1,356 research outputs found
Meromorphic solutions of nonlinear ordinary differential equations
Exact solutions of some popular nonlinear ordinary differential equations are
analyzed taking their Laurent series into account. Using the Laurent series for
solutions of nonlinear ordinary differential equations we discuss the nature of
many methods for finding exact solutions. We show that most of these methods
are conceptually identical to one another and they allow us to have only the
same solutions of nonlinear ordinary differential equations
Seven common errors in finding exact solutions of nonlinear differential equations
We analyze the common errors of the recent papers in which the solitary wave
solutions of nonlinear differential equations are presented. Seven common
errors are formulated and classified. These errors are illustrated by using
multiple examples of the common errors from the recent publications. We show
that many popular methods in finding of the exact solutions are equivalent each
other. We demonstrate that some authors look for the solitary wave solutions of
nonlinear ordinary differential equations and do not take into account the well
- known general solutions of these equations. We illustrate several cases when
authors present some functions for describing solutions but do not use
arbitrary constants. As this fact takes place the redundant solutions of
differential equations are found. A few examples of incorrect solutions by some
authors are presented. Several other errors in finding the exact solutions of
nonlinear differential equations are also discussed.Comment: 42 page
Effizientes Lösen von großskaligen Riccati-Gleichungen und ein ODE-Framework für lineare Matrixgleichungen
This work considers the iterative solution of large-scale matrix equations. Due to the size of the system matrices in large-scale Riccati equations the solution can not be calculated directly but is approximated by a low rank matrix ZYZ^*. Herein Z is a basis of a low-dimensional rational Krylov subspace. The inner matrix Y is a small square matrix. Two ways to choose this inner matrix are examined: By imposing a rank condition on the Riccati residual and by projecting the Riccati residual onto the Krylov subspace generated by Z. The rank condition is motivated by the well-known ADI iteration. The ADI solutions span a rational Krylov subspace and yield a rank-p residual. It is proven that the rank-p condition guarantees existence and uniqueness of such an approximate solution. Known projection methods are generalized to oblique projections and a new formulation of the Riccati residual is derived, which allows for an efficient evaluation of the residual norm. Further a truncated approximate solution is characterized as the solution of a Riccati equation, which is projected to a subspace of the Krylov subspace generated by Z. For the approximate solution of Lyapunov equations a system of ordinary differential equations (ODEs) is solved via Runge-Kutta methods. It is shown that the space spanned by the approximate solution is a rational Krylov subspace with poles determined by the time step sizes and the eigenvalues of the matrices of the Butcher tableau of the used Runge-Kutta method. The method is applied to a model order reduction problem. The analytical solution of the system of ODEs satisfies an algebraic invariant. Those Runge-Kutta methods which preserve this algebraic invariant are characterized by a simple condition on the corresponding Butcher tableau. It is proven that these methods are equivalent to the ADI iteration. The invariance approach is transferred to Sylvester equations.Diese Arbeit befasst sich mit der numerischen Lösung hochdimensionaler Matrixgleichungen mittels iterativer Verfahren. Aufgrund der Größe der Systemmatrizen in großskaligen algebraischen Riccati-Gleichung kann die Lösung nicht direkt bestimmt werden, sondern wird durch eine approximative Lösung ZYZ^* von geringem Rang angenähert. Hierbei wird Z als Basis eines rationalen Krylovraums gewählt und enthält nur wenige Spalten. Die innere Matrix Y ist klein und quadratisch. Es werden zwei Wege untersucht, die Matrix Y zu wählen: Durch eine Rang-Bedingung an das Riccati-Residuum und durch Projektion des Riccati-Residuums auf den von Z erzeugten Krylovraum. Die Rang-Bedingung wird durch die wohlbekannten ADI-Verfahren motiviert. Die approximativen ADI-Lösungen spannen einen Krylovraum auf und führen zu einem Riccati-Residuum vom Rang p. Es wird bewiesen, dass die Rang-p-Bedingung Existenz und Eindeutigkeit einer solchen approximativen Lösung impliziert. Aus diesem Ergebnis werden effiziente iterative Verfahren abgeleitet, die eine solche approximative Lösung erzeugen. Bisher bekannte Projektionsverfahren werden auf schiefe Projektionen erweitert und es wird eine neue Formulierung des Riccati-Residuums hergeleitet, die eine effiziente Berechnung der Norm erlaubt. Weiter wird eine abgeschnittene approximative Lösung als Lösung einer Riccati-Gleichung charakterisiert, die auf einen Unterraum des von Z erzeugten Krylovraums projiziert wird. Um die Lösung der Lyapunov-Gleichung zu approximieren wird ein System gewöhnlicher Differentialgleichungen mittels Runge-Kutta-Verfahren numerisch gelöst. Es wird gezeigt, dass der von der approximativen Lösung aufgespannte Raum ein rationaler Krylovraum ist, dessen Pole von den Zeitschrittweiten der Integration und den Eigenwerten der Koeffizientenmatrix aus dem Butcher-Tableau des verwendeten Runge-Kutta-Verfahrens abhängen. Das Verfahren wird auf ein Problem der Modellreduktion angewendet. Die analytische Lösung des Differentialgleichungssystems erfüllt eine algebraische Invariante. Diejenigen Runge-Kutta-Verfahren, die diese Invariante erhalten, werden durch eine Bedingung an die zugehörigen Butcher-Tableaus charakterisiert. Es wird gezeigt, dass diese speziellen Verfahren äquivalent zur ADI-Iteration sind. Der Invarianten-Ansatz wird auf Sylvester-Gleichungen übertragen
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System
Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy
leads to a multicomponent Burgers hierarchy. We show in particular that any
solution of the latter also solves a corresponding multicomponent (potential)
KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more
general relation between the multicomponent linear heat hierarchy and the
multicomponent KP hierarchy. From this results a construction of exact
solutions of the latter via a matrix linear system.Comment: 18 pages, 4 figure
A new operational matrix based on Bernoulli polynomials
In this research, the Bernoulli polynomials are introduced. The properties of
these polynomials are employed to construct the operational matrices of
integration together with the derivative and product. These properties are then
utilized to transform the differential equation to a matrix equation which
corresponds to a system of algebraic equations with unknown Bernoulli
coefficients. This method can be used for many problems such as differential
equations, integral equations and so on. Numerical examples show the method is
computationally simple and also illustrate the efficiency and accuracy of the
method
Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP Hierarchy
We use ideas of the geometry of bihamiltonian manifolds, developed by
Gel'fand and Zakharevich, to study the KP equations. In this approach they have
the form of local conservation laws, and can be traded for a system of ordinary
differential equations of Riccati type, which we call the Central System. We
show that the latter can be linearized by means of a Darboux covering, and we
use this procedure as an alternative technique to construct rational solutions
of the KP equations.Comment: Latex, 27 pages. To appear in Commun. Math. Phy
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