an application for a modified kdv equation by the decomposition method and finite element method a numerical solution to a modified korteweg de vries mkdv equation is obtained using a lumped galerkin method with quadratic b spline finite element method fem and using the adomian s decomposition method adm test problems concerning the motion and interaction of soliton solutions are used to compare the fem with the adm classical problems concerning the development motion and interaction of solitons are used to validate the methods the present methods extremely well in terms of accuracy efficiency simplicity stability and reliability

multirate sampling and input to state stable receding horizon control for nonlinear systems abstract in this paper a robust receding horizon control for multirate sampled data nonlinear systems with bounded disturbances is presented the proposed receding horizon control is based on the solution of bolza type optimal control problems for the approximate discrete time model of the nominal system low measurement rate is assumed it is shown that the multistep receding horizon controller that stabilizes the nominal approximate discrete time model also practically input to state stabilizes the exact discrete time system with disturbances

products of finitely many resolvents of maximal monotone mappings in reflexive banach spaces we propose two algorithms for finding common zeros of finitely many maximal monotone mappings in reflexive banach spaces these algorithms are based on the bregman distance related to a well chosen convex function and improve previous results finally we mention two applications of our algorithms for solving equilibrium problems and convex feasibility problems

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

homotopy perturbation method for fractional kdv equation in this paper the homotopy perturbation method is directly applied to derive approximate solutions of the fractional kdv equation the results reveal that the proposed method is very effective and simple for solving approximate solutions of fractional differential equations

splitting methods for the nonlocal fowler equation we consider a nonlocal scalar conservation law proposed by andrew c fowler to describe the dynamics of dunes and we develop a numerical procedure based on splitting methods to approximate its solutions we begin by proving the convergence of the well known lie formula which is an approximation of the exact solution of order one in time we next use the split step fourier method to approximate the continuous problem using the fast fourier transform and the finite difference method our numerical experiments confirm the theoretical results

numerical solution of complex modified korteweg de vries equation by petrov galerkin method abstract in this paper a petrov galerkin method is used to derive a numerical scheme for the complex modified korteweg de vries equation cmkdv where we have used cubic b splines as test functions and linear b splines as trial functions an implicit midpoint rule is used to discretize in time a block non linear pentadiagonal system is obtained we solve this system by newton s method the resulting scheme has a fourth order accuracy in space direction and second order in time direction it is unconditionally stable by von neumann method the exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the scheme the interaction of two solitary waves for different parameters is discussed

first order soliton perturbation theory for a generalized kdv model with stochastic forcing and damping we discuss an application of the expansion method to the soliton perturbation theory for a generalized kdv model with stochastic forcing and damping we find that the expansion method is a perturbation technique which allows one to retain a more representative linearization for a nonlinear differential equation than comparable methods of perturbation such as those relying on small parameters indeed one primary benefit of the method is that it tends to provide much more reasonable solutions with relatively few terms compared to standard perturbation methods due to the more accurate linearization we show that the method allows one to obtain first order perturbations in which the stochastic and nonlinear contributions separate allowing one to study each influence independently at least to first order we then apply the expansion method to the study of a stochastic generalized kdv model and we show that this procedure gives a natural extension to the perturbation theory for the traditional kdv model the actual method of computation is outlined and may be applied to a variety of generalized kdv models

identifiability of nonlinear systems with application to hiv aids models in this note we investigate different concepts of nonlinear identifiability in the generic sense we work in the linear algebraic framework necessary and sufficient conditions are found for geometrical identifiability algebraic identifiability and identifiability with known initial conditions relationships between different concepts are characterized constructive procedures are worked out for both generic geometrical and algebraic identifiability of nonlinear systems as an application of the theory developed we study the identifiability properties of a four dimensional model of hiv aids the questions answered in this study include the minimal number of measurement of the variables for a complete determination of all parameters and the best period of time to make such measurements this information will be useful in formulating guidelines for clinical practice

