solutions of nonlinear dirac equations abstract we study the dirac equation i t i c k 1 3 k k m c 2 g x and obtain existence and multiplicity results of stationary solutions for several classes of nonlinearities g r 3 c 4 r modeling various types of interaction a typical result states that if g x u depends periodically on x and is even in u the problem has infinitely many geometrically different localized solutions the arguments are variational the associated lagrangian functional is strongly indefinite and the palais smale condition does not hold we apply some recently developed critical point theorems

some filter regularization methods for a backward heat conduction problem in this paper a one dimensional backward heat conduction problem is investigated it is well known that such problem is ill posed some filter regularization methods are used to solve it convergence estimates under two a priori bound assumptions for the exact solution are given based on the conditional stabilities finally numerical examples are given to show that our used numerical methods are effective and stable

on fractional powers of generators of fractional resolvent families abstract we show that if a generates a bounded times resolvent family for some 0 2 then a generates an analytic times resolvent family for 0 2 2 and 0 2 and a generalized subordination principle is derived in particular if a generates a bounded times resolvent family for some 1 2 then a 1 generates an analytic c 0 semigroup such relations are applied to study the solutions of cauchy problems of fractional order and first order

nodal type bound states of schrodinger equations via invariant set and minimax methods for nonlinear schrodinger equations in the entire space we present new results on invariant sets of the gradient flows of the corresponding variational functionals the structure of the invariant sets will be built into minimax procedures to construct nodal type bound state solutions of nonlinear schrodinger type equations 2004 elsevier inc all rights reserved

a variational approach to optimal meshes a variational approach for the optimization of triangular or tetrahedral meshes is presented starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity it will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions in addition the injectivity of the deformed mesh is discussed thereby it is possible to construct suitable meshes for various numerical applications

a comparison of regularizations for an ill posed problem we consider numerical methods for a quasi boundary value regularization of the backward parabolic problem given by u t au 0 0 t t u t f where a is positive self adjoint and unbounded the regularization due to clark and oppenheimer perturbs the final value u t by adding u 0 where is a small parameter we show how this leads very naturally to a reformulation of the problem as a second kind fredholm integral equation which can be very easily approximated using methods previously developed by ames and epperson error estimates and examples are provided we also compare the regularization used here with that from ames and epperson

on a backward parabolic problem with local lipschitz source abstract we consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form u t a u t f u t t u 1 where a is a positive self adjoint unbounded operator and f is a local lipschitz function as known it is ill posed and occurs in applied mathematics e g in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology a new version of quasi reversibility method is described we show that the regularized problem with a regularization parameter 0 is well posed and that its solution u t converges on 0 1 to the exact solution u t as 0 these results extend some earlier works on the nonlinear backward problem

an extended pressure finite element space for two phase incompressible flows with surface tension we consider a standard model for incompressible two phase flows in which a localized force at the interface describes the effect of surface tension if a level set or vof method is applied then the interface which is implicitly given by the zero level of the level set function is in general not aligned with the triangulation that is used in the discretization of the flow problem this non alignment causes severe difficulties w r t the discretization of the localized surface tension force and the discretization of the flow variables in cases with large surface tension forces the pressure has a large jump across the interface in standard finite element spaces due to the non alignment the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure in many simulations these effects cause large oscillations of the velocity close to the interface so called spurious velocities in this paper for a simplified model problem we give an analysis that explains why known standard methods for discretization of the localized force term and for discretization of the pressure variable often yield large spurious velocities in the paper s grosz a reusken finite element discretization error analysis of a surface tension force in two phase incompressible flows preprint 262 igpm rwth aachen siam j numer anal accepted for publication we introduce a new and accurate method for approximation of the surface tension force in the present paper we use the extended finite element space xfem presented in n moes j dolbow t belytschko a finite element method for crack growth without remeshing int j numer meth eng 46 1999 131 150 t belytschko n moes s usui c parimi arbitrary discontinuities in finite elements int j numer meth eng 50 2001 993 1013 for the discretization of the pressure we show that the size of spurious velocities is reduced substantially provided we use both the new treatment of the surface tension force and the extended pressure finite element space

