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

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

strong convergence theorem by a hybrid method for nonexpansive mappings and lipschitz continuous monotone mappings in this paper we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone lipschitz continuous mapping the iterative process is based on two well known methods hybrid and extragradient we obtain a strong convergence theorem for three sequences generated by this process based on this result we also construct an iterative process for finding a common fixed point of two mappings such that one of these mappings is nonexpansive and the other is taken from the more general class of lipschitz pseudocontractive mappings

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

spectral splitting method for nonlinear schrodinger equations with singular potential we consider the time dependent one dimensional nonlinear schrodinger equation with pointwise singular potential by means of spectral splitting methods we prove that the evolution operator is approximated by the lie evolution operator where the kernel of the lie evolution operator is explicitly written this result yields a numerical procedure which is much less computationally expensive than multi grid methods previously used furthermore we apply the lie approximation in order to make some numerical experiments concerning the splitting of a soliton interaction among solitons and blow up phenomenon

approximate proximal algorithms for generalized variational inequalities with pseudomonotone multifunctions the purpose of this paper is to investigate the convergence of general approximate proximal algorithm resp general bregman function based approximate proximal algorithm for solving the generalized variational inequality problem for short gvi t w where t is a multifunction the general approximate proximal algorithm resp general bregman function based approximate proximal algorithm is to define new approximating subproblems on the domains w email protected email protected n 1 2 which form a general approximate proximate point scheme resp a general bregman function based approximate proximate point scheme for solving gvi t w it is shown that if t is either relaxed a pseudomonotone or pseudomonotone then the general approximate proximal point scheme resp general bregman function based approximate proximal point scheme generates a sequence which converges weakly to a solution of gvi t w under quite mild conditions

superposition coding for side information channels we present simple practical codes designed for the binary and gaussian dirty paper channels we show that the dirty paper decoding problem can be transformed into an equivalent multiple access decoding problem for which we apply superposition coding our concept is a generalization of the nested lattices approach of zamir shamai and erez in a theoretical setting our constructions are capable of achieving capacity using random component codes and maximum likelihood decoding we also present practical implementations of the constructions and simulation results for both dirty paper channels our results for the gaussian dirty paper channel are on par with the best known results for nested lattices we discuss the binary dirty tape channel for which we present a simple effective coding technique finally we propose a framework for extending our approach to general gel fand pinsker channels

capacity and lattice strategies for canceling known interference we consider the generalized dirty paper channel y x s n e x sup 2 spl les p sub x where n is not necessarily gaussian and the interference s is known causally or noncausally to the transmitter we derive worst case capacity formulas and strategies for strong or arbitrarily varying interference in the causal side information si case we develop a capacity formula based on minimum noise entropy strategies we then show that strategies associated with entropy constrained quantizers provide lower and upper bounds on the capacity at high signal to noise ratio snr conditions i e if n is weak relative to the power constraint p sub x these bounds coincide the optimum strategies take the form of scalar lattice quantizers and the capacity loss due to not having s at the receiver is shown to be exactly the shaping gain 1 2log 2 spl pi e 12 spl ap 0 254 bit we extend the schemes to obtain achievable rates at any snr and to noncausal si by incorporating minimum mean squared error mmse scaling and by using k dimensional lattices for gaussian n the capacity loss of this scheme is upper bounded by 1 2log2 spl pi eg spl lambda where g spl lambda is the normalized second moment of the lattice with a proper choice of lattice the loss goes to zero as the dimension k goes to infinity in agreement with the results of costa these results provide an information theoretic framework for the study of common communication problems such as precoding for intersymbol interference isi channels and broadcast channels

linearized difference schemes for a bbm equation with a fractional nonlocal viscous term this paper is concerned with the efficient finite difference schemes for a benjamin bona mahony equation with a fractional nonlocal viscous term by using the weighted shift grunwald letnikov and the fractional centered difference formulae to approximate the nonlocal fractional operators we design a class of linearized finite difference schemes for the presented nonlocal model the existence stability and convergence of the proposed numerical schemes are rigorously derived with the help of functional analysis theoretical analysis shows that the proposed numerical schemes are stable with second order accuracy numerical examples are presented to verify our theoretical analysis and to demonstrate the efficiency of the proposed numerical schemes

order estimates in time of splitting methods for the nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation u t i delta u f u 0 in two dimensions we show by an operator theoretic proof that the well known lie and strang formulae which are splitting methods are approximations of the exact solution of order 1 and 2 in time

analysis of splitting methods for reaction diffusion problems using stochastic calculus we consider linear and nonlinear reaction diffusion problems and their time dis cretization by splitting methods we give probabilistic interpretations of the splitting schemes and show how these representations allow to give error bounds for the deter ministic propagator under weak hypothesis on the reaction part to show these results we only use the it o formula and basic properties of solutions of stochastic differential equations eventually we show how probabilistic representations of splitting schemes can be used to derive hybrid numerical schemes based on monte carlo approxima tions of the splitting method itself

finite element method for a space fractional anti diffusive equation the numerical solution of a nonlinear and space fractional anti diffusive equation used to model dune morphodynamics is considered spatial discretization is effected using a finite element method whereas the crank nicolson scheme is used for temporal discretization the fully discrete scheme is analyzed to determine stability condition and also to obtain error estimates for the approximate solution numerical examples are presented to illustrate convergence results

