boundary value problem

On the response of continuous media to random excitations

This paper is concerned with the formulation of stochastic boundary value problems applied to a linearly elastic continuum subjected to a statistically correlated, multiple random excitation vector field. This formulation involves the description of the correlation tensor and the spectral density tensor of the field of the response of the solid in terms of the respective tensors of the excitation field.