Partial Integral Operators and Integro-Differential Equations

Appell, Jürgen; Kalitvin, A. S.; Zabrejko, Petr P.

doi:10.1201/9781482270402

Cited by 87 publications

(22 citation statements)

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“…Regarding other applications, see [8]. The classical results on the Barbashin equation are represented in the well-known book [9]. The recent results about various aspects of the theory of the Barbashin equation can be found, for instance, in [10][11][12][13][14] and the references given therein.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On Stability of Vector Nonlinear Integrodifferential Equations

Gil’

2016

International Journal of Engineering Mathematics

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On Stability of Vector Nonlinear Integrodifferential Equations

Gil’

2016

International Journal of Engineering Mathematics

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“…The solutions and properties of such problems is an active area of research. 44 However, it can be shown that solutions exist, that the solutions depend continuously on the initial conditions, and that the nonlocal equations converge to the local ones as the nonlocal terms become small enough. [45][46][47][48] Much less information is available about the properties of even linear nonlocal equations in the presence of boundaries, although they are frequently solved numerically.…”

Section: A Localization Of the Closure Problemsmentioning

confidence: 99%

A non-scale-invariant form for coarse-grained diffusion-reaction equations

Ostvar

Wood

2016

The Journal of Chemical Physics

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The process of mixing and reaction is a challenging problem to understand mathematically. Although there have been successes in describing the effective properties of mixing and reaction under a number of regimes, process descriptions for early times have been challenging for cases where the structure of the initial conditions is highly segregated. In this paper, we use the method of volume averaging to develop a rigorous theory for diffusive mixing with reactions from initial to asymptotic times under highly segregated initial conditions in a bounded domain. One key feature that arises in this development is that the functional form of the averaged differential mass balance equations is not, in general, scale invariant. Upon upscaling, an additional source term arises that helps to account for the initial configuration of the reacting chemical species. In this development, we derive the macroscopic parameters (a macroscale source term and an effectiveness factor modifying the reaction rate) defined in the macroscale diffusion-reaction equation and provide example applications for several initial configurations.

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“…The equation for ζ (s; q; x, y) can be constructed 16) with the condition lim t→0 c+i∞ c−i∞ dsΓ (s) t −s ζ (s; 0; x, y) = i2πδ (x − y), where the order of the integrate and the limit cannot be exchanged. This is a partial integro-differential equation [38]. Such an equation can be translated into a partial differential equation by taking derivative with respect to q:…”

Section: Equationsmentioning

confidence: 99%

An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions

Dai

Xie

2010

J. High Energ. Phys.

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In this paper, we provide an approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions and discuss the application of this approach in some physical problems. Concretely, we construct the equations for these three quantities; this allows us to achieve them by directly solving equations. In order to construct the equations, we introduce shifted local one-loop effective actions, shifted local vacuum energies, and local spectral counting functions. We solve the equations of one-loop effective actions, vacuum energies, and spectral counting functions for free massive scalar fields in R n , scalar fields in three-dimensional hyperbolic space H 3 (the Euclidean Anti-de Sitter space AdS 3 ), in H 3 /Z (the geometry of the Euclidean BTZ black hole), and in S 1 , and the Higgs model in a (1 + 1)-dimensional finite interval. Moreover, in the above cases, we also calculate the spectra from the counting functions. Besides exact solutions, we give a general discussion on approximate solutions and construct the general series expansion for one-loop effective actions, vacuum energies, and spectral counting functions. In doing this, we encounter divergences. In order to remove the divergences, renormalization procedures are used. In this approach, these three physical quantities are regarded as spectral functions in the spectral problem.

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Partial Integral Operators and Integro-Differential Equations

Cited by 87 publications

References 0 publications

On Stability of Vector Nonlinear Integrodifferential Equations

On Stability of Vector Nonlinear Integrodifferential Equations

A non-scale-invariant form for coarse-grained diffusion-reaction equations

An approach for the calculation of one-loop effective actions, vacuum energies, and spectral counting functions

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