weighted estimates for the helmholtz equation and some applications we characterize the radial functionsvin rnfor which the a priori inequality u1 2 2 k uv1 2 2 c v 1 2 k2 u 2holds with constant independent ofk the condition is forvto have the x rays transform everywhere bounded we apply these estimates to the well posedness of evolution schrodinger equations with time dependent drift terms and to the restriction of the fourier transform to euclidean spheres

a boundary value problem for the minimum time function a natural boundary value problem for the dynamic programming partial differential equation associated with the minimum time problem is proposed the minimum time function is shown to be the unique viscosity solution of this boundary value problem

constructing and predicting solitary pattern solutions for nonlinear time fractional dispersive partial differential equations building fractional mathematical models for specific phenomena and developing numerical or analytical solutions for these fractional mathematical models are crucial issues in mathematics physics and engineering in this work a new analytical technique for constructing and predicting solitary pattern solutions of time fractional dispersive partial differential equations is proposed based on the generalized taylor series formula and residual error function the new approach provides solutions in the form of a rapidly convergent series with easily computable components using symbolic computation software for method evaluation and validation the proposed technique was applied to three different models and compared with some of the well known methods the resultant simulations clearly demonstrate the superiority and potentiality of the proposed technique in terms of the quality performance and accuracy of substructure preservation in the construct as well as the prediction of solitary pattern solutions for time fractional dispersive partial differential equations

numerical algorithm for solving two point second order periodic boundary value problems for mixed integro differential equations in this study the numerical solution of fredholm volterra integro differential equations for two point second order periodic boundary value problems is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the periodic boundary conditions of the problem are satisfied the exact solution u x is represented in the form of series in the space w 2 3 in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 3 some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear equations

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

using reproducing kernel for solving a class of fractional partial differential equation with non classical conditions today most of the real physical world problems can be best modeled with fractional differential equations besides modeling the solution techniques and their reliabilities are most important therefore high accuracy solutions are always needed in this paper a new method is provided to solve fractional partial differential equation in a very favorable reproducing kernel space it s reproducing kernel function is discussed in detail from the examples considered here it can be easily seen that our method has small computational work fast convergence speed and high precision

viscosity solutions for hamilton jacobi equations with unbounded nonlinear term a simplified approach abstract in a previous paper the author defined a notion for hamilton jacobi equations with an unbounded nonlinear term ax du and for such solutions proved existence and uniqueness results here we define an equivalent notion of solution for such equations which in addition has the advantage of being simpler and more intuitive

reconstruction of singularities from full scattering data by new estimates of bilinear fourier multipliers we prove that the singularities of a complex valued potential q in the schrodinger hamiltonian q can be reconstructed from the linear born approximation for full scattering data by averaging in the extra variables we prove that with this procedure the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data in particular for q in w alpha 2 for 0 in 2d we recover the main singularity of q with an accuracy of one derivative in 3d the accuracy is epsilon 1 2 increasing with this gives a mathematical basis for diffraction tomography the proof is based on some new estimates for multidimensional bilinear fourier multipliers of independent interest

an adaptive numerical approach for the solutions of fractional advection diffusion and dispersion equations in singular case under riesz s derivative operator abstract the fractional diffusion and dispersion equations are reinterpreted in determining the effect of fluid flow and displacement processes through certain compressible phenomena and then reconstructed by considering the flow conductivity energy balance flow chambers with the interconnected pores and diffusion flow system the adaptive reproducing kernel approach is formulated and analyzed to investigate numerical solutions of fractional advection diffusion and dispersion equations in singular case on a finite domain with riesz s fractional derivative in such alternative representation the reproducing kernel functions are obtained to provide analytic and approximate solutions in desired hilbert spaces to enable the utilized approach more convergent analysis and error estimates are also given to assure our results some features with numerical experiments are presented to confirm the theoretical analysis and to illustrate the performance and effectiveness of the proposed scheme graphical and comparisons indicate the significant improvement of the algorithm in solving many singular fractional problems arising in physical issues