layered solutions for a semilinear elliptic system in a ball abstract we consider the following system of schrodinger poisson equations in the unit ball b 1 of r 3 e 2 v v v v p 0 4 v 2 with the dirichlet boundary conditions v 0 on b 1 here e 0 v b 1 r 1 p 7 3 we exhibit a family of positive radially symmetric solutions v e e such that v e concentrates on a sphere in the interior of b 1 as e 0 our approach relies upon a finite dimensional reduction by the lyapunov schmidt method which allows us to reduce the problem to maximizing a one dimensional functional the solution of such reduced problem also provides the radius of the concentration sphere

multiplicity and concentration of solutions for a quasilinear choquard equation in this paper we study a quasilinear choquard equation involving the p laplacian operator and a potential function v under suitable assumptions on v and the nonlinearity we prove the existence multiplicity and concentration of solutions for the equation by variational methods

a grid alignment finite element technique for incompressible multicomponent flows we present a numerical method for solving the multicomponent incompressible navier stokes equations the methods employs a moving grid technique and a projection scheme based on low order crouzeix raviart and p 1 finite elements the computational grid is aligned with the fluid interface at each time step to keep the interpolation order optimal the connectivity of the grid is fixed during the entire time simulation to ensure efficient assembling and parallelization the method is tested on several examples

semiclassical states for weakly coupled fractional schrodinger systems abstract in this paper we consider the nonlinear system of two weakly coupled fractional schrodinger equations with nonconstant potentials e 2 s s u p 1 x u u 2 p b u p 1 v p 1 u x r n e 2 s s v p 2 x v v 2 p b v p 1 u p 1 v x r n we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials

solutions of a nonlinear dirac equation with external fields we study the stationary dirac equation ic hbar sum 3 k 1 alpha k partial k u mc 2 beta u m x u r u x u where m x is a matrix potential describing the external field and r x u stands for an asymptotically quadratic nonlinearity modeling various types of interaction without any periodicity assumption for fixed our discussion includes the coulomb potential as a special case and for the semiclassical situation 0 we handle the scalar fields we obtain existence and multiplicity results of stationary solutions via critical point theory

a fractional landweber method for solving backward time fractional diffusion problem abstract in this paper we propose a fractional landweber regularization method for solving a backward time fractional diffusion problem choice rules for both a priori and a posterior regularization parameters are discussed in deriving the corresponding convergence proofs for the backward problem for verification on the theoretical estimates several numerical examples are constructed and compared with some existing works

solutions of perturbed p laplacian equations with critical nonlinearity in this paper we study a perturbed p laplacian equation under some given conditions on v x we prove that the equation has at least one positive solution provided that e e for any n n it has at least n pairs solutions if e en and suppose there exists an orthogonal involution t rn rn such that v x p x and k x are t invariant then it has at least one pair of solutions which change sign exactly once provided that e e where e and en are sufficiently small positive numbers moreover these solutions ue 0 in w1 p rn as e 0

on a backward cauchy problem associated with continuous spectrum operator the nonlinear backward cauchy problem ut au t f u t u t where a is a positive self adjoint unbounded operator which has a continuous spectrum and f is a lipschitz function being given is regularized by the well posed problem the new error estimates of the regularized solution are obtained this work extends to the nonlinear case earlier results by the authors 7 1 and by denche and bessila 8 13

regularization by a modified quasi boundary value method of the ill posed problems for differential operator equations of the first order abstract in this paper we consider the differential operator equation d u t d t a u t 0 with a a self adjoint unbounded operator coefficient which does not have a fixed sign the cauchy problem for the equation above with conditions of the form u 0 f or u t f is known to be an ill posed problem in this work we will use a modified quasi boundary value method we obtain an approximate non local problem depending on a small parameter 0 1 we show that the approximate problems are well posed and that their solutions converge if the original problem has a classical solution we also obtain a convergence result for these solutions

