A fourth algebraic order trigonometrically fitted predictor–corrector scheme for IVPs with oscillating solutions

A scheme of trigonometrically fitted predictor–corrector (P–C) Adams–Bashforth–Moulton methods is constructed in this paper. Our new P–C method is based on the third order Adams–Bashforth scheme (as predictor) and on the fourth order Adams–Moulton scheme (as corrector). We tested the efficiency of our newly developed scheme against well known methods, with excellent results. The numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions.     

Hamiltonian structures for the Ostrovsky–Vakhnenko equation

We obtain a bi-Hamiltonian formulation for the Ostrovsky–Vakhnenko (OV) equation using its higher order symmetry and a new transformation to the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. Central to this derivation is the relation between Hamiltonian structures when dependent and independent variables are transformed.     

A second‐order ensemble method based on a blended backward differentiation formula timestepping scheme for time‐dependent Navier–Stokes equations

We present a second‐order ensemble method based on a blended three‐step backward differentiation formula (BDF) timestepping scheme to compute an ensemble of Navier–Stokes equations. Compared with the only existing second‐order ensemble method that combines the two‐step BDF timestepping scheme and a special explicit second‐order Adams–Bashforth treatment of the advection term, this method is more accurate with nominal increase in computational cost. We give comprehensive stability and error analysis for the method. Numerical examples are also provided to verify theoretical results and demonstrate the improved accuracy of the method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 34–61, 2017     

Three methods for two‐sided bounds of eigenvalues—A comparison

We compare three finite element‐based methods designed for two‐sided bounds of eigenvalues of symmetric elliptic second order operators. The first method is known as the Lehmann–Goerisch method. The second method is based on Crouzeix–Raviart nonconforming finite element method. The third one is a combination of generalized Weinstein and Kato bounds with complementarity‐based estimators. We concisely describe these methods and use them to solve three numerical examples. We compare their accuracy, computational performance, and generality in both the lowest and higher order case.     

On Filippov algebroids and multiplicative Nambu–Poisson structures

We discuss relations of linear Nambu–Poisson structures to Filippov algebras and define a Filippov algebroid—a generalization of a Lie algebroid. We also prove results describing multiplicative Nambu–Poisson structures on Lie groups. In particular, it is shown that simple Lie groups do not admit multiplicative Nambu–Poisson structures of order >2.     

The streamline–diffusion method for a convection–diffusion problem with a point source

A linear singularly perturbed convection–diffusion problem with a point source is considered. The problem is solved using the streamline–diffusion finite element method on a class of Shishkin–type meshes. We prove that the method is almost optimal with uniform second order of convergence in the maximum norm. We also prove the existence of superconvergent points for the first derivative. Numerical experiments support these theoretical results.     

Cylindrically symmetric gravitational-wavelike space–times

We present Noether symmetries of a geodetic Lagrangian for a time-conformal cylindrically symmetric space–time. We introduce a time-conformal factor in the general cylindrically symmetric space–time to make it nonstatic and then find approximate Noether symmetries of the action of the corresponding Lagrangian. Taking the perturbation up to the first order, we find all Lagrangians for cylindrically symmetric space–times for which approximate Noether symmetries exist.     

Attractors for second order periodic lattices with nonlinear damping

We consider second order nonlinear lattices under the effect of nonlinear damping. The family we study is subject to cyclic boundary conditions and includes as distinguished examples the Fermi–Pasta–Ulam and sine-Gordon lattices. We prove global well posedness and existence of a global attractor.     

On the Feynman–Kac representation for solutions of the Cauchy problem for parabolic equations in divergence form

We consider the Cauchy problem for general second–order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem in terms of solutions of associated linear backward stochastic differential equations. Our representation may be considered as an extension of the classical Feynman–Kac formula.     

Uniqueness for Solutions of Fokker–Planck Equations on Infinite Dimensional Spaces

We develop a general technique to prove uniqueness of solutions for Fokker–Planck equations on infinite dimensional spaces. We illustrate this method by implementing it for Fokker–Planck equations in Hilbert spaces with Kolmogorov operators with irregular coefficients and both non-degenerate or degenerate second order part.     

