an extrapolation method with step size control for nonlinear volterra integral equations in 6 it has been shown that the midpoint rule applied to second kind volterra integral equations possesses an asymptotic expansion in even powers of the stepsizeh in this paper we describe an extrapolation method based on the midpoint rule together with a mechanism of step size control

boundary stabilization of linear elastodynamic systems it is proved that an elastic medium which occupies a bounded region in three dimensional euclidean space can be uniformly stabilized by means of traction forces applied on a portion of the boundary

well posedness of systems of linear elasticity with dirichlet boundary control and observation an open loop system of linear elasticity with dirichlet boundary control and collocated observation is considered the main result we obtained states that this system is well posed in the sense of salamon the result deduces the exponential stability of the closed loop system under proportional output feedback hence it answers positively an open question proposed by liu and krstic ima j appl math 65 2000 pp 109 121 moreover the well posedness together with the regularity of the open loop system in the sense of weiss that was obtained in our separate paper s g chai and b z guo feedthrough operator for linear elasticity system with boundary control and observation preprint school of computational and applied mathematics university of the witwatersrand south africa 2008 makes this infinite dimensional system parallel in many ways to a linear finite dimensional system in the framework of well posed and regular linear infinite dimensional systems

decay rates for the three dimensional linear system of thermoelasticity we consider the two and three dimensional system of linear thermoelasticity in a bounded smooth domain with dirichlet boundary conditions we analyze whether the energy of solutions decays exponentially uniformly to zero as t to infty first of all by a decoupling method we reduce the problem to an observability inequality for the lame system in linear elasticity and more precisely to whether the total energy of the solutions can be estimated in terms of the energy concentrated on its longitudinal component we show that when the domain is convex the decay rate is never uniform in fact the lack of uniform decay holds in a more general class of domains in which there exist rays of geometric optics of arbitrarily large length that are always reflected perpendicularly or almost tangentially on the boundary we also show that in three space dimensions the lack of uniform decay may also be due to a critical polarization of the energy on the transversal component of the displacement in two space dimensions we prove a sufficient and almost necessary condition for the uniform decay to hold in terms of the propagation of the transversal characteristic rays under the further assumption that the boundary of the domain does not have contacts of infinite order with its tangents we also give an example due to d hulin in which these geometric properties hold in three space dimensions we indicate without proof how a careful analysis of the polarization of singularities may lead to sharp sufficient conditions for the uniform decay to hold in two space dimensions we prove that smooth solutions decay polynomially in the energy space to a finite dimensional subspace of solutions except when the domain is a ball or an annulus finally we discuss some closely related controllability and spectral issues

feedthrough operator for linear elasticity system with boundary control and observation we are concerned with the feedthrough operator for the open loop system of linear elasticity with dirichlet boundary control and collocated observation this system has been known to be regular in the sense of g weiss in this paper the analytic expression of the corresponding feedthrough operator is presented by means of differential geometry and fourier transform methods

h 1 scattering for systems of n defocusing weakly coupled nls equations in low space dimensions abstract we prove that the scattering operators and wave operators are well defined in the energy space for the system of defocusing schrodinger equations i t u u 1 n u p 1 u p 1 u 0 1 n u 0 1 n u 0 1 n h 1 r d n with n 2 0 0 for p 2 if d 1 p 1 if d 2 and 1 p 2 if d 3

resolvents and solutions of singular volterra integral equations with separable kernels the volterra integral equation 1 x t a t 0 tb t s x s dswith a kernel of the form b t s p t q s is investigated where a p and q are functions that are defined a e on an interval 0 t and are measurable the main result of this paper states that if qa is lebesgue integrable on 0 t the sign of b t s does not change for almost all t s and if there is a function f that is continuous on 0 t except possibly at countably many points with b t t f t a e on 0 t then the function x defined by 2 x t a t 0 tr t s a s ds where 3 r t s b t s e s t b u u d u solves 1 a e on 0 t three diverse examples illustrate the efficacy of using 2 and 3 to calculate solutions of 1 two of the examples involve singular kernels the solution of one of them is nowhere continuous on 0 t

