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Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative

Luchko, Yury, Trujillo, Juan (2007)

Fractional Calculus and Applied Analysis

2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05The Caputo fractional derivative is one of the most used definitions of a fractional derivative along with the Riemann-Liouville and the Grünwald- Letnikov ones. Whereas the Riemann-Liouville definition of a fractional derivative is usually employed in mathematical texts and not so frequently in applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville fractional derivatives,...

Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems

Sen Huang (1995)

Studia Mathematica

Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem u ( n ) ( t ) = A u ( t ) , t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) u ( t ) = p ( t ) + A ʃ 0 t a ( t - s ) u ( s ) d s , t ∈ ℝ, where a ( · ) ∈ L l o c 1 ( ℝ ) and p(·) is a vector-valued polynomial of the form p ( t ) = ∑ j = 0 n 1 / ( j ! ) x j t j for some elements x j ∈ E . We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension of a theorem...

Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives

Paweł Domański (2004)

Banach Center Publications

This paper is an extended version of an invited talk presented during the Orlicz Centenary Conference (Poznań, 2003). It contains a brief survey of applications to classical problems of analysis of the theory of the so-called PLS-spaces (in particular, spaces of distributions and real analytic functions). Sequential representations of the spaces and the theory of the functor Proj¹ are applied to questions like solvability of linear partial differential equations, existence of a solution depending...

Conical diffraction by multilayer gratings: A recursive integral equation approach

Gunther Schmidt (2013)

Applications of Mathematics

The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ℝ 2 coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with...

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