nonlinear delay systems controllability abstract this research resolves some controllability questions of general nonlinear delay systems in function space in which earlier solutions of the optimal problem have been shown to exist by using a powerful open mapping theorem and gateaux differentiability we avoid the difficult requirement of frechet differentiability of an operator f wp 1 lp wp 1 which is associated with the nonlinear system properties of the linearized system help ensure the nonlinear system s controllability

exponentially fitted runge kutta methods abstract exponentially fitted runge kutta methods with s stages are constructed which exactly integrate differential initial value problems whose solutions are linear combinations of functions of the form x j exp x x j exp x r or i r j 0 1 j max where 0 j max s 2 1 the lower bound being related to explicit methods the upper bound applicable for collocation methods explicit methods with s 2 3 4 belonging to that class are constructed for these methods a study of the local truncation error is made out of which follows a simple heuristic to estimate the value error and step length control is introduced based on richardson extrapolation ideas some numerical experiments show the efficiency of the introduced methods it is shown that the same techniques can be applied to construct implicit exponentially fitted runge kutta methods

adapted numerical methods for advection reaction diffusion problems generating periodic wavefronts abstract the paper presents an adapted numerical integration for advection reaction diffusion problems the numerical scheme exploiting the a priori knowledge of the qualitative behaviour of the solution gains advantages in terms of efficiency and accuracy with respect to classic schemes already known in literature the adaptation is here carried out through the so called trigonometrical fitting technique for the discretization in space giving rise to a system of odes whose vector field contains both stiff and non stiff terms due to this mixed nature of the vector field an implicit explicit imex method is here employed for the integration in time based on the first order forward backward euler method the coefficients of the method here introduced rely on unknown parameters which have to be properly estimated in this work such an estimate is performed by minimizing the leading term of the local truncation error the effectiveness of this problem oriented approach is shown through a rigorous theoretical analysis and some numerical experiments

revised trigonometrically fitted two step hybrid methods with equation dependent coefficients for highly oscillatory problems for the numerical integration of highly oscillatory problems revised trigonometrically fitted two step hybrid methods rtftsh with equation dependent coefficients are considered the local truncation errors stability and phase properties of the new method are analyzed a feature of the new type of the methods is that the errors in the internal stages are assumed to contribute to the accuracy of the update a new revised method rtftsh4 of algebraic order four and phase lag order four is derived numerical experiments are reported to show that the new method rtftsh4 is much more efficient and robust than the standard fourth order method stftsh4

stochastic seir model with jumps abstract the paper establishes stochastic seir models with jumps obtains system 2 and system 3 by using two different disturbance manners respectively which are used to describe the wide spread of the infectious diseases due to the medical negligence etc by adding the first random perturbations we obtain lyapunov function and examine that the solutions of model 2 are stochastically stable with the average time summarize the conditions for persistence and extinction of the infectious diseases by adding another random disturbance we get the stochastic system 3 we prove that the positive solution of the system is stochastic asymptotically stable by using lyapunov method the conditions that the infectious diseases spread continually are obtained these studies are good for the prevention and control of the infectious diseases which are with the incubation period especially with long ones

controllability of semilinear impulsive nonautonomous systems in this paper we apply rothe s type fixed point theorem to prove the controllability of the following semilinear impulsive nonautonomous systems of differential equations where a t b t are continuous matrices of dimension n n and n m respectively the control function u belongs to and k 1 2 3 p under additional conditions we prove the following statement if the linear is controllable on 0 then the semilinear impulsive system is also controllable on 0 moreover we could exhibit a control steering the nonlinear system from an initial state z0 to a final state z1 at time 0

exponentially fitted singly diagonally implicit runge kutta methods it is the purpose of this paper to derive diagonally implicit exponentially fitted ef runge kutta methods for the numerical solution of initial value problems based on first order ordinary differential equations whose solutions are supposed to exhibit an exponential behaviour in addition to the standard approach for the derivation of ef methods we provide a revised constructive technique that takes into account the contribution to the error inherited from the computation of the internal stages the derived methods are then compared to those obtained by neglecting the contribution of the error associated to the internal stages as classically done in the standard derivation of multistage ef based methods standard and revised ef methods are then compared in terms of linear stability and numerical performances