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

infinitely many solutions to quasilinear schrodinger system in r n in this work we study the existence of multiple solutions to the quasilinear schrodinger system 0 1 p u a x u p 2 u d 1 f u x u v u m 2 u x r n q v b x v q 2 v d 1 f v x u v v m 2 v x r n u w 1 p r n v w 1 q r n where n 3 1 p q n 0 and m d q p r the potential functions a x b x l r n are positive in r n a major point is that we use the technique in chen 2015 to verify the p s conditions and then apply a version of mountain pass lemma to prove the existence of infinitely many solutions of system 0 1

numerical methods for a volterra integral equation with non smooth solutions we consider the numerical treatment of a singular volterra integral equation with an infinite set of solutions one of which is smooth and all others have infinite gradient at the origin this equation has been the subject of previous works where we have dealt with the approximation of the smooth solution here we present numerical methods which enable us to obtain approximations to any of the infinite class of solutions some numerical examples are given which illustrate the performance of the methods employed

on a graded mesh method for a class of weakly singular volterra integral equations in this paper a class of weakly singular volterra integral equations with an infinite set of solutions is investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solution of this class of equations has been a difficult topic to analyze and has received much previous investigation the aim of this paper is to improve the convergence rates by a graded mesh method the convergence rates are proved and a variety of numerical examples are provided to support the theoretical findings

rigorous convergence analysis of jacobi spectral galerkin methods for volterra integral equations with noncompact kernels abstract this article deals with the implementation of jacobi spectral and pseudospectral galerkin approaches for solving weakly singular volterra integral equations with noncompact kernels the computational domain of the main problem and intervals of the involved integrals are transformed into 1 1 by some appropriate change of variables therefore one can apply the jacobi galerkin method and impose the inner product conditions on the residual of the considered equation also jacobi gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the jacobi weight function in the pseudospectral galerkin method for both cases of spectral and pseudospectral galerkin schemes rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in l and weighted l 2 norms several numerical examples are also given to confirm the theoretical prediction

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

variable transformations in the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels extensions to fredholm integral equations abstract a number of techniques that use variable transformations in numerical integration have been developed recently cf sidi numerical integration iv h brass g hammerlin eds birkhauser basel 1993 pp 359 373 laurie j comput appl math 66 1996 337 344 the use of these transformations resulted in increasing the order of convergence of the trapezoidal and the midpoint quadrature rule in this paper the application of variable transformation techniques of sidi and laurie type to the numerical solution of second kind volterra integral equations with continuous and weakly singular kernels is considered since the transformations are such that the end points of integration need not be used as mesh points the methods introduced can be used for vie with both continuous and weakly singular kernel in a uniform way the methods have also the advantages of simplicity of application and of achieving high order of convergence the application of the idea to fredholm integral equations with continuous and weakly singular equations is also considered numerical results are included and they verify the expected increased order of convergence they were obtained by using the trapezoidal formula for the evaluation of the transformed integrals

spectral collocation methods for volterra integro differential equations with noncompact kernels the aim of this paper is to analyze spectral collocation methods for a class of volterra integro differential equations with noncompact kernels the spectral collocation method is proposed to solve the equations after the results on existence uniqueness and regularity of the solutions are established the convergence rates of the methods are proven and a numerical example is provided to verify the theoretical findings

solution of a class of volterra integral equations with singular and weakly singular kernels a class of volterra integral equations are solved in this paper where the involved kernel may be weakly singular singular and hypersingular for various cases existence and uniqueness of such volterra integral equations are established furthermore by reducing such equations to ordinary differential equations analytic solutions have been determined explicitly when restricting the solutions to certain special function classes the number of suitable solutions is given

the exact solution of a class of volterra integral equation with weakly singular kernel abstract in this paper the weakly singular volterra integral equations with an infinite set of solutions are investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation the aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory applying weighted integral we provide a new definition for reproducing kernel space and obtain reproducing kernel function using the good properties of reproducing kernel function the only smooth solution is exactly expressed in the form of series the n term approximate solution is obtained by truncating the series meanwhile we prove that the derivative of approximation converges to the derivative of exact solution uniformly the final numerical examples compared with other methods show that the method is efficient

