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Differential Operators (differential + operators)
Selected AbstractsOn stochastic modelling of linear circuitsINTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, Issue 3 2010Tarun Kumar Rawat Abstract In this paper, the deterministic modelling of linear circuits is replaced by stochastic modelling by including variance in the parameters (resistance, inductance and capacitance). Our method is based on results from the theory of stochastic differential equations. This method is general in the following sense. Any electrical circuit that consists of resistances, inductances and capacitances can be modelled by ordinary differential equations, in which the parameters of the differential operators are the functions of circuit elements. The deterministic ordinary differential equation can be converted into a stochastic differential equation by adding noise to the input potential source and to the circuit elements. The noise added in the potential source is assumed to be a white noise and that added in the parameters is assumed to be a correlated process because these parameters change very slowly with time and hence must be modelled as a correlated process. In this paper, we model a series RLC circuit by using the proposed method. The stochastic differential equation that describes the concentration of charge in the capacitor of a series RLC circuit is solved. Numerical simulations in MATLAB are obtained using the Euler,Maruyama method. Copyright © 2008 John Wiley & Sons, Ltd. [source] Feedback control of dissipative PDE systems using adaptive model reductionAICHE JOURNAL, Issue 4 2009Amit Varshney Abstract The problem of feedback control of spatially distributed processes described by highly dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original infinite dimensional PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is the Karhunen-Loève expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. The proposed approach is applied to representative examples of dissipative PDEs, with both linear and nonlinear spatial differential operators, to demonstrate its effectiveness of the proposed methodology. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source] Resolvent estimates in W,1,p related to strongly coupled-linear parabolic systems with coupled nonsmooth capacitiesMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2007Annegret Glitzky Abstract We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in W,1, p spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain Lp estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. Copyright © 2007 John Wiley & Sons, Ltd. [source] On the asymptotic positivity of solutions for the extended Fisher,Kolmogorov equation with nonlinear diffusionMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 8 2002M. V. Bartuccelli Abstract The objective of this paper aims to prove positivity of solutions for a semilinear dissipative partial differential equation with non-linear diffusion. The equation is a generalized model of the well-known Fisher,Kolmogorov equation and represents a class of dissipative partial differential equations containing differential operators of higher order than the Laplacian. It arises in a variety of meaningful physical situations including gas flows, diffusion of an electron,ion plasma and the dynamics of biological populations whose mobility is density dependent. In all these situations, the solutions of the equation must be positive functions. Copyright © 2002 John Wiley & Sons, Ltd. [source] Solvability on the Heisenberg groupMATHEMATISCHE NACHRICHTEN, Issue 4 2006G. Eremiev Karadzhov Abstract We solve in various spaces the linear equations L,g = f , where L, belongs to a class of transversally elliptic second order differential operators on the Heisenberg group with double characteristics and complex-valued coefficients, not necessarily locally solvable. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Spectral analysis of fourth order differential operators IMATHEMATISCHE NACHRICHTEN, Issue 1-2 2006Horst Behncke Abstract We study the spectral theory of differential operators of the form on ,2w(0, ,). By means of asymptotic integration, estimates for the eigenfunctions andM -matrix are derived. Since the M -function is the Stieltjes transform of the spectral measure, spectral properties of , are directly related to the asymptotics of the eigenfunctions. The method of asymptotic integration, however, excludes coefficients which are too oscillatory or whose derivatives decay too slowly. Consequently there is no singular continuous spectrum in all our cases. This was found earlier for Sturm,Liouville operators, for which theWKB method provides a good approximation. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Deficiency indices and spectral theory of third order differential operators on the half lineMATHEMATISCHE NACHRICHTEN, Issue 12-13 2005Horst Behncke Abstract We investigate the spectral theory of a general third order formally symmetric differential expression of the form acting in the Hilbert space ,2w(a ,,). A Kummer,Liouville transformation is introduced which produces a differential operator unitarily equivalent to L . By means of the Kummer,Liouville transformation and asymptotic integration, the asymptotic solutions of L [y ] = zy are found. From the asymptotic integration, the deficiency indices are found for the minimal operator associated with L . For a class of operators with deficiency index (2, 2), it is further proved that almost all selfadjoint extensions of the minimal operator have a discrete spectrum which is necessarily unbounded below. There are however also operators with continuous spectrum. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Properties of discrete Chebyshev collocation differential operators in curvilinear geometriesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 8 2008P. Pironkov Abstract The properties of discrete systems resulting from spectral Chebyshev collocation discretizations are investigated with respect to the solution efficiency of corresponding solvers. Complex geometries are encountered by a mapping technique to connect computational and physical domains. Several representative transformation techniques are considered. The influences of the differential operators, the boundary conditions, the geometry, and the number of grid points are systematically studied. The convergence properties of the BiCGSTAB method when iteratively solving the discrete systems are investigated. Copyright © 2008 John Wiley & Sons, Ltd. [source] Numerical valuation of options under Kou's modelPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007Jari ToivanenArticle first published online: 6 AUG 200 Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. Also for pricing American options similar iterations can be employed. A numerical experiment demonstrates that the described method is very efficient as accurate option prices can be computed in a few milliseconds on a PC. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Mechanics with variable-order differential operatorsANNALEN DER PHYSIK, Issue 11-12 2003C.F.M. Coimbra Abstract This work presents the novel concept of Variable-Order (VO) Calculus through the description of a simple problem in Mechanics. A mathematical definition for the VO-differential operator that is suitable to mechanical modelling is proposed, and an example concerning the effect of nonuniform viscoelastic frictional forces is described. A numerical method for the solution of Variable Order Differential Equations (VODEs) is proposed. The physical model under study requires mathematical tools that lie beyond the traditional methods of Constant-Order (CO) differential equations. The VO-Calculus formulation is compared to a CO-Calculus model in order to show the limitations of the latter in resolving the transition between the relevant dynamic regimes. [source] Pseudospectra of semiclassical (pseudo-) differential operatorsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2004Nils Dencker First page of article [source] | |||||||||||||