particulate flows with the subspace projection method a novel finite element method for the 3d simulation of many particles in a newtonian carrier liquid is presented the methods features the celebrated one domain approach a subspace projection method to account for the rigid body motion within the particles and operator splitting combined with local mesh refinement the method results in a fast and accurate though conceptually simple to implement algorithm validation is achieved using the sedimentation of one particle and comparing the resulting drag coefficient with theoretical and experimental results furthermore a viscometer is considered where the effective viscosity of a particle laden fluid is compared with analytic results

multiple solutions for a semiclassical schrodinger equation abstract this paper is concerned with the semiclassical nonlinear schrodinger equation e 2 u v x u f x u x r n u x 0 as x where e 0 is a small parameter under certain hypotheses on v and a general spectral assumption the existence and multiplicity of semiclassical solutions are obtained for asymptotically linear nonlinearity via variational methods

a modified integral equation method of the semilinear backward heat problem abstract the paper is concerned with the non linear backward heat equation in the rectangle domain the problem is severely ill posed we shall use a modified integral equation method to regularize the nonlinear problem the error estimates of holder type of the regularized solutions are obtained numerical results are presented to illustrate the accuracy and efficiency of the method this work is a generalization of many earlier papers including the recent paper d d trong n h tuan regularization and error estimate for the nonlinear backward heat problem using a method of integral equation nonlinear anal 71 9 2009 4167 4176

a hybrid level set front tracking finite element approach for fluid structure interaction and two phase flow applications we present a hybrid level set front tracking approach suitable for fluid structure interaction and two phase flow applications our approach aims at extending geometrical flexibility of standard mesh moving front tracking methods by introducing an additional implicit level set representation of the geometry under consideration the computational mesh is automatically aligned to the implicitly described geometry by minimizing a nonlinear constrained functional resulting triangulations approximate the geometry accurately while being optimal in a certain sense due to the mesh alignment finite element spaces defined on these triangulations may be easily adjusted to account for special solution properties such as discontinuities across interfaces in order to demonstrate the flexibility of the proposed approach we apply it to a simplified one way coupled fluid structure interaction problem inspired by the flow induced by a moving cardiac valve furthermore we evaluate the approach by solving a two phase flow benchmark problem

the truncation method for a two dimensional nonhomogeneous backward heat problem abstract we consider the backward heat problem u t u xx u yy f x y t x y t 0 t u x y t g x y x y with the homogeneous dirichlet condition on the rectangle 0 0 where the data f and g are given approximately the problem is severely ill posed using the truncation method for fourier series we propose a simple regularized solution which not only works on a very weak condition on the exact data but also attains due to the smoothness of the exact solution explicit error estimates which include the approximation ln 1 3 2 in h2 some numerical examples are given to illuminate the effect of our method

on spurious velocities in incompressible flow problems with interfaces we show that a non physical velocity may appear in the finite element calculation of incompressible two phase flows subjected to an external local force there are different sources responsible for this phenomenon approximation of the incompressibility constraint the interface and the local external force furthermore we demonstrate that this non physical velocity does not vanish with the expected order of convergence we evaluate different concepts and give recommendations for a proper handling of this phenomenon

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

regularization and convergence for ill posed backward evolution equations in banach spaces abstract this work is concerned with a mathematical study of ill posed backward evolution equations associated with densely defined linear differential operators in banach spaces a general approach is presented to investigate the convergence and stability of a class of regularized solutions for ill posed backward evolution equations associated with sectorial or half strip operators generalized concepts of qualification pairs and index functions are introduced to characterize the explicit convergence rates of the concerned regularized solutions applications of our results to general backward evolution equations are also investigated

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

semiclassical states for nonlinear schrodinger equations with sign changing potentials we establish the existence and multiplicity of semiclassical bound states of the following nonlinear schrodinger equation e2 u v x u g x u for x rn u x 0as x where v changes sign and g is superlinear with critical or supercritical growth as u