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

approximate analytical solution of the nonlinear fractional kdv burgers equation in this paper explicit and approximate solutions of the nonlinear fractional kdv burgers equation with time space fractional derivatives are presented and discussed the solutions of our equation are calculated in the form of rabidly convergent series with easily computable components the utilized method is a numerical technique based on the generalized taylor series formula which constructs an analytical solution in the form of a convergent series five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution the results reveal that the method is very effective and simple in determination of solution of the fractional kdv burgers equation

necessary and sufficient conditions for viability for semilinear differential inclusions given a set a in a banach space x we define the tangent set and the quasi tangent set to k atc g k concepts more general than the one of tangent vector introduced by bouligand 1930 and severi 1931 both notions prove very suitable in the study of viability problems referring to differential inclusions namely we establish several new necessary and even necessary and sufficient conditions for viability referring to both differential inclusions and semilinear evolution inclusions conditions expressed in terms of the tangency concepts introduced

on l p resolvent estimates and the density of eigenvalues for compact riemannian manifolds we address an interesting question raised by dos santos ferreira kenig and salo forum math 2014 about regions mathcal r g subset mathbb c for which there can be uniform l frac 2n n 2 to l frac 2n n 2 resolvent estimates for delta g zeta zeta in mathcal r g where delta g is the laplace beltrami operator with metric g on a given compact boundaryless riemannian manifold of dimension n geq 3 this is related to earlier work of kenig ruiz and the third author duke math j 55 329 347 1987 for the euclidean laplacian in which case the region is the entire complex plane minus any disc centered at the origin presently we show that for the round metric on the sphere s n the resolvent estimates in dos santos ferreira et al in forum math 2014 involving a much smaller region are essentially optimal we do this by establishing sharp bounds based on the distance from zeta to the spectrum of delta s n in the other direction we also show that the bounds in dos santos ferreira et al in forum math 2014 can be sharpened logarithmically for manifolds with nonpositive curvature and by powers in the case of the torus mathbb t n mathbb r n mathbb z n with the flat metric the latter improves earlier bounds of shen int math res not 1 1 31 2001 the work of dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 was based on hadamard parametrices for delta g zeta 1 ours is based on the related hadamard parametrices for cos t sqrt delta g and it follows ideas in sogge in ann math 126 439 447 1987 of proving l p multiplier estimates using small time wave equation parametrices and the spectral projection estimates from sogge in j funct anal 77 123 138 1988 this approach allows us to adapt arguments in berard math z 155 249 276 1977 and hlawka monatsh math 54 1 36 1950 to obtain the aforementioned improvements over dos santos ferreira et al in forum math 2014 and shen in int math res not 1 1 31 2001 further improvements for the torus are obtained using recent techniques of the first author bourgain in israel j math 193 1 441 458 2013 and his work with guth bourgain and guth in geom funct anal 21 1239 1295 2011 based on the multilinear estimates of bennett carbery and tao math z 2 261 302 2006 our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of sqrt delta g and moreover a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals as opposed to those in sogge in j funct anal 77 123 138 1988 for unit length intervals we show that the resolvent estimates are sensitive to clustering within the spectrum which is not surprising given sommerfeld s original conjecture sommerfeld in physikal zeitschr 11 1057 1066 1910 about these operators

multi bump solutions for a strongly indefinite semilinear schrodinger equation without symmetry or convexity assumptions abstract in this paper we study the following semilinear schrodinger equation with periodic coefficient u v x u f x u u h 1 r n the functional corresponding to this equation possesses strongly indefinite structure the nonlinear term f x t satisfies some superlinear growth conditions and need not be odd or increasing in t using a new variational reduction method and a generalized morse theory we proved that this equation has infinitely many geometrically different solutions furthermore if the solutions of this equation under some energy level are isolated then we can show that this equation has infinitely many m bump solutions for any positive integer m 2

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

on the bellman equation for the minimum time problem in infinite dimensions we consider the time optimal control problem for a semilinear parabolic control system where the target is the closed ball with center 0 and radius rgeq0 in a hilbert space x in particular we allow the origin of x to be the target using an appropriate kruzkov type transformation we give an existence and uniqueness result for the associated hamilton jacobi bellman equation even when the reachable set is not the whole space

new numerical methods for the riesz space fractional partial differential equations in this paper we consider the numerical solution of the riesz space fractional diffusion equation and advection dispersion equation first a system of ordinary differential equations is obtained from the above equations with respect to the space variable by using the improved matrix transform method furthermore we use the 2 2 pade approximation to compute the exponential matrix in the analytic solution of the ordinary differential equation and get two difference schemes second using the matrix analysis method we prove that the two difference schemes are unconditionally stable finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

multiple solutions of schrodinger equations with indefinite linear part and super or asymptotically linear terms abstract based on new information concerning strongly indefinite functionals without palais smale conditions we study existence and multiplicity of solutions of the schrodinger equation u v x u g x u for x r n u x 0 as x where v and g are periodic with respect to x and 0 lies in a gap of v supposing g is asymptotically linear as u and symmetric in u we obtain infinitely many geometrically distinct solutions we also consider the situation where g is super linear with mild assumptions different from those studied previously and establish the existence and multiplicity