a class of explicit high order exponentially fitted two step methods for solving oscillatory ivps abstract the derivation of new exponentially fitted ef modified two step hybrid mtsh methods for the numerical integration of oscillatory second order ivps is analyzed these methods are modifications of classical two step hybrid methods so that they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions exp t exp t c or equivalently sin t cos t when i r where represents an approximation of the main frequency of the problem the ef conditions and the conditions for this class of ef schemes to have algebraic order p with p 8 are derived with the help of these conditions we construct explicit efmtsh methods with algebraic orders seven and eight which require five and six function evaluation per step respectively these new efmtsh schemes are optimal among the two step hybrid methods in the sense that they reach a certain order of accuracy with minimal computational cost per step in order to show the efficiency of the new high order explicit efmtsh methods in comparison to other ef and standard two step hybrid codes from the literature some numerical experiments with several orbital and oscillatory problems are presented

mixed collocation methods for y f x y inline equatio abstract the collocation methods introduced here are based on linear combinations of trigonometric functions and powers the motivation is to provide better approximations for oscillatory solutions of initial value problems for differential equations of the special form y f x y the resulting methods for two or more collocation points are implicit runge kutta nystrom methods with coefficients which depend on both the fitted angular frequency and the steplength algebraic and trigonometric order conditions are considered and the stability properties of some methods are examined particular mixed collocation methods and other methods for the same class of problems are compared by applying them to a variety of test problems

exponentially fitted two step hybrid methods for y f x y it is the purpose of this paper to derive two step hybrid methods for y f x y with oscillatory or periodic solutions specially tuned to the behaviour of the solution through the usage of the exponential fitting technique the construction of two step exponentially fitted hybrid methods is shown and their properties are discussed some numerical experiments confirming the theoretical expectations are provided

exponentially fitted numerov methods the error in the estimate of the kth eigenvalue of a regular sturm liouville problem obtained by numerov s method with mesh length h is o k 6h 4 it is shown that the error can be reduced to o k 3h 4 by using one of the three versions of the exponentially fitted numerov method numerical examples demonstrate the usefulness of this approach even for low values of k

symmetric trigonometrically fitted two step hybrid methods for oscillatory problems abstract in this paper we propose a general class of trigonometrically fitted two step hybrid tftsh methods for solving numerically oscillatory second order initial value problems the tftsh methods integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set exp i w t exp i w t or equivalently the set cos w t sin w t where w represents an approximation of the main frequency of the problem by introducing the generalized b2 series the necessary and sufficient conditions for tftsh methods of up to arbitrarily high order p are derived we also investigate the symmetry of tftsh methods and analyze the symmetric conditions of tftsh methods based on the order conditions and symmetric conditions a diagonally implicit two stage symmetric tftsh method with order four is constructed some numerical experiments are provided to confirm the theoretical expectations

the optimal exponentially fitted numerov method for solving two point boundary value problems second order boundary value problems are solved by means of exponentially fitted numerov methods these methods which depend on a parameter can be constructed following a six step flow chart of ixaru and vanden berghe l gr ixaru g vanden berghe exponential fitting kluwer academic publishers dordrecht 2004 special attention is paid to the expression of the error term of such methods an algorithm concerning the choice of the best suited method and its parameter is discussed several numerical examples are given to sustain the theory

exponentially fitted gauss laguerre quadrature rule for integrals over an unbounded interval new quadrature formulae are introduced for the computation of integrals over the whole positive semiaxis when the integrand has an oscillatory behaviour with decaying envelope the new formulae are derived by exponential fitting and they represent a generalization of the usual gauss laguerre formulae their weights and nodes depend on the frequency of oscillation in the integrand and thus the accuracy is massively increased rules with one up to six nodes are treated with details numerical illustrations are also presented