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

extrapolation method of iterated collocation solution for two dimensional nonlinear volterra integral equations in this paper we study the numerical solution of two dimensional nonlinear volterra integral equation by collocation and iterated collocation method asymptotic error expansion of iterated collocation solution is obtained using richardson s extrapolation we show that the accuracy of numerical solution in the iterated collocation method can be improved greatly some numerical examples are given to illustrate this theory

special set and solutions of coupled nonlinear schrodinger equations a special set of n coupled nonlinear schrodinger equations which consists of 2n interaction types each of which is characterized by a specific array of interaction parameters is presented it is shown that every lame function of order n n is a solution for one or more components for one or more interaction types of this special set simple rules that relate interaction and solution types and interesting features of these relationships are presented

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

well posedness of boundary control systems continuity of the input output map for boundary control systems is shown through the system transfer function our approach transforms the question of continuity of the input output map of a boundary control system to uniform boundedness of the solution to a related elliptic problem this is shown for a class of boundary control systems with dirichlet neumann or robin boundary control

well posedness and regularity of naghdi s shell equation under boundary control and observation abstract a system of naghdi s shell equation with dirichlet boundary control and collocated observation is considered results on the associated nonhomogeneous boundary value problem are presented based on these results it is shown that the system is well posed in the sense of d salamon and regular in the sense of g weiss the expression of the corresponding feedthrough operator is explicitly found by means of riemannian geometric method and partial fourier transform these properties make this partial differential control system parallel in many ways finite dimensional ones in the general framework of well posed and regular infinite dimensional systems

numerical analysis of a singular integral equation in this paper we give details of the numerical analysis of an integral equation of the form 1 y t f t email protected 0ts m 1t my s email protected 0 the distinctive feature of the equation is the presence of a singularity at t 0 for all values of m 0 and at s 0 for all values of t 0 for 0 m 1 this means that conventional analytical and numerical theory does not apply in fact for 0 m 1 the equation has an infinite family of solutions as background we give details of the analytical results on existence and uniqueness of solutions to 1 and we give new results on the use of numerical schemes that will yield approximations to any specific solution we conclude with some numerical results that show that our methods enable us to find approximations of any chosen order to any of the infinite class of exact solutions to 1

volterra integral equations and a new gronwall inequality part ii the nonlinear case we consider nonlinear singular volterra integral equations of the second kind we generalise the transformation method introduced in part i of this paper 6 to cope with both the nonlinearity and slightly more general singular kernels we also consider a particular class of nonlinear equation for which the solution behaviour is known using this a priori knowledge we propose a modification of the transformation technique which results in a numerical method with good asymptotic stability properties applying the general theory of part i of this paper we prove convergence of this scheme

implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity abstract uniform exponential decay of solution is established for the elastodynamic system of elasticity using boundary feedback control energy dissipation is introduced via linear velocity feedbacks acting through a portion of the boundary as traction forces two primary goals are achieved first these results are proven without the imposition of strong geometric assumptions on the controlled portion of the boundary thus extending earlier work which required that the domain be star shaped second the feedback is only a function of velocity as opposed to also containing the tangential derivative of the displacement resulting in a physically viable feedback proof is based on the multiplier method and relies critically on sharp trace estimates for the tangential derivative of the displacement on the boundary as well as on unique continuation results for the corresponding static system

existence and uniqueness of solutions of abel integral equations with power law non linearities abstract in this article a class of nonlinear abel type integral equations is studied these equations have at least the trivial solution but we are interested in a continuous positive solution we show that there exists a unique positive solution in an order interval of a cone in a banach space