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

a partial answer to a conjecture of b simon concerning unique continuation in his survey paper on schrijdinger semigroups s b simon formulated the following conjecture see s p 5193 let ve kfp the local kato class of potentials then the time independent schrddinger operator h d v has the unique continuation property ucp by this it is meant that given any connected open subset 52 c r the only solution u of hu 0 in 52 that can vanish in an open subset 52 c sz is u 0 a measurable function v on r is said to belong to the kato class k if

inverse scattering problem for a two dimensional random potential we study an inverse problem for the two dimensional random schrodinger equation q k 2 u 0 the potential q x is assumed to be a gaussian random function whose covariance operator is a classical pseudodifferential operator we show that the backscattered field obtained from a single realization of the random potential q determines uniquely the principal symbol of the covariance operator of q the analysis is carried out by combining harmonic and microlocal analysis with stochastic methods

l p l q estimates for electromagnetic helmholtz equation abstract in space dimension n 3 we consider the electromagnetic schrodinger hamiltonian h i a x 2 v and the corresponding helmholtz equation i a x 2 u u v x u f in r n we extend the well known l p l q estimates for the solution of the free helmholtz equation to the case when the electromagnetic hamiltonian h is considered

decay estimates and strichartz estimates of fourth order schrodinger operator abstract we study time decay estimates of the fourth order schrodinger operator h 2 v x in r d for d 3 and d 5 we analyze the low energy and high energy behaviour of resolvent r h z and then derive the jensen kato dispersion decay estimate and local decay estimate for e i t h p a c under suitable spectrum assumptions of h based on jensen kato type decay estimate and local decay estimate we obtain the l 1 l estimate of e i t h p a c in 3 dimension by ginibre argument and also establish the endpoint global strichartz estimates of e i t h p a c for d 5 furthermore using the local decay estimate and the georgescu larenas soffer conjugate operator method we prove the jensen kato type decay estimates for some functions of h

unique continuation and absence of positive eigenvalues for schrodinger operators on considere l operateur de laplace sur r n et une fonction v x sur un sous ensemble connexe ouvert de r n on montre que si n 3 une propriete de prolongement unique est vraie pour v l loc n 2 r n dans l espace de sobolev h loc 2 q r n avec q 2n n 2

uniform resolvent estimates for schrodinger operator with an inverse square potential abstract we study the uniform resolvent estimates for schrodinger operator with a hardy type singular potential let l v v x where is the usual laplacian on r n and v x v 0 r 2 where r x x x and v 0 c 1 s n 1 is a real function such that the operator v 0 n 2 2 4 is a strictly positive operator on l 2 s n 1 we prove some new uniform weighted resolvent estimates and also obtain some uniform sobolev estimates associated with the operator l v

crank nicolson method for the fractional diffusion equation with the riesz fractional derivative we examine a numerical method to approximate to a fractional diffusion equation with the riesz fractional derivative in a finite domain which has second order accuracy in time and space level in order to approximate the riesz fractional derivative we use the fractional centered derivative approach we determine the error of the riesz fractional derivative to the fractional centered difference we apply the crank nicolson method to a fractional diffusion equation which has the riesz fractional derivative and obtain that the method is unconditionally stable and convergent numerical results are given to demonstrate the accuracy of the crank nicolson method for the fractional diffusion equation with using fractional centered difference approach

regularity results for the minimum time function of a class of semilinear evolution equations of parabolic type under suitable controllability and smoothness assumptions the minimum time function t x of a semilinear control system is proved to be locally lipschitz continuous and semiconcave on the controllable set these properties are then applied to derive optimality conditions relating optimal trajectories to the superdifferential of t

numerical algorithm for parabolic problems with non classical conditions the parabolic problems with non classical conditions are discussed in a reproducing kernel space in this paper a reproducing kernel space is constructed in which the non classical conditions of the parabolic problems are satisfied based on the reproducing kernel space a new technique for solving the non classical parabolic problems is presented some examples are displayed to demonstrate the validity and applicability of the proposed method

regularity of the minimum time function and minimum energy problems the linear case we prove new results on the continuity properties and on reachable sets of the minimum time function associated with a linear dynamical system in a separable hilbert space h part of the results are new also in the finite dimensional case the regularity results are stated thanks to the result we obtain on the connection between the minimum time function and the minimum energy