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

a modified quasi boundary value method for ill posed problems in this paper we study a final value problem for first order abstract differential equation with positive self adjoint unbounded operator coefficient this problem is ill posed perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter we show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution we also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions finally we give explicit convergence rates

a new regularized method for two dimensional nonhomogeneous backward heat problem we consider the problem of finding from the final data u x y t g x y the initial data u x y 0 of the temperature function u x y t x y i 0 p x 0 p t 0 t satisfying the following systemu t u x x u y y f x y t x y t ix 0 t u 0 y t u p y t u x 0 t u x p t 0 x y t ix 0 t the problem is severely ill posed in this paper a simple and convenient new regularization method for solving this problem is considered meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

a nonlinear parabolic equation backward in time regularization with new error estimates abstract consider a nonlinear backward parabolic problem in the form u t a u t f t u t u t g where a is a positive self adjoint unbounded operator based on the fundamental solution to the parabolic equation we propose to solve this problem by the fourier truncated method which generates a well posed integral equation then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven our regularizing scheme can be considered a new regularization with the advantage of a relatively small amount of computation compared with the quasi reversibility or quasi boundary value regularizations error estimates for this method are provided together with a selection rule for the regularization parameter these errors show that our method works effectively

on a class of singularly perturbed elliptic equations in divergence form existence and multiplicity results the main purpose of this paper is to study the existence of single peaked positive solutions of the singularly perturbed elliptic equation e2div j x u v x u f u inrn where j is a symmetric uniformly elliptic matrix and v is a positive potential possibly unbounded from above if f u up then solutions concentrate at non degenerate critical points of x v x p 1p 1 n2 detj x 1 2

on the axisymmetric backward heat equation with non zero right hand side regularization and error estimates abstract in this paper we consider the time inverse problem for the axisymmetric heat equation with non zero right hand side this problem is ill posed the solution if it exists does not depend continuously on the final data we use a modified quasi boundary value to regularize the inhomogeneous problem numerical results are presented to illustrate the accuracy and efficiency of the method

identification of the initial condition in backward problem with nonlinear diffusion and reaction abstract in this paper we study the backward in time problem for nonlinear parabolic equations associated with nonlinear diffusion coefficient the problem is ill posed in the sense of hadamard under an a priori assumption on the exact solution in gevrey space we propose a new regularization method for stabilising the ill posed problem our results extend some earlier works on backward problems for nonlinear parabolic equations

a monolithic conservative level set method with built in redistancing abstract we introduce a new level set method for representing evolving interfaces in the case of divergence free velocity fields the new method satisfies a conservation principle conservation is important for many applications such as modeling two phase incompressible flow in the present implementation the conserved quantity is defined as the integral of a smoothed characteristic function the new approach embeds level sets into a volume of fluid formulation the evolution of an approximate signed distance function is governed by a conservation law for its smoothed sign the non linear level set transport equation is regularized by adding a flux correction term that assures a non singular jacobian and penalizes deviations from a distance function the result is a locally conservative level set method with built in elliptic redistancing the continuous model is monolithic in the sense that the level set transport model the volume of fluid law of mass conservation and the minimization problem that preserves the approximate distance function property are incorporated into a single equation there is no need for any extra stabilization artificial compression flux limiting redistancing mass correction and other numerical fixes which are commonly used in level set or volume of fluid methods in addition there is just one free parameter that controls the strength of regularization and penalization in the model the accuracy and conservation properties of the monolithic finite element level set method are illustrated by the results of numerical studies for passive advection of free interfaces

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

existence and multiplicity of semiclassical solutions for a schrodinger equation abstract we establish the existence and multiplicity of solutions for the semiclassical nonlinear schrodinger equation e 2 u v x u g x u for x r n u x 0 as x where e 0 is a small parameter and the potential v changes sign and may not be bounded from below and g x u is asymptotically linear in u as u