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

convergence analysis of spectral collocation methods for a class of weakly singular volterra integral equations abstract in this paper we discuss the application of spectral jacobi collocation methods to a certain class of weakly singular volterra integral equations first we use some function transformations and variable changes to transform the equation into a volterra integral equation defined on the standard interval 1 1 then the jacobi gauss quadrature formula is used to approximate the integral operator for the spectral jacobi collocation method a rigorous error analysis in both the l and weighted l 2 norms is given under the assumption that both the kernel function and the source function are sufficiently smooth finally some numerical examples are provided to illustrate the theoretical results

boundary observability controllability and stabilization of linear elastodynamic systems in 1988 lions obtained observability and exact controllability results for linear homogeneous isotropic elastodynamic systems siam rev 30 1988 pp 1 68 applying some new identities we extend his theorems to nonisotropic systems in 1991 lagnese obtained uniform stabilizability results for two dimensional linear homogeneous isotropic systems by applying a somewhat artificial feedback nonlinear anal 16 1991 pp 35 54 then he asked whether analogous results hold for a natural and physically implementable boundary feedback using some new identities and applying a method introduced in 1987 by zuazua and the second author j math pures appl 69 1990 pp 33 54 we give an affirmative answer to this question in all dimensions and also for nonisotropic systems moreover we obtain good decay estimates finally applying a recent general method of uniform stabilization we construct boundary feedbacks leading to arbitrarily large energy decay rates

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

positive solutions to some systems of coupled nonlinear schrodinger equations abstract we study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time independent schrodinger equations u j j u j j u j 3 k 1 k j n j k u j u k 2 u j w 1 2 r n j 1 n where n 1 2 3 j j 0 for j 1 n the coupling parameters j k k j r for j k 1 n j k precisely we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors additionally with more restrictive conditions on the coupled parameters we show that the bound states founded are positive

numerical solutions of weakly singular volterra integral equations using the optimal homotopy asymptotic method the purpose of this paper is to apply a numerical technique namely the optimal homotopy asymptotic method oham for finding the approximate solutions of a class of volterra integral equations with weakly singular kernels this method uses simple computations with quite acceptable approximate solutions which has close agreement with exact solutions illustrative examples are included to demonstrate the validity and applicability of the present method and a comparison has been made with existing results

existence and nonexistence of bound state solutions for schrodinger systems with linear and nonlinear couplings abstract we study the schrodinger systems with linear and nonlinear coupling terms doubly coupled nonlinear schrodinger system for short which arise naturally in nonlinear optics and in the hartree fock theory for bose einstein condensates among other physical problems first for small linear coupling constant we get existence of a nontrivial bound state solution to the system via perturbation method furthermore we prove each component of the bound state solution is nonnegative by energy estimate second we establish a version of pohozaev nehari identity and prove a nonexistence result for the more general system when the spatial dimension n 4

well posedness of a structural acoustics control model with point observation of the pressure abstract we consider a controlled and observed partial differential equation pde which describes a structural acoustics interaction physically this pde describes an acoustic chamber with a flexible chamber wall the control is applied to this flexible wall and the class of controls under consideration includes those generated by piezoceramic patches the observation we consider is point measurements of acoustic pressure inside the cavity mathematically the model consists of a wave equation coupled through boundary trace terms to a structurally damped plate or beam equation and the point controls and observations for this system are modeled by highly unbounded operators we analyze the map from the control to the observation since the properties of this map are central to any control design which is based upon this observation we also show there exists an appropriate state space x so that if the initial state is in x and the control is in l2 then the state evolves continuously in x and the observation is in l2 the analysis of this system entails a microlocal analysis of the wave component of the system and the use of pseudodifferential machinery

