Search results for " partial D"

Higher order matrix differential equations with singular coefficient matrices

2015

In this article, the class of higher order linear matrix differential equations with constant coefficient matrices and stochastic process terms is studied. The coefficient of the highest order is considered to be singular; thus, rendering the response determination of such systems in a straightforward manner a difficult task. In this regard, the notion of the generalized inverse of a singular matrix is used for determining response statistics. Further, an application relevant to engineering dynamics problems is included.

Global integrability of the gradients of solutions to partial differential equations

1994

Stochastic integro-differential and differential equations of non-linear systems excited by parametric Poisson pulses

1997

Abstract The connection between stochastic integro-differential equation and stochastic differential equation of non-linear systems driven by parametric Poisson delta correlated processes is presented. It is shown that the two different formulations are fully equivalent in the case of external excitation. In the case of parametric type excitation the two formulation are equivalent if the non-linear argument in the integral representation is related by means of a series to the corresponding non-linear parametric term in the stochastic differential equation. Differential rules for the two representations to find moment equations of every order of the response are also compared.

Einstein-Smoluchowsky equation handled by complex fractional moments

2014

In this paper the response of a non linear half oscillator driven by α-stable white noise in terms of probability density function (PDF) is investigated. The evolution of the PDF of such a system is ruled by the so called Einstein-Smoluchowsky equation involving, in the diffusive term, the Riesz fractional derivative. The solution is obtained by the use of complex fractional moments of the PDF, calculated with the aid of Mellin transform operator. It is shown that solution can be found for various values of stability index α and for any nonlinear function of the drift term in the stochastic differential equation.

On ordinary differential equations with interface conditions

1968

Adaptive Wavelet Methods for SPDEs

2014

We review a series of results that have been obtained in the context of the DFG-SPP 1324 project “Adaptive wavelet methods for SPDEs”. This project has been concerned with the construction and analysis of adaptive wavelet methods for second order parabolic stochastic partial differential equations on bounded, possibly nonsmooth domains \(\mathcal{O}\subset \mathbb{R}^{d}\). A detailed regularity analysis for the solution process u in the scale of Besov spaces \(B_{\tau,\tau }^{s}(\mathcal{O})\), 1∕τ = s∕d + 1∕p, α > 0, p ≥ 2, is presented. The regularity in this scale is known to determine the order of convergence that can be achieved by adaptive wavelet algorithms and other nonlinear appro…

Hydrodynamics and Stochastic Differential Equation with Sobolev Coefficients

2013

In this chapter, we will explain how the Brenier’s relaxed variational principle for Euler equation makes involved the ordinary differential equations with Sobolev coefficients and how the investigation on stochastic differential equations (SDE) with Sobolev coefficients is useful to establish variational principles for Navier–Stokes equations. We will survey recent results on this topic.

Linear Systems Excited by Polynomials of Filtered Poission Pulses

1997

The stochastic differential equations for quasi-linear systems excited by parametric non-normal Poisson white noise are derived. Then it is shown that the class of memoryless transformation of filtered non-normal delta correlated process can be reduced, by means of some transformation, to quasi-linear systems. The latter, being excited by parametric excitations, are frst converted into ltoˆ stochastic differential equations, by adding the hierarchy of corrective terms which account for the nonnormality of the input, then by applying the Itoˆ differential rule, the moment equations have been derived. It is shown that the moment equations constitute a linear finite set of differential equatio…

Set-valued and fuzzy stochastic differential equations driven by semimartingales

2013

Abstract In the paper we present set-valued and fuzzy stochastic integrals with respect to semimartingale integrators as well as their main properties. Then we study the existence of solutions to set-valued and fuzzy-set-valued stochastic differential equations driven by semimartingales. The stability of solutions is also established.

A digital device for the diagnosis of insulation systems

2004

A complete innovative measurement system (D2SI - device for the diagnosis of insulation systems) for both acquisition, analysis and processing of partial discharge signals generated in insulation systems under voltage stress (IEC standard), was designed and realized; several measurements on practical insulation systems demonstrated the full and correct equipment operation.