a hybrid level set front tracking approach for finite element simulations of two phase flows abstract in this paper we give details on the numerical realization of a new finite element method for the simulation of two phase flows which was recently introduced in basting and weismann 2013 the main ingredient is a hybrid representation of the interface between the fluid phases an implicit description of the interface is given by a level set function and an explicit representation is obtained from aligning edges of the computational mesh to the implicitly described interface this step is done by a black box optimization based mesh smoothing approach which does not change the topology of the mesh while guaranteeing optimal mesh quality furthermore we make use of quadratic isoparametric elements to increase the approximation quality of the discrete interface due to the alignment discontinuities of the solution variables pressure can be captured accurately while a variational treatment of the curvature allows for a precise approximation of surface tension we present our time discretization scheme for the coupled navier stokes level set equations and discuss our space discretization based on the so called subspace projection method spm to account for discontinuities across the interface we present two numerical examples for which reference solutions exist we consider the oscillation of a single droplet and provide our results for an established two phase flow benchmark problem

a parallel method for backward parabolic problems based on the laplace transformation a parallel method for time discretization of backward parabolic problems is proposed the problem is reformulated to a set of helmholtz type problems with a parameter on a suitably chosen contour in the complex plane after solving the resulting elliptic equations which can be solved in parallel we obtain a regularized solution with high frequency terms cut off by the inverse laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed the quality of the solution is improved significantly compared to those obtained by other methods two different numerical inversions of laplace transforms with an arbitrary high order of accuracy and spectral accuracy respectively are used error estimates and numerical examples are presented

final value problem for nonlinear time fractional reaction diffusion equation with discrete data abstract in this paper we study the problem of finding the solution of a multi dimensional time fractional reaction diffusion equation with nonlinear source from the final value data we prove that the present problem is not well posed then regularized problems are constructed using the truncated expansion method in the case of two dimensional and the quasi boundary value method in the case of multi dimensional finally convergence rates of the regularized solutions are given and investigated numerically

on fractional schrodinger equation in rn with critical growth in this paper we study the following nonlinear fractional schrodinger equation with critical exponent h2 u v x u u 2 2u u q 2u x rn where h is a small positive parameter 0 2 0 is a parameter the potential v rn r is a positive continuous function satisfying some natural assumptions by using variational methods we obtain the existence of solutions in the following case if 2 0 such that for all 0 we show that it has one nontrivial solution and there exist at least cat nontrivial solutions if max 2 4 n 2 0

modified quasi reversibility method for final value problems in banach spaces this paper is concerned with the final value problem associated with a linear operator a in a banach space where a is the generator of a uniformly bounded analytic semigroup based on the delaubenfels functional calculus we use new quasi reversibility method introduced by boussetila and rebbani recently to form an approximate problem we obtain some results in a banach space similar to those in a hilbert space

regularization of an inverse nonlinear parabolic problem with time dependent coefficient and locally lipschitz source term abstract we consider a backward problem of finding a function u satisfying a nonlinear parabolic equation in the form u t a t a u t f t u t subject to the final condition u t here a is a positive self adjoint unbounded operator in a hilbert space h and f satisfies a locally lipschitz condition this problem is ill posed using quasi reversibility method we shall construct a regularized solution u e from the measured data a e and e we show that the regularized problems are well posed and that their solutions converge to the exact solutions error estimates of logarithmic type are given and a simple numerical example is presented to illustrate the method as well as verify the error estimates given in the theoretical parts

regularization and error estimate for the nonlinear backward heat problem using a method of integral equation abstract we consider the inverse time problem for the nonlinear heat equation in the form u t u x x f x t u x t x t 0 0 t u 0 t u t 0 t 0 t the nonlinear problem is severely ill posed we shall use the method of integral equation to regularize the problem and to get some error estimates we show that the approximate problems are well posed and that their solution u x t converges on 0 t if and only if the original problem has a unique solution we obtain several other results including some explicit convergence rates some numerical tests illustrate that the proposed method is feasible and effective

material distribution resembled level set method for optimal shape design of stokes flow inspired by the work of borrvall and petersson 1 we propose a novel algorithm for optimal shape design problem of two dimensional stokes flow in order to minimize a given objective functional the setting of the optimization problem is in the most general form which makes this work a nontrivial extension of 1 borrowed idea of material distribution method from shape or topology optimization in solid mechanics we employ an implicit coupled level set method to track the interface boundary of the fluid flow although our target application in this work is optimal shape design of stokes flow the proposed method can be generalized to solve other shape or topology optimization problems in fluid mechanics need no further additional effort the promising features of the proposed algorithm are illustrated by two numerical examples