semitrivial vs fully nontrivial ground states in cooperative cubic schrodinger systems with d 3 equations abstract in this work we consider the weakly coupled schrodinger cubic system u i i u i i u i 3 u i j i b i j u j 2 u i h 1 r n r i 1 d where 1 n 3 i i 0 and b i j b j i 0 for i j this system admits semitrivial solutions that is solutions u u 1 u d with null components we provide optimal qualitative conditions on the parameters i i and b i j under which the ground state solutions have all components nontrivial or conversely are semitrivial this question had been clarified only in the d 2 equations case for d 3 equations prior to the present paper only very restrictive results were known namely when the above system was a small perturbation of the super symmetrical case i and b i j b we treat the general case uncovering in particular a much more complex and richer structure with respect to the d 2 case

uniform exponential energy decay of wave equations in a bounded region with l2 0 l2 γ feedback control in the dirichlet boundary conditions etant donne un ouvert borne r n n 2 on considere l equation d onde w tt w dans o avec des conditions initiales w o x w o x l 2 w 1 o x w 1 x h 1 et des conditions aux limites non homogenes de type dirichlet w t u t sur o etant la frontiere de

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

symmetry properties in systems of semilinear elliptic equations abstract we investigate symmetry properties of solutions of systems of semilinear elliptic equations the two main tools we use consist of the maximum principle and the device of moving parallel planes to a critical position and then showing that the solution is symmetric about the limiting plane an application to the belousov zhabotinskii chemical reaction is given

solitary and self similar solutions of two component system of nonlinear schrodinger equations conventionally to learn wave collapse and optical turbulence one must study finite time blow up solutions of one component self focusing nonlinear schrodinger equations nlse here we consider simultaneous blow up solutions of two component systems of self focusing nlse by studying the associated self similar solutions we prove two components of solutions blow up at the same time these self similar solutions may come from solitary wave solutions with multi bumps forming abundant geometric patterns which cannot be found in one component self focusing nlse our results may provide the first step to investigate optical turbulence in two component systems of nlse

numerical modelling of qualitative behaviour of solutions to convolution integral equations we consider the qualitative behaviour of solutions to linear integral equations of the form 1 y t g t email protected 0 tk t s y s ds where the kernel k is assumed to be either integrable or of exponential type after a brief review of the well known paley wiener theory we give conditions that guarantee that exact and approximate solutions of 1 are of a specific exponential type as an example we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular volterra equation with infinitely many solutions we show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour

symmetric ground state solutions of m coupled nonlinear schrodinger equations abstract we prove the existence of radial and radially decreasing ground states of an m coupled nonlinear schrodinger equation with a general nonlinearity

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

collocation methods for second kind volterra integral equations with weakly singular kernels in this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of volterra integral equations vies are smooth if the exact solutions are not smooth however suitable transformations can be made so that the new vies possess smooth solutions spline collocation methods with uniform meshes applied to these new vies are then shown to be able to yield optimal global convergence rates the general theory is applied to a typical case i e the integral kernels consisting of the singular term t s

a nystrom interpolant for some weakly singular nonlinear volterra integral equations we consider a second kind weakly singular nonlinear volterra hammerstein integral equation defined by a compact operator and derive a nystrom type interpolant of the solution based on gauss radau nodes we prove the convergence of the interpolant and derive convergence estimates for equations with nonlinearity of algebraic kind we improve the rate of convergence by using a smoothing transformation some numerical examples are given

multiplicity of localized solutions of nonlinear schrodinger systems for infinite attractive case abstract in this paper we consider the following nonlinear schrodinger system 2 u v 1 x u u v 2 in r n u h 1 r n 2 v v 2 x v u 2 v in r n v h 1 r n where n 3 0 1 v i c 1 r n r 0 inf x r n v i x sup x r n v i x i 1 2 and there exists a bounded open set r n with smooth boundary such that inf x min v 1 x v 2 x 0 we prove the existence of a given number of semi positive solutions localized in here we say u v is semi positive if either u 0 or v 0