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

a direct theory for the perturbed unstable nonlinear schrodinger equation a direct perturbation theory for the unstable nonlinear schrodinger equation with perturbations is developed the linearized operator is derived and the squared jost functions are shown to be its eigenfunctions then the equation of linearized operator is transformed into an equivalent 4 4 matrix form with first order derivative in t and the eigenfunctions into a four component row adjoint functions and the inner product are defined orthogonality relations of these functions are derived and the expansion of the unity in terms of the four component eigenfunctions is implied the effect of damping is discussed as an example

halpern s iteration for bregman strongly nonexpansive mappings in reflexive banach spaces we investigate strong convergence for bregman strongly nonexpansive mappings by halpern s iteration in the framework of reflexive banach spaces using the obtained result convergence for a family of bregman strongly nonexpansive mappings is also discussed as applications we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive banach spaces

well posedness of stochastic kdv bo equation driven by fractional brownian motion the cauchy problem for the korteweg de vries benjamin ono equation driven by cylindrical fractional brownian motion is discussed in this paper fractional brownian motion is a family of processes b h it is known that the smaller the value of hurst parameter h is the worse of the regularity of fbm is using bourgain restriction method we obtain the lower bound of the hurst parameter h for the driving processes b h with h 38 we prove local existence results with initial value in classical sobolev spaces of negative indices i e h s with s 18

receding horizon control via bolza type optimization this paper deals with the stabilization problem of nonlinear control systems a variant of the receding horizon control method is proposed which is based on the solution of a certain bolza problem the stabilizing property of the method is proved in the case of lipschitz continuous value function this version of receding horizon scheme can be considered as a straightforward generalization of the so called fake riccati equation technique for nonlinear constrained systems

two strong convergence theorems for bregman strongly nonexpansive operators in reflexive banach spaces abstract we study the convergence of two iterative algorithms for finding common fixed points of finitely many bregman strongly nonexpansive operators in reflexive banach spaces both algorithms take into account possible computational errors we establish two strong convergence theorems and then apply them to the solution of convex feasibility variational inequality and equilibrium problems

numerical simulation of the stochastic korteweg de vries equation abstract in this work we numerically investigate the influence of a homogeneous noise on the evolution of solitons for the korteweg de vries equation our numerical method is based on finite elements and least squares we present numerical experiments for different values of noise amplitude and describe different types of behaviours

stabilization of sampled data nonlinear systems by receding horizon control via discrete time approximations abstract results on stabilizing receding horizon control of sampled data nonlinear systems via their approximate discrete time models are presented the proposed receding horizon control is based on the solution of bolza type optimal control problems for the parametrized family of approximate discrete time models this paper investigates both situations when the sampling period t is fixed and the integration parameter h used in obtaining approximate model can be chosen arbitrarily small and when these two parameters coincide but they can be adjusted arbitrary sufficient conditions are established which guarantee that the controller that renders the origin to be asymptotically stable for the approximate model also stabilizes the exact discrete time model for sufficiently small integration and or sampling parameters

quasi stationary perturbations of the kdv soliton in this paper we show how the results of quasi stationary perturbation theory relate to those based upon squared eigenfunction expansions we show that the two give the same results for a quasi stationary approximation and also show how standard adiabatic perturbation theory connects to the quasi stationary theory

dissipative conservative galerkin method using discrete partial derivatives for nonlinear evolution equations a new method is proposed for designing galerkin schemes that retain the energy dissipation or conservation properties of nonlinear evolution equations such as the cahn hilliard equation the korteweg de vries equation or the nonlinear schrodinger equation in particular as a special case dissipative or conservative finite element schemes can be derived the key device there is the new concept of discrete partial derivatives as examples of the application of the present method dissipative or conservative galerkin schemes are presented for the three equations with some numerical experiments

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

approximate solution for system of differential difference equations by means of the homotopy analysis method abstract in this paper the homotopy analysis method is successfully applied to solve approximate solutions of the differential difference system the solution of another relativistic toda lattice system is considered comparisons made between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective in solving differential difference system

the stochastic damped kdv equation the author applies singular perturbation theory to the study of a damped kdv soliton under the influence of space and time dependent external noise he finds that asymptotically the shape of the averaged solution approaches that of a gaussian packet whose amplitude decays and width grows in the same way as in the case of purely time dependent noise

