The Experts below are selected from a list of 342078 Experts worldwide ranked by ideXlab platform
Jinrong Wang - One of the best experts on this subject based on the ideXlab platform.
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representation of solution of a riemann liouville fractional differential equation with pure delay
2018Co-Authors: Jinrong WangAbstract:Abstract This paper derives a representation of a solution to the initial value problem for a linear fractional delay differential equation with Riemann–Liouville derivative. We apply the method of variation of constants to obtain the representation of a solution via a delayed Mittag-Leffler type Matrix Function.
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exploring delayed mittag leffler type Matrix Functions to study finite time stability of fractional delay differential equations
2018Co-Authors: Jinrong WangAbstract:Abstract In this paper, we introduce a concept of delayed two parameters Mittag-Leffler type Matrix Function, which is an extension of the classical Mittag-Leffler Matrix Function. With the help of the delayed two parameters Mittag-Leffler type Matrix Function, we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations via the variation of constants method. In addition, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations. Thereafter, we present finite time stability results of nonlinear fractional delay differential equations under mild conditions on nonlinear term. Finally, an example is presented to illustrate the validity of the main theorems.
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finite time stability of fractional delay differential equations
2017Co-Authors: Jinrong WangAbstract:Abstract In this paper, we firstly introduce a concept of delayed Mittag-Leffler type Matrix Function, an extension of Mittag-Leffler Matrix Function for linear fractional ODEs, which help us to seek explicit formula of solutions to fractional delay differential equations by using the variation of constants method. Secondly, we present the finite time stability results by virtue of delayed Mittag-Leffler type Matrix. Finally, an example is given to illustrate our theoretical results.
Sergei V Rogosin - One of the best experts on this subject based on the ideXlab platform.
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exact conditions for preservation of the partial indices of a perturbed triangular 2 2 Matrix Function
2020Co-Authors: Victor M Adukov, Gennady Mishuris, Sergei V RogosinAbstract:The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of Matrix Functions. This paper is devoted to a study of...
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regular approximate factorization of a class of Matrix Function with an unstable set of partial indices
2018Co-Authors: Gennady Mishuris, Sergei V RogosinAbstract:From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk.XIII, 3-72. (Russian).), it is well known that the set of partial indices of a non-singular Matrix Function may change depending on the properties of the original Matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original Matrix Function, there exists another Matrix Function possessing a different set of partial indices. As a result, the factorization of Matrix Functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing Matrix Functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting Matrix Function, allow to construct another family of Matrix Functions with the same origin that preserves the non-stable partial indices and is close to the original set of the Matrix Functions.
Gennady Mishuris - One of the best experts on this subject based on the ideXlab platform.
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exact conditions for preservation of the partial indices of a perturbed triangular 2 2 Matrix Function
2020Co-Authors: Victor M Adukov, Gennady Mishuris, Sergei V RogosinAbstract:The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of Matrix Functions. This paper is devoted to a study of...
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numerical factorization of a Matrix Function with exponential factors in an anti plane problem for a crack with process zone
2019Co-Authors: P Livasov, Gennady MishurisAbstract:In this paper, we consider an interface mode III crack with a process zone located in front of the fracture tip. The zone is described by imperfect transmission conditions. After application of the Fourier transform, the original problem is reduced to a vectorial Wiener-Hopf equation whose kernel contains oscillatory factors. We perform the factorization numerically using an iterative algorithm and discuss convergence of the method depending on the problem parameters. In the analysis of the solution, special attention is paid to its behaviour near both ends of the process zone. Qualitative analysis was performed to determine admissible values of the process zone's length for which equilibrium cracks exist. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.
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regular approximate factorization of a class of Matrix Function with an unstable set of partial indices
2018Co-Authors: Gennady Mishuris, Sergei V RogosinAbstract:From the classic work of Gohberg & Krein (1958 Uspekhi Mat. Nauk.XIII, 3-72. (Russian).), it is well known that the set of partial indices of a non-singular Matrix Function may change depending on the properties of the original Matrix. More precisely, it was shown that if the difference between the largest and the smallest partial indices is larger than unity then, in any neighbourhood of the original Matrix Function, there exists another Matrix Function possessing a different set of partial indices. As a result, the factorization of Matrix Functions, being an extremely difficult process itself even in the case of the canonical factorization, remains unresolvable or even questionable in the case of a non-stable set of partial indices. Such a situation, in turn, has became an unavoidable obstacle to the application of the factorization technique. This paper sets out to answer a less ambitious question than that of effective factorizing Matrix Functions with non-stable sets of partial indices, and instead focuses on determining the conditions which, when having known factorization of the limiting Matrix Function, allow to construct another family of Matrix Functions with the same origin that preserves the non-stable partial indices and is close to the original set of the Matrix Functions.
Alexander Sakhnovich - One of the best experts on this subject based on the ideXlab platform.