homoclinic orbits for a nonperiodic hamiltonian system abstract in this paper we prove the existence and multiplicity of homoclinic orbits for first order hamiltonian systems of the form z j h z t z where h z is asymptotically linear at and is not assumed to be periodic

a modified method of stochastic regularization for an ill posed cauchy problem abstract this paper is concerned with the ill posed cauchy problem associated with a densely defined linear operator in a banach space we investigate a method of stochastic regularization for this ill posed cauchy problem by using fractional powers and functional calculus for the sectorial operators

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

on a backward heat problem with time dependent coefficient regularization and error estimates in this paper we consider a homogeneous backward heat conduction problem which appears in some applied subjects this problem is ill posed in the sense that the solution if it exists does not depend continuously on the final data a new regularization method is applied to formulate regularized solutions which are stably convergent to the exact ones with holder estimates a numerical example shows that the computational effect of the method is all satisfactory

logarithmic well posed approximation of the backward heat equation in banach space abstract in this paper we apply a logarithmic approximation introduced by boussetila and rebbani in order to study an ill posed cauchy problem in banach space the proposed method induces a well posed problem whose approximation parameter yields continuous dependence on modeling throughout properties of holomorphic semigroups are utilized culminating in the motivating application which is the backward heat equation in l p r 1 p

ill posed problems associated with sectorial operators in banach space abstract ill posed problems of the form u t a u h 0 t t u 0 are studied in banach space under two related scenarios both of which consider a well posed logarithmic approximation in the first scenario a generates a bounded holomorphic semigroup of angle 2 and so a is sectorial of angle 0 in the second where may be relaxed to any value in 0 2 we impose an additional assumption that e a is sectorial regularization results for the original ill posed problem are then established in the linear case where h h t and subsequently extended to nonlinear problems where h is replaced by h t u t we compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations both linear and nonlinear in l p r n 1 p

future sequential regularization methods for iii posed volterra equations applications to the inverse heat conduction problem abstract we develop a theoretical context in which to study the future sequential regularization method developed by beck for the inverse heat conduction problem in the process we generalize beck s ideas and view that method as one in a large class of regularization methods in which the solution of an ill posed first kind volterra equation is seen to be the limit of a sequence of solutions of well posed second kind volterra equations such techniques are important because standard regularization methods such as tikhonov regularization tend to transform a naturally sequential volterra problem into a full domain fredholm problem destroying the underlying causal nature of the volterra model and leading to inefficient global approximation strategies in contrast the ideas we present here preserve the original volterra structure of the problem and thus can lead to easily implemented localized approximation strategies theoretical properties of these methods are discussed and proofs of convergence are given

a kernel based method for the approximate solution of backward parabolic problems a method for approximating the solution of self adjoint parabolic equations backwards in time is presented the technique uses the green s function of the parabolic equation to formulate the problem as a first kind fredholm integral equation and then does a regularized least squares solution of the resulting discrete system error estimates and examples are presented direct knowledge of the kernel of the differential operator is not required

an approximate solution for nonlinear backward parabolic equations we consider the backward parabolic equation ut au f t u t 0 t t u t g where a is a positive unbounded operator and f is a nonlinear function satisfying a lipschitz condition with an approximate datum g the problem is severely ill posed using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensional subspaces according to some a priori assumptions on the regularity of the exact solution we obtain several explicit error estimates including an error estimate of holder type for all t 0 t an example on heat equations and numerical experiments are given

solitary charged waves interacting with the electrostatic field abstract in this paper we study the semiclassical limit for the following system of schrodinger maxwell equations in the unit ball b 1 of r 3 2 2 m v e v v 4 e v 2 with the boundary conditions u 0 g on b 1 here m e 0 v b 1 r g b 1 r this system has been introduced by v benci d fortunate in v benci d fortunate an eigenvalue problem for the schrodinger maxwell equations topol methods nonlinear anal 11 1998 283 293 as a model describing standing charged waves for the schrodinger equation in presence of an electrostatic field we exhibit a family of positive solutions v such that v concentrates as 0 around some points of the boundary b 1 which are proved to be minima for g