a comparative study of numerical solutions of a class of kdv equation abstract in this work we present a comparative study of meshless method modified bernstein polynomials bp and b spline finite element method bs fem for the numerical solution of two different models of korteweg de vries kdv equation the multiquadric mq and gaussian ga radial basis functions rbfs are used due to exponential convergence rate excellent agreement is found between exact and rbfs solutions and same accuracy is obtained as bernstein polynomials and b spline finite element method

a split step fourier method for the complex modified korteweg de vries equation abstract in this study the complex modified korteweg de vries cmkdv equation is solved numerically by three different split step fourier schemes the main difference among the three schemes is in the order of the splitting approximation used to factorize the exponential operator the space variable is discretized by means of a fourier method for both linear and nonlinear subproblems a fourth order runge kutta scheme is used for the time integration of the nonlinear subproblem classical problems concerning the motion of a single solitary wave with a constant polarization angle are used to compare the schemes in terms of the accuracy and the computational cost furthermore the interaction of two solitary waves with orthogonal polarizations is investigated and particular attention is paid to the conserved quantities as an indicator of the accuracy numerical tests show that the split step fourier method provides highly accurate solutions for the cmkdv equation

long time dynamics of korteweg de vries solitons driven by random perturbations this paper deals with the transmission of a soliton in a random medium described by a randomly perturbed korteweg de vries equation different kinds of perturbations are addressed depending on their specific time or position dependences with or without damping we derive effective evolution equations for the soliton parameter by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus original results are derived that are very different compared to a randomly perturbed nonlinear schrodinger equation first the emission of a soliton gas is proved to be a very general feature second some perturbations are shown to involve a speeding up of the soliton instead of the decay that is usually observed in random media

multisoliton solution of a class of coupled soliton equations abstract the n soliton solutions of the korteweg devries kdv and boussinesq equations with a source are obtained explicitly by using the bilinear transformation method also the interactions of two soliton solutions are studied and the explicit forms for the one and two soliton solutions are obtained for the two systems

a posteriori error estimates for discontinuous galerkin methods for the generalized korteweg de vries equation we construct analyze and numerically validate a posteriori error estimates for conservative discontinuous galerkin dg schemes for the generalized korteweg de vries gkdv equation we develop the concept of dispersive reconstruction i e a piecewise polynomial function which satisfies the gkdv equation in the strong sense but with a computable forcing term enabling the use of a priori error estimation techniques to obtain computable upper bounds for the error both semidiscrete and fully discrete approximations are treated

sufficient conditions for stabilization of sampled data nonlinear systems via discrete time approximations given a parameterized by sampling period t family of approximate discrete time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all t sufficiently small we present conditions which guarantee that the same family of controllers semi globally practically stabilizes the exact discrete time model of the plant for sufficiently small sampling periods when the family of controllers is locally bounded uniformly in the sampling period the inter sample behavior can also be uniformly bounded so that the time varying sampled data model of the plant is uniformly semi globally practically stabilized the result justifies controller design for sampled data nonlinear systems based on the approximate discrete time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied our analysis is applicable to a wide range of time discretization schemes and general plant models

a general framework to design stabilizing nonlinear model predictive controllers abstract we propose a new model predictive control mpc framework to generate feedback controls for time varying nonlinear systems with input constraints we provide a set of conditions on the design parameters that permits to verify a priori the stabilizing properties of the control strategies considered the supplied sufficient conditions for stability can also be used to analyse the stability of most previous mpc schemes the class of nonlinear systems addressed is significantly enlarged by removing the traditional assumptions on the continuity of the optimal controls and on the stabilizability of the linearized system some important classes of nonlinear systems including some nonholonomic systems can now be stabilized by mpc in addition we can exploit increased flexibility in the choice of design parameters to reduce the constraints of the optimal control problem and thereby reduce the computational effort in the optimization algorithms used to implement mpc

