Linear Equations of the Second Order of Parabolic Type

Ильин, Арлен Михайлович; Kalashnikov, A. S.; Oleinik, O A

doi:10.1070/rm1962v017n03abeh004115

Cited by 272 publications

(94 citation statements)

References 40 publications

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“…It is known that under certain restrictions, fundamental solutions of such equations are transition probability densities. Numerous results concerning the existence of fundamental solutions in the case of equations with time-dependent coefficients can be found in [11,12,23,29,33]. In the present section we will discuss what follows from the results obtained in Sections 6 and 7 for such transition probability densities.…”

Section: Examplesmentioning

confidence: 85%

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Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

Gulisashvili

2008

Trans. Amer. Math. Soc.

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Abstract. We study the inheritance of properties of free backward propagators associated with transition probability functions by backward FeynmanKac propagators corresponding to functions and time-dependent measures from non-autonomous Kato classes. The inheritance of the following properties is discussed: the strong continuity of backward propagators on the space L r , the (L r − L q )-smoothing property of backward propagators, and various generalizations of the Feller property. We also prove that a propagator on a Banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded.

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Section: Examplesmentioning

confidence: 85%

“…n ) and satisfies (88) (see, e.g., [11,12,23,33]). The fundamental solution p is simultaneously a transition probability density.…”

Section: Subclasses Of the Classes P * F And P * Mmentioning

confidence: 99%

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

Gulisashvili

2008

Trans. Amer. Math. Soc.

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“…Das Maximumprinzip far rein parabolische Ungleichungen in unbeschr~inkten Gebieten wird in [1,2,9] diskutiert und im Fallen = 1 in [6] etwas weiter entwickelt. Die vorliegende Arbeit ist teilweise eine Fortsetzung yon [6], und fiir weiteres tiber die Geschichte dieses Themas wird der Leser auf [6] hingewiesen.…”

Section: Introductionunclassified

Das Maximumprinzip in unbeschr�nkten Gebieten f�r parabolische Ungleichungen mit Funktionalen

Redheffer

Walter

1977

Math. Ann.

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“…Conditions (5.5), (5.16) 4.3. When conditions (5.5), (5.16a) are valid, the solution of the difference scheme (5.14), (3.16) satisfies the estimate (this fact is established with using the majorant function technique; see, e.g., [9,[12][13][14]):…”

Section: On Conditioning Of a Schwarz Methods For Singularly Perturbedmentioning

confidence: 99%

On Conditioning of a Schwarz Method for Singularly Perturbed Convection–diffusion Equations in the Case of Disturbances in the Data of the Boundary-value Problem

Шишкин

2003

Computational Methods in Applied Mathematics

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-In this paper we discuss conditioning of a discrete Schwarz method on piecewise-uniform meshes with an example of a one-dimensional singularly perturbed boundary-value problem. We consider a Dirichlet problem for singularly perturbed ordinary differential equations with convection terms and a small perturbation parameter ε. To solve the problem numerically we use an ε-uniformly convergent (in the maximum norm) difference scheme on special piecewise-uniform meshes. For this base scheme we construct a decomposition scheme based on a Schwarz technique with overlapping subdomains, which converges ε-uniformly with respect to both the number of mesh points and the number of iterations. The step-size of such special meshes is extremely small in the neighborhood of the layer and changes sharply on its boundary, that (as was shown by A.A. Samarskii) can generally lead to a loss of well-conditioning of the above schemes. For the decomposition scheme we study the conditioning of the system (difference scheme) and the conditioning of the system matrix (difference operator), and also the influence of perturbations in the data of the boundary-value problem on disturbances of its numerical solutions. We derive estimates for the disturbances of the numerical solutions (in the maximum norm) depending on the subdomain in which the disturbance of the data appears. It is shown that the condition number of the difference operator associated with the Schwarz method, just as for the base scheme, is not ε-uniformly bounded. However, these difference schemes are well-conditioned ε-uniformly (with the ε-uniform estimate for the condition number being the same as for the schemes on uniform meshes for regular problems) when the right-hand side of the discrete equations is considered in a "natural" norm, i.e., in the maximum norm with a special weight multiplier (that is, ε ln N for ε = O (ln N ) −1 in the neighborhood of the boundary layer, where N defines the number of mesh points). In the case of the boundary-value problem with perturbed data we give conditions under which the solution of the iterative scheme based on the overlapping Schwarz method is convergent ε-uniformly to the solution of this Dirichlet problem as the number of mesh points and the number of iterations increase.

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Linear Equations of the Second Order of Parabolic Type

Cited by 272 publications

References 40 publications

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

Das Maximumprinzip in unbeschr�nkten Gebieten f�r parabolische Ungleichungen mit Funktionalen

On Conditioning of a Schwarz Method for Singularly Perturbed Convection–diffusion Equations in the Case of Disturbances in the Data of the Boundary-value Problem

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