the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

localized turing patterns in nonlinear optical cavities abstract the subcritical turing instability is studied in two classes of models for laser driven nonlinear optical cavities in the first class of models the nonlinearity is purely absorptive with arbitrary intensity dependent losses in the second class the refractive index is real and is an arbitrary function of the intracavity intensity through a weakly nonlinear analysis a ginzburg landau equation with quintic nonlinearity is derived thus the maxwell curve which marks the existence of localized patterns in parameter space is determined in the particular case of the lugiato lefever model the analysis is continued to seventh order yielding a refined formula for the maxwell curve and the theoretical curve is compared with recent numerical simulation by gomila et al d gomila a scroggie w firth bifurcation structure of dissipative solitons physica d 227 2007 70 77

collocation software for boundary value differential algebraic equations we describe the methods and implementation of a general purpose code coldae this code can solve boundary value problems for nonlinear systems of semi explicit differential algebraic equations daes of index at most 2 fully implicit index 1 boundary value dae problems can be handled as well the code coldae is an extension of the package colnew colsys for solving boundary value odes the implemented method is piecewise polynomial collocation at gaussian points extended as needed by the projection method of ascher petzold for general semi explicit index 2 problems as well as for fully implicit index 1 problems we define a em selective projected collocation method and demonstrate its use the mesh selection procedure of colsys is modified for the case of index 2 constraints we also discuss shooting for initial guesses the power and generality of the code are demonstrated by examples

asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order abstract in this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear gradient like evolution equations of mixed first and second order the proof of convergence is based on the lojasiewicz simon inequality the construction of an appropriate lyapunov functional and some differential inequalities applications are given to nonautonomous semilinear wave and heat equations with dissipative dynamical boundary conditions a nonlinear hyperbolic parabolic partial differential equation a damped wave equation and some coupled system

density profile equation with p laplacian analysis and numerical simulation analytical properties of a nonlinear singular second order boundary value problem in ordinary differential equations posed on an unbounded domain for the density profile of the formation of microscopic bubbles in a nonhomogeneous fluid are discussed especially sufficient conditions for the existence and uniqueness of solutions are derived two approximation methods are presented for the numerical solution of the problem one of them utilizes the open domain matlab code bvpsuite the results of numerical simulations are presented and discussed

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

collocation methods for index 1 daes with a singularity of the first kind we study the convergence behavior of collocation schemes applied to approximate solutions of bvps in linear index 1 daes which exhibit a critical point at the left boundary such a critical point of the dae causes a singularity within the inherent ode system we focus our attention on the case when the inherent ode system is singular with a singularity of the first kind apply polynomial collocation to the original dae system and consider different choices of the collocation points such as equidistant gaussian or radau points we show that for a well posed boundary value problem for daes having a sufficiently smooth solution the global error of the collocation scheme converges with the order o h s where s is the number of collocation points superconvergence cannot be expected in general due to the singularity not even for the differential components of the solution the theoretical results are illustrated by numerical experiments

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

about macroscopic models of instability pattern formation abstract various macroscopic models to describe instability pattern formation are discussed in this paper they are similar to the ginzburg landau envelope equation but they can remain valid away from the bifurcation and are based on the technique of fourier series with slowly varying coefficients we focus on two questions the need to take phase changes into account and the boundary conditions to be associated with macroscopic models the analysis is carried out on the basis of numerical simulations for the problem of a compressed beam on a nonlinear foundation that is quite similar to the well known swift hohenberg equation the first macroscopic model involves a real envelope so that the phase is assumed to be constant the second model is also macroscopic and it is a sort of ginzburg landau equation with a complex envelope the third one follows from a multi scaled approach with a numerical bridging between the full model near the boundary and a macroscopic model in the bulk

stability of singular boundary value problems and their discretization by finite differences stability bounds for the solution of nonlinear singular boundary value problem for systems of ordinary differential equations of second order are derived the stability of a finite difference method is related to the stability of the analytical problem being solved

stable localized solutions of arbitrary length for the quintic swift hohenberg equation abstract we show that localized solutions of arbitrary length are stable over a finite parameter interval of subcritical values for the quintic swift hohenberg equation with a destabilizing cubic term this equation is thought to model a weakly hysteretic transition to stationary patterns we argue that the stabilization of the localized states of arbitrary length can be traced back to the interaction between long wavelength modulations and spatial variations on the length scale of one unit cell these results are critically compared with other known mechanisms to stabilize localized states in various situations we also discuss for which experimental systems the states predicted here could be detected including e g the stationary onset of binary fluid convection