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

the boundary forced mkdv equation an unconditionally stable numerical algorithm for the modified korteweg de vries equation based on the b spline finite element method is described the algorithm is validated through a single soliton simulation in further numerical experiments forced boundary conditions u u0 are applied at the end x 0 and the generated states of solitary waves are studied by long impulse experiments these are shown to be generated periodically with period tb proportional to u 30 and to have a limiting amplitude proportional to u0 this limit is achieved by all waves after the first provided the experiment proceeds long enough the temporal development of the derivatives u 0 t u 0 t and u 0 t is also periodic with period tb the effect of negative forcing is to generate a train of negative waves the solitary wave states generated by applying a positive impulse followed immediately by an negative impulse of equal amplitude and duration is dependent on the period of forcing the solitary waves generated by these various forcing functions possess many of the attributes of free solitons

korteweg de vries solitons under additive stochastic perturbations the evolution of solitonic solutions of the korteweg de vries equation subject to additive noise is investigated using numerical techniques various types of additive white gaussian noise are considered the averaged solution amplitudes exhibit in all cases algebraic decay verifying wadati s universality conjecture if the noise is time dependent or position and time dependent algebraic decay is obtained for intermediate times too these intermediate time results agree well with the outcome of an experiment on ion acoustic soliton propagation in a noisy plasma the distribution of soliton first passage times in a noisy medium is also discussed

hiv dynamics modeling data analysis and optimal treatment protocols we present an overview of some concepts and methodologies we believe useful in modeling hiv pathogenesis after a brief discussion of motivation for and previous efforts in the development of mathematical models for progression of hiv infection and treatment we discuss mathematical and statistical ideas relevant to structured treatment interruptions sti among these are model development and validation procedures including parameter estimation data reduction and representation and optimal control relative to sti results from initial attempts in each of these areas by an interdisciplinary team of applied mathematicians statisticians and clinicians are presented

strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications abstract the purpose of this article is to prove strong convergence theorems for common fixed points of two countable families of weak relatively nonexpansive mappings in banach spaces in order to get the strong convergence theorems the monotone hybrid algorithms are presented and are used to approximate the common fixed points using this result we also discuss the problem of strong convergence concerning the maximal monotone operators in a banach space the results of this article modify and improve the results of matsushita and takahashi s matsushita w takahashi a strong convergence theorem for relatively nonexpansive mappings in a banach space j approx theory 134 2005 257 266 and the results of plubtieng and ungchittrakool s plubtieng k ungchittrakool strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a banach space j approx theory 149 2007 103 115 and the results of su et al y su z wang and h xu strong convergence theorems for a common fixed point of two hemi relatively nonexpansive mappings nonlinear anal 71 2009 5616 5628 and many others

modified block iterative algorithm for quasi ϕ asymptotically nonexpansive mappings and equilibrium problem in banach spaces abstract the purpose of this paper is to propose a modified block iterative algorithm for find a common element of the set of common fixed points of an infinite family of quasi asymptotically nonexpansive mappings and the set of an equilibrium problem under suitable conditions some strong convergence theorems are established in a uniformly smooth and strictly convex banach space with the kadec klee property as an application at the end of the paper a numerical example is given the results presented in the paper improve and extend the corresponding results in qin et al convergence theorems of common elements for equilibrium problems and fixed point problem in banach spaces j comput appl math 225 2009 20 30 zhou et al convergence theorems of a modified hybrid algorithm for a family of quasi asymptotically nonexpansive mappings j appl math compt 17 march 2009 doi 10 1007 s12190 009 0263 4 takahashi and zembayshi strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in banach spaces nonlinear anal 70 2009 45 57 wattanawitoon and kumam strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi nonexpansive mappings nonlinear anal hybrid syst 3 2009 11 20 and matsushita and takahashi a strong convergence theorem for relatively nonexpansive mappings in banach spaces j approx theory 134 2005 257 266 and others

operator splitting methods for generalized korteweg de vries equations we apply the method of operator splitting on the generalized korteweg de vries kdv equation ut f u x uxxx 0 by solving the nonlinear conservation law ut f u x 0 and the linear dispersive equation ut uxxx 0 sequentially we prove that if the approximation obtained by operator splitting converges then the limit function is a weak solution of the generalized kdv equation convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems

hiv dynamics analysis and robust multirate mpc based treatment schedules abstract analysis and control of human immunodeficiency virus hiv infection have attracted the interests of mathematicians and control engineers during the recent years several mathematical models exist and adequately explain the interaction of the hiv infection and the immune system up to the stage of clinical latency as well as viral suppression and immune system recovery after treatment therapy however none of these models can completely exhibit all that is observed clinically and account the full course of infection besides model inaccuracies that hiv models suffer from some disturbances uncertainties from different sources may arise in the modelling in this paper we study the basic properties of a 6 dimensional hiv model that describes the interaction of hiv with two target cells cd4 t cells and macrophages the disturbances are modelled in the hiv model as additive bounded disturbances highly active antiretroviral therapy haart is used the control input is defined to be dependent on the drug dose and drug efficiency we developed treatment schedules for hiv infected patients by using robust multirate model predictive control mpc based method the mpc is constructed on the basis of the approximate discrete time model of the nominal model we established a set of conditions which guarantee that the multirate mpc practically stabilizes the exact discrete time model with disturbances the proposed method is applied to the stabilization of the uninfected steady state of the hiv model the results of simulations show that after initiation of haart with a strong dosage the viral load drops quickly and it can be kept under a suitable level with mild dosage of haart moreover the immune system is recovered with some fluctuations due to the presence of disturbances

linearly implicit methods for the nonlinear schrodinger equation in nonhomogeneous media the nonlinear schrodinger equation in one dimensional cartesian coordinates is studied numerically by means of three linearly implicit finite difference methods that result in block tridiagonal matrices at each time level as a function of the damping pumping and changes in the refraction index the numerical experiments performed to determine the effects of the time step spatial step size and implicitness parameters of the dispersion and nonlinear terms show that the most accurate results are obtained when both the dispersion and nonlinear terms are treated by means of a second order accurate trapezoidal discretization and full linearization of the nonlinear terms it is also shown that if either the dispersion or the nonlinear terms are treated with a first order accurate approximation substantial errors are found and these errors are largest when the nonlinear terms are discretized by means of a first order difference and do not decrease substantially as the number of grid points is increased it has been observed that when solitons move from a medium to another one characterized by a higher refraction index the amplitude and speed of the soliton increase whereas its width decreases upon approaching the interface the amplitude of the soliton was found to increase downstream while some radiation was observed upstream both the increase in amplitude and radiation were found to increase with the change in the refraction index the effect of the distance where the index of refraction changes was found to be small if this distance is smaller than the width of the soliton for solitons propagating from a medium to another one of lower refraction index the amplitude and speed of the soliton decrease whereas its width increases the amplitude of the soliton increases downstream and spreads towards the boundary where it is reflected when the index of refraction decreases below a threshold it was found that the soliton may become trapped at the interface the results presented in this paper indicate that theories based on treating solitons as particles perturbation methods inverse scattering and variational formulations can provide accurate results only when the change in the refraction index is sufficiently small

an analytical numerical method for solving the korteweg de vries equation in this paper an analytical numerical method is applied to the one dimensional korteweg de vries equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times two test problem with known exact solutions are studied to demonstrate the accuracy of the present method the obtained results are compared with the exact solution of each problem and are found to be in good agreement with each other the numerical scheme is also compared with earlier work and shown to be accurate and efficient

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

a modified halpern type iteration algorithm for a family of hemi relatively nonexpansive mappings and systems of equilibrium problems in banach spaces in this paper we prove strong convergence theorems by the hybrid method for a family of hemi relatively nonexpansive mappings in a banach space our results improve and extend the corresponding results given by qin et al xiaolong qin yeol je cho shin min kang haiyun zhou convergence of a modified halpern type iteration algorithm for quasi f nonexpansive mappings appl math lett 22 2009 1051 1055 and at the same time our iteration algorithm is different from the kimura and takahashi algorithm which is a modified mann type iteration algorithm yasunori kimura wataru takahashi on a hybrid method for a family of relatively nonexpansive mappings in banach space j math anal appl 357 2009 356 363 in addition we succeed in applying our algorithm to systems of equilibrium problems which contain a family of equilibrium problems

a direct approach to studying soliton perturbations starting with an integrable nonlinear evolution equation the author investigates perturbations about a one soliton solution through the inversion of a linear equation for the first order correction to the soliton solution this inversion differs from past methods as the proposed method takes place in coordinate space not spectral space while it employs some of the tools of inverse scattering theory the method is applied to the korteweg de vries nonlinear schrodinger and sine gordon equations the first order corrections are then obtained