Weighted core–EP inverse of an operator between Hilbert spaces

We introduce and study the weighted core–EP inverse of an operator between two Hilbert spaces as a generalization of the weighted core–EP inverse for a rectangular matrix. Several new properties of weighted core–EP inverses are given and some known results are extended. Using a weighted operator, the core–EP pre-order and the minus partial order of corresponding operators, we define new pre-orders on the set of all Wg–Drazin invertible operators between two Hilbert spaces. As consequences of our results, we present a new characterization and new representations of the core–EP inverse, new characterizations of the core–EP pre-order and extend the core–EP pre-order to a partial order.     

Electro–Reaction–Diffusion Systems Including Cluster Reactions of Higher Order

In this paper we consider electro–reaction–diffusion systems modelling the transport of charged species in two–dimensional heterostructures. Our aim is to investigate the case that besides of reactions with source terms of at most second order so called cluster reactions of higher order are involved. We prove the unique solvability of the model equations and show the global boundedness and asymptotic properties of the solution. In order to get necessary a priori estimates we apply an anisotropic iteration scheme followed by usual Moser iterations. Then existence is obtained by cutting off the reaction terms.     

Rota–Baxter Operators on Quadratic Algebras

We prove that all Rota–Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota–Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota–Baxter operators and the solutions to the alternative Yang–Baxter equation on the Cayley–Dickson algebra. We also investigate the Rota–Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.     

A nonautonomous Beverton–Holt equation of higher order

In this paper, we discuss a certain nonautonomous Beverton–Holt equation of higher order. After a brief introduction to the classical Beverton–Holt equation and recent results, we solve the higher-order Beverton–Holt equation by rewriting the recurrence relation as a difference system of order one. In this process, we examine the existence and uniqueness of a periodic solution and its global attractivity. We continue our analysis by studying the corresponding second Cushing–Henson conjecture, i.e., by relating the average of the unique periodic solution to the average of the carrying capacity.     

High order variational numerical schemes with application to Nash–MFG vaccination games

This paper introduces high-order explicit Runge–Kutta numerical schemes in metric spaces. We show that our approach reduces to the corresponding Runge–Kutta schemes if the ambient space is Hilbert. We apply these schemes to compute the Nash equilibrium in a mean field vaccination game. Numerical simulations show improvement in the speed of convergence towards the Nash equilibrium; the numerical scheme has high order (2–4) in time.     

A canonical duality approach for the solution of affine quasi-variational inequalities

We propose a new formulation of the Karush–Kunt–Tucker conditions of a particular class of quasi-variational inequalities. In order to reformulate the problem we use the Fisher–Burmeister complementarity function and canonical duality theory. We establish the conditions for a critical point of the new formulation to be a solution of the original quasi-variational inequality showing the potentiality of such approach in solving this class of problems. We test the obtained theoretical results with a simple heuristic that is demonstrated on several problems coming from the academy and various engineering applications.     

A Calderón–Zygmund operator of higher order Schrödinger type

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We show that an operator of higher order Schrödinger type is a Calderón–Zygmund operator. We also show that there exist potentials which satisfy our assumptions but are not nonnegative polynomials.     

Controllability of quasi-linear Hamiltonian NLS equations

We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash–Moser–Hörmander implicit function theorem as a black box.     

Free point processes and free extreme values

We continue here the study of free extreme values begun in Ben Arous and Voiculescu (Ann Probab 34:2037–2059, 2006). We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure (Voiculescu in Lecture notes in mathematics. Springer, Heidelberg, pp. 279–349, 1998; Barndorff-Nielsen and Thorbjornsen in Probab Theory Relat Fields 131:197–228, 2005). We relate this convergence to the free extremal laws introduced in Ben Arous and Voiculescu (Ann Probab 34:2037–2059, 2006) and give the limit laws for free order statistics.