Rudolf Scherer

The Grünwald-Letnikov method for fractional differential equations

This paper is devoted to the numerical treatment of fractional differential equations. Based on the Grunwald-Letnikov definition of fractional derivatives, finite difference schemes for the approximation of the solution are discussed. The main properties of these explicit and implicit methods concerning the stability, the convergence and the error behavior are studied related to linear test equations. The asymptotic stability and the absolute stability of these methods are proved. Error representations and estimates for the truncation, propagation and global error are derived. Numerical experiments are given.

Reflected and transposed Runge-Kutta methods

By means of a reflection and a transposition principle one can show interesting interrelations between several classes of implicit Runge-Kutta methods. These relations are very useful in connection with the study of stability properties and the construction of implicit Runge-Kutta methods.

Predictor-corrector methods for a linear stochastic oscillator with additive noise

The predictor-corrector methods P(EC)^k with equidistant discretization are applied to the numerical integration of a linear stochastic oscillator. Their ability in preserving the symplecticity, the linear growth property of the second moment, and the oscillation property of the solution of this stochastic system is studied. Their mean-square orders of convergence are discussed. Numerical experiments are performed.

A generalization of the positive real lemma

The positive real lemma (also called the Kalman-Yacubovich-Popov lemma) characterizes the positive realness of the transfer function matrix of a linear dynamic system by algebraic conditions. In the case of a pole-zero cancellation appearing in the transfer function matrix, or in other words, missing the assumptions of controllability and observability, there exist generalized versions, which are discussed and proven applying the Kalman canonical decomposition. >

Algebraic characterization of A-stable Runge-Kutta methods

Important stability concepts for Runge-Kutta methods are A-stability and B-stability. Whereas A-stability is described by an analytic property of the stability function, B-stability can be characterized by algebraic conditions related to the generating matrix of the method. In this paper, we deduce an algebraic characterization of A-stable Runge-Kutta methods similar to that of B-stable methods. This result gives us information about the gap between the set of A- and B-stable Runge-Kutta methods. Further results, concerning the W-transformation, provide the possibility to derive A-stable methods. To a giver positive quadrature formula we construct all corresponding A-stable Runge-Kutta methods of a certain type.

A necessary condition forB-stability

A necessary condition forB-stability is derived. Then it is shown that the Runge-Kutta methods of Lobatto type IIIA and IIIB and some other methods are notB-stable.

A Generalized W-Transformation for Constructing Symplectic Partitioned Runge-Kutta Methods

This paper deals with the construction of implicit symplectic partitioned Runge–Kutta methods (PRKM) of high order for separable and general partitioned Hamiltonian systems. The main tool is a generalized W-transformation for PRKM based on different quadrature formulas. Methods of high order and special properties can be determined using the transformed coefficient matrices. Examples are given.

Stability Analysis of Equilibrium Manifolds for a Two-Predators One-Prey Model

Abstract The objective of this paper is to analyze the stability of equilibrium manifolds for a ratio-dependent two-predators one-prey model. Some model results are presented first with the bifurcation without parameters method, and then the method was used to study bifurcation along the equilibrium manifold for the model. The model does not lose stability even when some equilibria are locally unstable because the equilibrium manifold is stable when treated as a whole. The ecological implications of the results are discussed.

Protter-Morawetz multidimensional problems

About 50 years ago M.H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter-Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain—the nonhomogeneous wave equation is studied in a (3+1)-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in ℝ2. In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex O of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.

GAUSS-RUNGE-KUTTA-NYSTROM METHODS

Symplectic Runge-Kutta-Nyström methods are frequently used to integrate secondorder systems of the special formÿ=f(y), where the functionf is the gradient of a scalar field multiplied by a regular matrix. In this paper Gauss-Runge-Kutta-Nyström methods, i.e., methods of the highest order, are discussed. It is proved that these methods are always symmetric and that symmetry is equivalent to symplecticness. Furthermore, it is shown that for each stage number the symplectic Gauss-Runge-Kutta-Nyström methods are given by a family of methods with one free parameter.