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Riccati equation

About: Riccati equation is a research topic. Over the lifetime, 10428 publications have been published within this topic receiving 210015 citations. The topic is also known as: Riccati's differential equation.


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Journal ArticleDOI
TL;DR: In this article, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system, and a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm.
Abstract: The differential equations governing transfer and stiffness matrices and acoustic impedance for a functionally graded generally anisotropic magneto-electro-elastic medium have been obtained. It is shown that the transfer matrix satisfies a linear 1st order matrix differential equation, while the stiffness matrix satisfies a nonlinear Riccati equation. For a thin nonhomogeneous layer, approximate solutions with different levels of accuracy have been formulated in the form of a transfer matrix using a geometrical integration in the form of a Magnus expansion. This integration method preserves qualitative features of the exact solution of the differential equation, in particular energy conservation. The wave propagation solution for a thick layer or a multilayered structure of inhomogeneous layers is obtained recursively from the thin layer solutions. Since the transfer matrix solution becomes computationally unstable with increase of frequency or layer thickness, we reformulate the solution in the form of a stable stiffness-matrix solution which is obtained from the relation of the stiffness matrices to the transfer matrices. Using an efficient recursive algorithm, the stiffness matrices of the thin nonhomogeneous layer are combined to obtain the total stiffness matrix for an arbitrary functionally graded multilayered system. It is shown that the round-off error for the stiffness-matrix recursive algorithm is higher than that for the transfer matrices. To optimize the recursive procedure, a computationally stable hybrid method is proposed which first starts the recursive computation with the transfer matrices and then, as the thickness increases, transits to the stiffness matrix recursive algorithm. Numerical results show this solution to be stable and efficient. As an application example, we calculate the surface wave velocity dispersion for a functionally graded coating on a semispace.

50 citations

Journal ArticleDOI
TL;DR: It is shown that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view.
Abstract: In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse, and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton--Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual, and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure. In the context of linear-quadratic regulator problems, we show that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view. Finally, the new results provide theoretical ground for recently proposed modifications of projection methods onto rational Krylov subspaces.

50 citations

Journal ArticleDOI
TL;DR: In this paper, the use of state dependent differential Riccati equations and numerical integration to propagate their solutions forward in time is explored, and examples illustrating the usefulness of these methods are given.

50 citations

Journal ArticleDOI
TL;DR: A matrix linear ordinary differential equation of the first order whose coefficients depend on an additional parameter having two irregular first order singular points and the monodromy data of this equation as and are computed as discussed by the authors.
Abstract: A matrix linear ordinary differential equation of the first order is considered whose coefficients depend on an additional parameter having two irregular first order singular points and . The monodromy data of this equation as and are computed. These computations are used to find the asymptotics of the "degenerate" fifth Painleve equation, which is equivalent to the "complete" third one. This is possible due to the connection of these Painleve equations with isomonodromy deformations of the coefficients of the matrix linear equation. Backlund transformations and their application to asymptotic problems are considered in detail.Bibliography: 42 titles.

50 citations

Journal ArticleDOI
TL;DR: In this paper, an extended Fan sub-equation method is used to seek some new and more general traveling wave solutions of nonlinear Schrodinger equation (NLSE), which is preferable to use this method to solve NLSE because this method gives us all the solutions obtained previously by the application of at least four methods (the method of using Riccati equation, or auxiliary ordinary differential equation method, or first kind elliptic equation or the generalized Ricciati equation as mapping equation) in a unified manner.
Abstract: An extended Fan sub-equation method is used to seek some new and more general traveling wave solutions of nonlinear Schrodinger equation (NLSE). The important fact of this method is to take the full advantage of clear relationship among general elliptic equation involving five parameters and other existing sub-equations involving three parameters. It is preferable to use this method to solve NLSE because this method gives us all the solutions obtained previously by the application of at least four methods (the method of using Riccati equation, or auxiliary ordinary differential equation method, or first kind elliptic equation or the generalized Riccati equation as mapping equation) in a unified manner. So it is shown that this method is concise and its applications are promising.

50 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023153
2022335
2021203
2020240
2019223
2018231