rothe s fixed point theorem and controllability of semilinear nonautonomous systems abstract in this paper we apply rothe s fixed point theorem to prove the controllability of the following semilinear system of ordinary differential equations z t a t z t b t u t f t z t u t t 0 z 0 z 0 where z t r n u t r m a t b t are continuous matrices of dimensions n n and n m respectively the control function u belongs to l 2 l 2 0 r m and the nonlinear function f 0 r n r m r n is continuous and there are a b c r and 1 2 1 such that f t z u r n a z r n b u r m c u r m z r n under this condition we prove the following statement if the linear z t a t z t b t u t is controllable then the semilinear system is also controllable on 0 moreover we could exhibit a control steering the nonlinear system from an initial state z 0 to a final state z 1 at time 0

on partial s controllability of semilinear partially observable systems in this paper the partial s controllability of semilinear partially observable control systems is investigated the essence of study is the presence of partial observations we define a weakened partial s controllability concept for partially observable systems a sufficient condition for this controllability concept is proved the method of proof differs from the traditional proofs by fixed point theorems the obtained result is demonstrated with examples

diagonally implicit runge kutta nystro m methods for oscillatory problems implicit runge kutta nystrom rkn methods are constructed for the integration of second order differential equations possessing an oscillatory solution based on a linear homogeneous test model we analyse the phase errors or dispersion introduced by these methods and derive so called dispersion relations diagonally implicit rkn methods of relatively low algebraic order are constructed which have a high order of dispersion up to 10 application of these methods to a number of test examples linear as well as nonlinear yields a greatly reduced phase error when compared with conventional dirkn methods

phase fitted and amplification fitted two step hybrid methods for y f x y this paper provides a theoretical framework for a new type of phase fitted and amplification fitted two step hybrid ftsh methods which is introduced by the author in h van de vyver a phase fitted and amplification fitted explicit two step hybrid method for second order periodic initial value problems internat j modern phys c 17 2006 663 675 the methods constitute a modification of dissipative two step hybrid methods in the sense that two free parameters are added to eliminate the phase lag and the amplification error the methods are useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize the algebraic order zero stability stability and phase properties are examined the theory is illustrated with sixth order explicit ftsh methods numerical results carried out on an assortment of test problems show the relevance of the theory

p stable general nystrom methods for y f y t we focus our attention on the family of general linear methods glms for the numerical solution of second order ordinary differential equations odes these are multivalue methods introduced in d ambrosio et al 2012 3 with the aim to provide an unifying approach for the analysis of the properties of accuracy of numerical methods for second order odes our investigation is addressed to providing the building blocks useful to analyze the linear stability properties of glms for second order odes thus we present the extension of the classical notions of stability matrix stability polynomial stability and periodicity interval a stability and p stability to the family of glms special attention will be given to the derivation of highly stable glms whose stability properties depend on the stability polynomial of indirect runge kutta nystrom methods based on gauss legendre collocation points which are known to be p stable in this way we are able to provide p stable glms whose order of convergence is greater than that of the corresponding rkn method without heightening the computational cost we finally provide and discuss examples of p stable irreducible glms satisfying the mentioned features

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

exponentially fitted explicit runge kutta nystrom methods exponentially fitted runge kutta nystrom efrkn methods for the numerical integration of second order ivps with oscillatory solutions are derived these methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions exp t exp t c or equivalently sin t cos t when i r explicit efrkn methods with two and three stages and algebraic orders 3 and 4 are constructed in addition a 4 3 embedded pair of explicit efrkn methods based on the fsal technique is obtained which permits to introduce an error and step length control with a small cost added some numerical experiments show the efficiency of our explicit efrkn methods when they are compared with other exponential fitting type codes proposed in the scientific literature

an eight step semi embedded predictor corrector method for orbital problems and related ivps with oscillatory solutions for which the frequency is unknown our new linear symmetric semi embedded predictor corrector method sepcm presented here is based on the multistep symmetric method of quinlan and tremaine 1990 with eight steps and eighth algebraic order and constructed to solve numerically the two dimensional kepler problem it can also be used to integrate related ivps with oscillatory solutions for which the frequency is unknown firstly we present a sepcm see panopoulos and simos 2013 36 37 in pair form this form has the advantage that reduces the computational expense from this form we construct a new symmetric eight step method the new scheme has constant coefficients and algebraic order ten we tested the efficiency of our newly developed scheme against some well known methods from the literature we measure the efficiency of the methods and conclude that the new scheme is the most efficient of all the compared methods and for all the problems solved