the minimum time function for semilinear evolutions we study the bellman equation for the minimum time problem associated with a semilinear evolution system from the point of view of contingent solutions for this purpose new contingent derivatives are defined which involve functions instead of vectors as directions our analysis is based on flow invariance results for appropriate differential inclusions

a novel expansion iterative method for solving linear partial differential equations of fractional order in this manuscript we implement a relatively new analytic iterative technique for solving time space fractional linear partial differential equations subject to given constraints conditions based on the generalized taylor series formula the solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations this method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics physics and engineering some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality generality and accuracy for solving such equations with different constraints conditions numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm

dual variational methods and nonvanishing for the nonlinear helmholtz equation we set up a dual variational framework to detect real standing wave solutions of the nonlinear helmholtz equation u k2u q x u p 2u u w2 p rn with n 3 2 n 1 n 1 p 2nn 2 and nonnegative q l rn we prove the existence of nontrivial solutions for periodic q as well as in the case where q x 0 as x in the periodic case a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation the solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation

the hamilton jacobi bellman equation for time optimal control in this paper several assertions concerning viscosity solutions of the hamilton jacobi bellman equation for the optimal control problem of steering a system to zero in minimal time are proved first two rather general uniqueness theorems are established asserting that any positive viscosity solution of the hjb equation must in fact agree with the minimal time function near zero if also a boundary condition introduced by bardi siam j control optim 27 1988 pp 776 785 is satisfied then the agreement is global additionally the holder continuity of any subsolution of the hjb equation is proved in the case where the related dynamics satisfy a hormander type hypothesis this last assertion amounts to a half derivative analogue of a theorem of crandall and lions traps amer math soc 277 1 983 pp 1 42 concerning lipschitz viscosity solutions

a priori estimates for operational differential inclusions we prove a set valued gronwall lemma and a relaxation theorem for the semilinear differential inclusion x ax f t x x 0 x0 where a is the infinitesimal generator of a c0 semigroup on a separable banach space x and f 0 t x x is a set valued map this allows us to investigate infinitesimal generators of reachable sets and variational inclusions the results are applied to a semilinear optimal control problem with end point constraints

delay asymptotic solutions for the time fractional delay type wave equation abstract a new version of the wave equation endowed with time fractional derivative and multiple time delays is considered a new form of the generalized maclaurin fractional series is suggested to extract analytical solution of the model graphical investigations on the relation between the fractional derivative and the time delay are established finally we reported some remarks regarding the new model

on the periodic and asymptotically periodic nonlinear helmholtz equation abstract in the first part of this paper the existence of infinitely many l p standing wave solutions for the nonlinear helmholtz equation u u q x u p 2 u in r n is proven for n 2 and 0 under the assumption that q be a nonnegative periodic and bounded function and the exponent p lies in the helmholtz subcritical range in a second part the existence of a nontrivial solution is shown in the case where the coefficient q is only asymptotically periodic

mathematical analysis of dengue fever outbreak by novel fractional operators with field data abstract an epidemiological model is proposed for three different types of novel fractional order derivative operators known as the caputo the caputo fabrizio and the atangana baleanu caputo operators the classical model is fractionalized while taking care of the dimensional consistency for each ordinary differential equation of the model the true field data obtained through some authentic sources for the dengue fever outbreak in cape verde islands in 2009 is the major motivation behind analysis of the present research study the proposed model being nonlinear possesses the possibility of having no closed form solution it is therefore existence and uniqueness for the solutions of the models are investigated via fixed point theory the residuals computed via least squares approach for all types of cases under consideration reveal the better performance of the models under fractional order derivative operators proving that the dynamics of the disease dengue virus can be well understood if non local effects are taken into consideration within the model based upon the results obtained the efficiency rates of the fractional order operators under the caputo caputo fabrizio and the atangana baleanu caputo are higher than that of the existing classical model

the dynamic programming equation for the time optimal control problem in infinite dimensions the existence and uniqueness of a viscosity solution for the bellman equation associated with the time optimal control problem for a semilinear evolution equation in hilbert space is provided applications to time optimal control problems governed by parabolic equations are given