an interface fitted subspace projection method for finite element simulations of particulate flows abstract a novel finite element method for the direct numerical simulation of particles in a newtonian carrier fluid is presented the proposed method is based on a fictitious or one domain formulation and a subspace projection method to account for the rigid body motion of the particles underlying equations are posed in an ale arbitrary lagrangian eulerian formulation allowing for moving computational meshes the mesh is adapted to the particles by a novel mesh smoothing approach guaranteeing both mesh optimality and a sharp representation of the particles boundaries we show that by using quadratic taylor hood finite elements for the discretization of the navier stokes equations and isoparametric elements to represent the geometry second order convergence with respect to the energy norm can be achieved as opposed to a sharp h 1 2 result which holds if no mesh adaptation is performed we present numerical examples in 2d confirming the theoretical convergence results furthermore we validate the approach by simulating the sedimentation process of a single particle and show the potential of the method by simulating a lid driven cavity flow with 100 particles

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

regularization for ill posed cauchy problems associated with generators of analytic semigroups this paper is concerned with the ill posed cauchy problem associated with a densely defined linear operator a in a banach space our main result is that if a is the generator of an analytic semigroup then there exists a family of regularizing operators for such an ill posed cauchy problem by using the quasi reversibility method fractional powers and semigroups of linear operators the applications to ill posed partial differential equations are also given

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

a non local boundary value problem method for parabolic equations backward in time abstract the ill posed parabolic equation backward in time u t a u 0 0 t t u t f subject to the constraint u 0 e with the positive self adjoint unbounded operator a and e 0 being given is regularized by the well posed non local boundary value problem u t a u 0 0 t t u 0 u t f 0 the error estimates of holder type of the regularized solutions are obtained these estimates improve the related results by mel nikova denche and bessila

optimal error bound and simplified tikhonov regularization method for a backward problem for the time fractional diffusion equation in this paper we consider a backward problem for a time fractional diffusion equation such a problem is ill posed the optimal error bound for the problem under a source condition is analyzed a simplified tikhonov regularization method is utilized to solve the problem and its convergence rates are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule respectively numerical examples show that the proposed regularization method is effective and stable and both parameter choice rules work well

a backward parabolic equation with a time dependent coefficient regularization and error estimates we consider the problem of determining the temperature u x t for x t 0 p x 0 t in the parabolic equation with a time dependent coefficient this problem is severely ill posed i e the solution if it exists does not depend continuously on the given data in this paper we use a modified method for regularizing the problem and derive an optimal stability estimation a numerical experiment is presented for illustrating the estimate

a subspace projection method for the implementation of interface conditions in a single drop flow problem we present a mesh moving method within a finite element context where the interface conditions of a two phase flow problem are conveniently included in suitable chosen subspaces of the general trial and testfunction spaces a weak formulation of the two phase flow problem including species transport is derived where the problem specific function spaces are replaced by appropriate projections operating on standard function spaces the transfer of the variational formulation to a finite element method is straightforward and results in one set of equations for both fluidic phases this subspace projection method is applicable to a wide range of interfacial conditions for multiphase flow problems the method is validated for single drop flow problems including species transfer furthermore an application to the simulation of stagnant caps is presented

multi bump solutions and multi tower solutions for equations on rn let 0 be a small parameter in this paper we study existence of multiple multi bump positive solutions for the semilinear schrodinger equation u u 1 a x u p 2u u h1 rn where n 1 2 2 if n 1 or n 2 a c rn a x 0 for x rn and lim x a x 0 we also study existence of multiple multi tower positive solutions for the prescribed scalar curvature equation u 1 k x un 2n 2 u d1 2 rn where n 3 k c 0 k r 0 for r 0 limr 0k r 0 and limr k r 0