computational synergetics and mathematical innovation it is demonstrated how the computer used in a heuristic mode has greatly augmented our understanding of the mathematics of nonlinear dynamical process examples are given of recent work in soliton mathematics mainly but also nonlinear iterative mappings and low order nonlinear ordinary differential equation systems the role of good graphics in enhancing the discovery and retention of new mathematical properties of equations is illustrated several examples are given of the present frontier of research in two space dimensional nonlinear evolutionary problems

numerical simulation of focusing stochastic nonlinear schrodinger equations abstract in this paper we numerically investigate nonlinear schrodinger equations with a stochastic contribution which is of white noise type and acts either as a potential multiplicative noise or as a forcing term additive noise in the subcritical case we recover similar results as in the case of the korteweg de vries equation in the critical or supercritical case we observe that depending on its smoothness the noise may have different effects spatially smooth noises amplify blow up phenomena whereas delta correlated multiplicative noises prevent blow up formation note that in this latter case very few results are known both from a theoretical and a numerical point of view

strong convergence theorems for bregman totally quasi asymptotically nonexpansive mappings in reflexive banach spaces the purpose of this article is by using the shrinking projection method introduced by takahashi kubota and takeuchi to propose an iteration algorithm for bregman total quasi f asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of reflexive banach spaces as applications we apply our results to a system of equilibrium problems and zero point problem of maximal monotone mappings in reflexive banach spaces the results presented in the paper improve and extend the corresponding results of reich and sabach 2010 12 suantai et al 2012 13 nilsrakoo and saejung 2011 11 qin et al 2009 5 wang et al 2011 6 su et al 2010 7 martinez yanes and xu 2006 3 and others

the stochastic korteweg de vries equation on a bounded domain this paper considers the stochastic korteweg de vries equation on a bounded domain the existence of a weak martingale solution is discussed by the galerkin s approximation and compactness method based on this result we establish the unique continuation property for the stochastic korteweg de vries equation

strong convergence of the cq method for fixed point iteration processes strong convergence theorems are obtained for the cq method for an ishikawa iteration process a contractive type iteration process for nonexpansive mappings and the proximal point algorithm for maximal monotone operators in hilbert spaces

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

regular article finite difference schemes for u t x α δg δu that inherit energy conservation or dissipation property we propose a new procedure for designing by rote finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations such as the korteweg de vries kdv equation and the cahn hilliard equation the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly since the inherited properties are kept even if the time mesh size changes in the time evolution process we can use some appropriate time mesh adaptive methods to obtain numerical solutions through the derived schemes because of these properties the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions and sufficiently flexible to treat the inheritance of the energy conservation and dissipation properties are verified numerically for the kdv equation and the cahn hilliard equation

strong convergence theorems for relatively nonexpansive mappings in a banach space abstract in this paper we prove two strong convergence theorems of modified ishikawa iteration and modified halpern iteration for relatively nonexpansive mappings in a banach space our results extend and improve the recent ones announced by nakajo takahashi kim martinez yanes xu and some others

a conservative spectral method for several two dimensional nonlinear wave equations a conservative spectral method is proposed to solve several two dimensional nonlinear wave equations the conventional fast fourier transform is used to approximate the spatial derivatives and a three level difference scheme with a free parameter is to advance the solution in time our time discretization is semi implicit in the sense that the linear terms are treated implicitly while the nonlinear terms are evaluated only by previous time levels and thus treated explicitly however the cost of the algorithm is no greater than that of a fully explicit method because the linear boundary value problem that must be solved at each time step is almost trivial in a spectral spatial discretization a linear stability analysis shows that the method leads to a less restrictive stability condition than the corresponding explicit one the method is conservative and the ratio of the numerical dispersion to the physical dispersion is of the order o t2 applications of our method to the kadomtsev petviashvili and the zakharov kuznetsov equations exhibit excellent results

blow up for the stochastic nonlinear schrodinger equation with multiplicative noise we study the influence of a multiplicative gaussian noise white in time and correlated in space on the blow up phenomenon in the supercritical nonlinear schrodinger equation we prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability