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inverse problems and nonlinear evolution equations solutions darboux matrices and weyl titchmarsh Functions
2013Co-Authors: Alexander Sakhnovich, L A Sakhnovich, Inna Ya RoitbergAbstract:This monograph fits the clearly need for books with a rigorous treatment of the inverse problems for non-classical systems and that of initial-boundary-value problems for integrable nonlinear equations. The authors develop a unified treatment of explicit and global solutions via the transfer Matrix Function in a form due to Lev A. Sakhnovich. The book primarily addresses specialists in the field. However, it is self-contained and starts with preliminaries and examples, and hence also serves as an introduction for advanced graduate students in the field.
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semiseparable integral operators and explicit solution of an inverse problem for a skew self adjoint dirac type system
2010Co-Authors: Bernd Fritzsche, Bernd Kirstein, Alexander SakhnovichAbstract:Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl Matrix Function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl Function and a procedure to solve the inverse problem is given. The case of the generalized Weyl Functions of the form \({\phi(\lambda)\,{\rm exp}\{-2i{\lambda}D\}}\), where \({\phi}\) is a strictly proper rational Matrix Function and D = D* ≥ 0 is a diagonal Matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.
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semiseparable integral operators and explicit solution of an inverse problem for the skew self adjoint dirac type system
2009Co-Authors: Bernd Fritzsche, Bernd Kirstein, Alexander SakhnovichAbstract:Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl Matrix Function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl Function and a procedure to solve the inverse problem is given. The case of the generalized Weyl Functions of the form $\phi(\lambda)\exp\{-2i\lambda D\}$, where $\phi$ is a strictly proper rational Matrix Function and $D=D^* \geq 0$ is a diagonal Matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.
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discrete canonical system and non abelian toda lattice backlund darboux transformation weyl Functions and explicit solutions
2007Co-Authors: Alexander SakhnovichAbstract:A version of the iterated Backlund–Darboux transformation, where Darboux Matrix takes a form of the transfer Matrix Function from the system theory, is constructed for the discrete canonical system and non-Abelian Toda lattice. Results on the transformations of the Weyl Functions, insertion of the eigenvalues, and construction of the bound states are obtained. A wide class of the explicit solutions is given. An application to the semi-infinite block Jacobi matrices is treated. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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dirac type system on the axis explicit formulae for Matrix potentials with singularities and soliton positon interactions
2003Co-Authors: Alexander SakhnovichAbstract:The Jost solutions and the scattering Matrix Function are constructed explicitly for the Dirac type system on the axis in the case of the rectangular Matrix pseudo-exponential potentials. It is shown that the evolution of these potentials is related to the Matrix soliton–positon interaction. The subclass of 'positons' is characterized in terms of the parameter Matrix α - 'generalized' eigenvalue of the Backlund–Darboux transformation.
Andre C M Ran - One of the best experts on this subject based on the ideXlab platform.
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wiener hopf indices of unitary Functions on the unit circle in terms of realizations and related results on toeplitz operators
2017Co-Authors: G J Groenewald, M A Kaashoek, Andre C M RanAbstract:We provide new formulas for the Wiener–Hopf factorization indices of a rational Matrix Function R which has neither poles nor zeros on the unit circle. In addition, we recover recent results on the Fredholm characteristics of the Toeplitz operator with symbol R via the method of matricial coupling. Furthermore, we present an alternative formula for the index in terms of the Fourier coefficients of R.
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right invertible multiplication operators and stable rational Matrix solutions to an associate bezout equation i the least squares solution
2011Co-Authors: A E Frazho, M A Kaashoek, Andre C M RanAbstract:In this paper a state space formula is derived for the least squares solution X of the corona type Bezout equation G(z)X(z) = Im. Here G is a (possibly non-square) stable rational Matrix Function. The formula for X is given in terms of the matrices appearing in a state space representation of G and involves the stabilizing solution of an associate discrete algebraic Riccati equation. Using these matrices, a necessary and sufficient condition is given for right invertibility of the operator of multiplication by G. The formula for X is easy to use in Matlab computations and shows that X is a rational Matrix Function of which the McMillan degree is less than or equal to the McMillan degree of G.
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the non symmetric discrete algebraic riccati equation and canonical factorization of rational Matrix Functions on the unit circle
2010Co-Authors: A E Frazho, M A Kaashoek, Andre C M RanAbstract:Canonical factorization of a rational Matrix Function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati difference equation is also discussed.
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local minimal factorizations of rational Matrix Functions in terms of null and pole data formulas for factors
1993Co-Authors: M A Kaashoek, Andre C M Ran, Leiba RodmanAbstract:It is known that local minimal factorizations of a rational Matrix Function can be described in terms of local null and pole data (expressed in the form of left null-pole triples and their corestrictions) of this Function. In this paper we give formulas for the factors in a local minimal factorization that corresponds to a given corestriction of the left null-pole triple.