fitted reproducing kernel hilbert space method for the solutions of some certain classes of time fractional partial differential equations subject to initial and neumann boundary conditions latterly many problems arising in different fields of science and engineering can be reduced by applying some appropriate discretization to a series of time fractional partial differential equations unlike the normal case derivative the differential order in such equations is with a fractional order which will lead to new challenges for numerical simulation the purpose of this analysis is to introduce the reproducing kernel hilbert space method for treating classes of time fractional partial differential equations subject to neumann boundary conditions with parameters derivative arising in fluid mechanics chemical reactions elasticity anomalous diffusion and population growth models the method provides appropriate representation of the solutions in convergent series formula with accurately computable components numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n term of the exact solutions finally the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time fractional neumann problems

a computational method for solving periodic boundary value problems for integro differential equations of fredholm volterra type abstract in this paper the numerical solution of periodic fredholm volterra integro differential equations of first order is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the periodic condition of the problem is satisfied the exact solution u x is represented in the form of series in the space w 2 2 in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 2 some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear equations

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

the minimal time function in infinite dimensions the goal of this paper is to study regularity properties of the minimal time function t cdot associated with linear control systems in infinite dimensions this paper proves that various regularity properties of t cdot hold on the whole reachable set provided they hold around the target some methods to obtain estimates on t cdot around the target are given and applications to both distributed and boundary control problems are presented

high order algorithms for riesz derivative and their applications ii in this paper we firstly develop two high order approximate formulas for the riesz fractional derivative secondly we propose a temporal second order numerical method for a fractional reaction dispersion equation where we discretize the riesz fractional derivative by using two numerical schemes we prove that the numerical methods for a spatial riesz fractional reaction dispersion equation are both unconditionally stable and convergent and the orders of convergence are o 2 h 6 and o 2 h 8 in which and h are spatial and temporal step sizes respectively finally we test our numerical schemes and observe that the numerical results are in good agreement with the theoretical analysis

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

unique continuation from infinity for linear waves abstract we prove various uniqueness results from null infinity for linear waves on asymptotically flat space times assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities we derive that the solution must vanish in an open set in the interior we find that the parts of infinity where we must impose a vanishing condition depend strongly on the background geometry in particular for backgrounds with positive mass such as schwarzschild or kerr the required assumptions are much weaker than the ones in the minkowski space time the results are nearly optimal in many respects they can be considered analogues of uniqueness from infinity results for second order elliptic operators this work is partly motivated by questions in general relativity

improved eigenvalue bounds for schrodinger operators with slowly decaying potentials we extend a result of davies and nath j comput appl math 148 1 1 28 2002 on the location of eigenvalues of schrodinger operators with slowly decaying complex valued potentials to higher dimensions in this context we also discuss various examples related to the laptev and safronov conjecture laptev and safronov in commun math phys 292 1 29 54 2009

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

non trivial l q solutions to the ginzburg landau equation it is shown that the contour problem for the stationary ginzburg landau equation where x r with r x is well posed in l 4 n for a class of small data f l 2 s n 1

singular solutions of elliptic equations and the determination of conductivity by boundary measurements abstract we improve some results on uniqueness and stability in the determination of the coefficient a in the equation i div a grad u 0 in when all possible pairs of dirichlet and neumann data on are known we also treat cases of anisotropic equations our method relies on the construction of solutions to i having an isolated singularity with prescribed asymptotic behaviour

lyapunov pairs for multi valued semi linear evolutions abstract we give here a characterization of a lyapunov pair for a multi valued semi linear evolution equation on a banach space by means of an appropriate lower contingent derivative the contingent derivative introduced in this paper is related to a new concept of tangency introduced recently in o c rj m necula i i vrabie necessary and sufficient conditions for viability for semilinear differential inclusions trans amer math soc 361 2009 343 390 as an application we give a controllability result and a lipschitz estimate of the corresponding minimum time function under a petrov like condition

solving fredholm integro differential equations using reproducing kernel hilbert space method in this study the numerical solution of fredholm integro differential equation is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the initial condition of the problem is satisfied the exact solution ux is represented in the form of series in the space w 2 2 a b in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 2 a b some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear fredholm integro differential equations

sharp resolvent estimates outside of the uniform boundedness range in this paper we are concerned with resolvent estimates for the laplacian delta in euclidean spaces uniform resolvent estimates for delta were shown by kenig et al duke math j 55 2 329 347 1987 who established rather a complete description of the lebesgue spaces allowing such estimates however the problem of obtaining sharp l p l q bounds depending on z has not been considered in a general framework which admits all possible p q in this paper we present a complete picture of sharp l p l q resolvent estimates which may depend on z we also obtain the sharp resolvent estimates for the fractional laplacians and a new result for the bochner riesz operators of negative index