strong convergence theorems by halpern mann iterations for relatively nonexpansive mappings in banach spaces abstract in this paper we modify halpern and mann s iterations for finding a fixed point of a relatively nonexpansive mapping in a banach space consequently a strong convergence theorem for a nonspreading mapping is deduced using a concept of duality theorems we also obtain analogue results for certain generalized nonexpansive and generalized nonexpansive type mappings finally we discuss two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a banach space

a finite difference method for the korteweg de vries and the kadomtsev petviashvili equations a linearized implicit finite difference method for the korteweg de vries equation is proposed and straightforwardly extended to the kadomtsev petviashvili equation we investigate the order of accuracy of the method and prove the method to be unconditionally linearly stable the numerical experiments for the korteweg de vries and the kadomtsev petviashvili equations are carried out with various conditions numerical results for the collision of two lump type solitary wave solutions to the kadomtsev petviashvili equation are also reported

fully discrete galerkin methods for the korteweg de vries equation abstract fully discrete schemes for the numerical simulation of solutions of the periodic initial value problem for the korteweg de vries equation are introduced implemented and tested of special interest are stable schemes featuring rates of convergence of order higher than two in both the spatial and temporal variable a careful analysis of the relative and absolute efficiency of these schemes is carried out and one of the schemes is applied to demonstrate that solutions of certain generalized korteweg de vries equations apparently may develop singularities in finite time

a differential quadrature algorithm for simulations of nonlinear schrodinger equation numerical simulations of nonlinear schrodinger equation are studied using differential quadrature method based on cosine expansion propogation of a soliton interaction of two solitons birth of standing and mobile solitons and bound state solutions are simulated the accuracy of the method dq is measured using maximum error norm the results are compared with some earlier works the lowest two conserved quantities are computed numerically for all cases

solitary wave solutions of the mkdv equation abstract a new numerical solution to the modified korteweg devries minus equation is obtained using a lumped galerkin method with quadratic b spline finite elements the motion interaction and generation of solitary waves are studied using the method new results concerning the decay of quasi soliton initial conditions are given

white noise driven korteweg de vries equation abstract we consider a stochastic korteweg de vries equation on the real line the noise is additive we use function spaces similar to those introduced by bourgain to prove well posedness results for the korteweg de vries equation in l 2 r we are able to handle a noise which is locally white in space and time more precisely it is a space time white noise multiplied by an l 2 function of the space variable due to the lack of a priori estimates we can only get a local existence result in time however we obtain the global existence of l 2 r solutions when the covariance operator of the noise is hilbert schmidt in l 2 r

numerical modeling of nonlinear acoustic waves in a tube connected with helmholtz resonators acoustic wave propagation in a one dimensional waveguide connected with helmholtz resonators is studied numerically finite amplitude waves and viscous boundary layers are considered the model consists of two coupled evolution equations a nonlinear pde describing nonlinear acoustic waves and a linear ode describing the oscillations in the helmholtz resonators the thermal and viscous losses in the tube and in the necks of the resonators are modeled by fractional derivatives a diffusive representation is followed the convolution kernels are replaced by a finite number of memory variables that satisfy local ordinary differential equations a splitting method is then applied to the evolution equations their propagative part is solved using a standard tvd scheme for hyperbolic equations whereas their diffusive part is solved exactly various strategies are examined to compute the coefficients of the diffusive representation finally an optimization method is preferred to the usual quadrature rules the numerical model is validated by comparisons with exact solutions the properties of the full nonlinear solutions are investigated numerically in particular the existence of acoustic solitary waves is confirmed

numerical studies of the stochastic korteweg de vries equation we present numerical solutions of the stochastic korteweg de vries equation for three cases corresponding to additive time dependent noise multiplicative space dependent noise and a combination of the two we employ polynomial chaos for discretization in random space and discontinuous galerkin and finite difference for discretization in physical space the accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and monte carlo simulation results of particular interest is the interplay of spatial discretization error with the stochastic approximation error which is examined for different orders of spatial and stochastic approximation

numerical solution of korteweg de vries equation by galerkin b spline finite element method a finite element solution of the kdv equation is presented to demonstrate the efficiency of the method two test problems are considered the numerical solutions of the kdv equation are compared with both the exact solutions and other numerical solutions in the literature the numerical solutions are found to be in good agreement with the exact solutions