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

a close to capacity dirty paper coding scheme the writing on dirty paper channel model offers an information theoretic framework for precoding techniques for canceling arbitrary interference known at the transmitter it indicates that lossless precoding is theoretically possible at any signal to noise ratio snr and thus dirty paper coding may serve as a basic building block in both single user and multiuser communication systems we design an end to end coding realization of a system materializing a significant portion of the promised gains we employ multidimensional quantization based on trellis shaping at the transmitter coset decoding is implemented at the receiver using virtual bits combined with iterative decoding of capacity approaching codes we achieve an improvement of 2db over the best scalar quantization scheme code design is done using the exit chart technique

an outer approximation method for the variational inequality problem we study two outer approximation schemes applied to the variational inequality problem in reflexive banach spaces first we propose a generic outer approximation scheme and its convergence analysis unifies a wide class of outer approximation methods applied to the constrained optimization problem as is standard in this setting boundedness and optimality of weak limit points are proved to hold under two alternative conditions i boundedness of the feasible set or ii coerciveness of the operator to develop a convergence analysis where i and ii do not hold we consider a second scheme in which the approximated subproblems use a coercive approximation of the original operator under conditions alternative to both i and ii we obtain standard convergence results furthermore when the space is uniformly convex we establish full strong convergence of the second scheme to a solution

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in this paper we show strong convergence theorems for nonexpansive mappings and nonexpansive semigroups in hilbert spaces by the hybrid method in the mathematical programming

stability of operator splitting methods for systems with indefinite operators advection diffusion reaction systems this brief paper presents an a stability result for operator splitting type time integration methods applied to advection diffusion reaction equations with possibly indefinite source terms these results extend our earlier work on diffusion reaction systems d l ropp j n shadid stability of operator splitting methods for systems with indefinite operators reaction diffusion systems j comput phys 203 2 2005 449 466 the a stability result presents sufficient conditions that control both low and high wave number instabilities a corollary shows that if l stable methods are used for the diffusion term the high wave number instability will be controlled more easily numerical results are presented that verify second order convergence for the operator splitting methods and demonstrate control of instabilities on a chemotaxis problem by use of an l stable diffusion integrator

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

optimal modulation for known interference we present a symbol by symbol approach to the problem of canceling known interference at the transmitter in a communication system in the envisioned system the modulator maps an information symbol taken from a finite alphabet and an interference symbol from the complex field onto a transmitted constellation point our scheme is based on joint optimization of a modulator and demodulator subject to a constraint on the average transmit power the demodulator picks the information symbol as a function of the received symbol that minimizes the average error probability we emphasize that our focus is on transmission in a single complex dimension and hence the proposed technique is a modulation rather than a coding scheme we illustrate that the new scheme outperforms tomlinson harashima precoding which is a classical but suboptimal solution to the one dimensional known interference precoding problem in our simulations the new approach is able to perform close to the no interference bound

implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and lipschitz continuous monotone mappings in this paper we introduce some implicit iterative algorithms for finding a common element of the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone lipschitz continuous mapping these implicit iterative algorithms are based on two well known methods extragradient and approximate proximal methods we obtain some weak convergence theorems for these implicit iterative algorithms based on these theorems we also construct some implicit iterative processes for finding a common fixed point of two mappings such that one of these two mappings is taken from the more general class of lipschitz pseudocontractive mappings and the other mapping is asymptotically nonexpansive

strong convergence of modified mann iterations for asymptotically nonexpansive mappings and semigroups abstract the mann iterations for nonexpansive mappings have in general only weak convergence in a hilbert space we modify an iterative method of mann s type introduced by nakajo and takahashi strong convergence theorems for nonexpansive mappings and nonexpansive semigroups j math anal appl 279 2003 372 379 for nonexpansive mappings and prove strong convergence of our modified mann s iteration processes for asymptotically nonexpansive mappings and semigroups

discontinuous galerkin method for fractional convection diffusion equations we propose a discontinuous galerkin method for fractional convection diffusion equations with a superdiffusion operator of order alpha 1 alpha 2 defined through the fractional laplacian the fractional operator of order alpha is expressed as a composite of first order derivatives and a fractional integral of order 2 alpha the fractional convection diffusion problem is expressed as a system of low order differential integral equations and a local discontinuous galerkin method scheme is proposed for the equations we prove stability and optimal order of convergence cal o h k 1 for the fractional diffusion problem and an order of convergence of cal o h k frac 1 2 is established for the general fractional convection diffusion problem the analysis is confirmed by numerical examples

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

operator splitting for the kdv equation we provide a new analytical approach to operator splitting for equations of the type u t au b u where a is a linear operator and b is quadratic a particular example is the korteweg de vries kdv equation u t uu x u xxx 0 we show that the godunov and strang splitting methods converge with the expected rates if the initial data are sufficiently regular

asymptotically strict pseudocontractive mappings in the intermediate sense abstract it is proved that the modified mann iteration process x n 1 1 n x n n t n x n n n where n is a sequence in 0 1 with n 1 for some 0 1 converges weakly to a fixed point of an asymptotically strict pseudocontractive mapping t in the intermediate sense which is not necessarily lipschitzian we also develop cq method for this modified mann iteration process which generates a strongly convergent sequence