analytical and numerical aspects of certain nonlinear evolution equations 1v numerical modified korteweg de vries equation various numerical methods are used in order to approximate the korteweg de vries equation namely i zabusky kruskal scheme ii hopscotch method iii a scheme due to goda iv a proposed local scheme v a proposed global scheme vi a scheme suggested by kruskal vii split step fourier method by tappert viii an improved split step fourier method and ix pseudospectral method by fornberg and whitham comparisons between our proposed scheme which is developed using notions of the inverse scattering transform and the other utilized schemes are obtained

numerical solutions for fractional kdv burgers equation by adomian decomposition method abstract based on the symbolic computation system maple adomian decomposition method developed for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional kdv burgers equation the fractional derivatives are described in the caputo sense according to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional kdv burgers equation with time and space fractional derivatives finally the solutions of our model equation are calculated in the form of convergent series with easily computable components

asymptotic transmission of solitons through random media this paper contains a study of the transmission of a soliton through a slab of nonlinear and random media a random nonlinear schrodinger equation is considered where the randomness holds in the potential and the nonlinear coefficient using the inverse scattering transform we exhibit several asymptotic behaviors corresponding to the limit when the amplitudes of the random fluctuations go to zero and the size of the slab goes to infinity the mass of the transmitted soliton may tend to zero exponentially as a function of the size of the slab or following a power law or else the soliton may keep its mass while its velocity decreases at a logarithmic rate or even more slowly numerical simulations are in good agreement with the theoretical results

numerical methods for the generalized zakharov system we present two numerical methods for the approximation of the generalized zakharov system zs the first one is the time splitting spectral tssp method which is explicit time reversible and time transverse in variant if the generalized zs is keeps the same decay rate of the wave energy as that in the generalized zs gives exact results for the plane wave solution and is of spectral order accuracy in space and second order accuracy in time the second one is to use a local spectral method the discrete singular convolution dsc for spatial derivatives and the fourth order runge kutta rk4 for time integration which is of high the same as spectral order accuracy in space and can be applied to deal with general boundary conditions in order to test accuracy and stability we compare these two methods with other existing methods fourier pseudospectral method fps and wavelet galerkin method wg for spatial derivatives combining with the rk4 for time integration as well as the standard finite difference method fd for solving the zs with a solitary wave solution furthermore extensive numerical tests are presented for plane waves solitary wave collisions in 1d as well as a 2d problem of the generalized zs numerical results show that tssp and dsc are spectralorder accuracy in space and much more accurate than fd and for stability tssp requires k o h dsc rk4 requires k o h2 for fixed acoustic speed where k is the time step and h is the spatial mesh size

a new hybrid method for solving a generalized equilibrium problem solving a variational inequality problem and obtaining common fixed points in banach spaces with applications abstract the purpose of this paper is to prove by using a new hybrid method a strong convergence theorem for finding a common element of the set of solutions for a generalized equilibrium problem the set of solutions for a variational inequality problem and the set of common fixed points for a pair of relatively nonexpansive mappings in a banach space as applications we utilize our results to obtain some new results for finding a solution of an equilibrium problem a fixed point problem and a common zero point problem for maximal monotone mappings in banach spaces

global properties of delayed hiv dynamics models with differential drug efficacy in cocirculating target cells the objective of this work is to investigate the qualitative behaviors of three hiv dynamical models with two types of cocirculating target cells and intracellular discrete time delay in the two types of target cells the drug efficacy is assumed to be different the models take into account both short lived infected cells and long lived chronically infected cells the incidence rate of infection is given by bilinear and saturation functional response in the first and second model respectively while it is given by a general function in the third model lyapunov functionals are constructed and lasalle s invariance principle is applied to prove the global asymptotic stability of all equilibria of the models in the first two models we have derived the basic reproduction number r0 it has been proven that the disease free equilibrium is globally asymptotically stable gas when r0 1 and the endemic equilibrium is gas when r0 1 for the third model we have derived r0 and established a set of sufficient conditions for global stability of both disease free and endemic equilibria of the model at the end theoretical results have been checked by numerical simulations