asymptotic analysis and upper semicontinuity with respect to rotational inertia of attractors to von karman plates with geometrically localized dissipation and critical nonlinearity abstract in this study we consider the von karman evolution equations accounting for rotational inertial forces along with a nonlinear localized damping under suitable hypotheses we prove the existence regularity and finite dimensionality of a compact attractor a r the main result of the paper is upper semicontinuity of attractors with respect to rotational inertia terms the main obstacles are criticality of the nonlinear term of the equation combined with geometrically constrained dissipation the flux multiplier used for propagation of the damping does not cooperate with the nonlinear term which is of critical exponent a new method based on uniform compactness is developed in order to handle such cases the method allows to establish uniform smoothness of the elements from the attractor which is the key ingredient in proving that the attractor is upper semicontinuous with respect to the rotational inertia 0 and conclude that it converges to the attractor a r obtained in the case 0 in geredeli et al 2013 23 when 0

a finite difference moving mesh method based on conservation for moving boundary problems we propose a velocity based moving mesh method in which we move the nodes so as to preserve local mass fractions consequently the mesh evolves to be finer where the solution presents rapid changes naturally providing higher accuracy without the need to add nodes we use an integral approach which avoids altering the structure of the original equations when incorporating the velocity and allows the solution to be recovered algebraically we apply our method to a range of one dimensional moving boundary problems the porous medium equation richards equation and the crank gupta problem we compare our results to exact solutions where possible or to results obtained from other methods and find that our approach can be very accurate 1 relative error with as few as ten or twenty nodes

pattern formation in systems with slowly varying geometry a class of equations is considered which describe the evolution of patterns in problems with a geometry varying slightly as characterized by a small parameter tilde nu ll 1 and slowly on a spatial scale hat epsilon x for hat epsilon ll 1 the model inspired by the problem of sedimentation in meandering rivers is generalized for an analytical study of nonlinear dynamics of periodic and quasi periodic solutions which bifurcate from a basic solution two cases prove to be of interest slightly slowly varying geometry and slightly very slowly varying geometry nonlinear modulation equations are studied foreach case yielding analytical results complimentary to previous linear and numerical analyses for slightly slowly varying geometry the basic solution loses its stability to quasi periodic solutions whose existence and stability are studied in the context of the nonlinear modulation equation in the slightly very slowly varying case there are stationary spatially periodic solutions of th

monge ampere simulation of fourth order pdes in two dimensions with application to elastic electrostatic contact problems abstract we present an efficient moving mesh method for the simulation of fourth order nonlinear partial differential equations pdes in two dimensions using the parabolic monge ampere pma equation pma methods have been successfully applied to the simulation of second order problems but not on systems with higher order equations which arise in many topical applications our main application is the resolution of fine scale behavior in pdes describing elastic electrostatic interactions the pde system considered has multiple parameter dependent singular solution modalities including finite time singularities and sharp interface dynamics we describe how to construct a dynamic mesh algorithm for such problems which incorporates known self similar or boundary layer scalings of the underlying equation to locate and dynamically resolve fine scale solution features in these singular regimes we find a key step in using the pma equation for mesh generation in fourth order problems is the adoption of a high order representation of the transformation from the computational to physical mesh we demonstrate the efficacy of the new method on a variety of examples and establish several new results and conjectures on the nature of self similar singularity formation in higher order pdes

wave number selection in convection and related problems the wave number selection process induced by grain boundaries is studied beyond the framework of amplitude equations it is shown that the process is an intrinsic property of the problem being associated with its spatial symmetry the process always results in discretization of possible values of the wave number if the control parameter is small enough so that pinning of the grain boundary to the small scale underlying structure may be neglected the system possesses an extra restricting condition the lyapunov functional density for both coexisting structures must coincide in this case the selected wave number provides a local minimum of the lyapunov functional for the spatially uniform patterns as an application of the theory we consider an extended swift hohenberg equation for rolls with subcritical bifurcation

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

analysis of a new error estimate for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for the numerical solution of boundary value problems for nonlinear systems of ordinary differential equations with a singularity of the first kind the estimate for the global error of an approximation obtained by collocation with piecewise polynomial functions is based on the defect correction principle we prove that for collocation methods which are not superconvergent the error estimate is asymptotically correct as an essential prerequisite we derive convergence results for collocation methods applied to nonlinear singular problems

an l stability analysis for the finite difference solution of one dimensional linear convection diffusion equations on moving meshes the stability of three moving mesh finite difference schemes is studied in the l norm for one dimensional linear convection diffusion equations these schemes use central finite differences for spatial discretization and the q method for temporal discretization and they are based on conservative and non conservative forms of transformed partial differential equations the stability conditions obtained consist of the cfl condition and the mesh speed related conditions the cfl condition is independent of the mesh speed and has the same form as that for fixed meshes the mesh speed related conditions restrict how fast the mesh can move the conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine illustrative numerical results are presented

convergence of collocation schemes for boundary value problems in nonlinear index 1 daes with a singular point we analyze the convergence behavior of collocation schemes ap plied to approximate solutions of bvps in nonlinear index 1 daes which exhibit a critical point at the left boundary such a critical point of the dae causes a singularity in the inherent nonlinear ode system in particular we focus on the case when the inherent ode system is singular with a singularity of the rst kind and apply polynomial collocation to the original dae system we show that for a certain class of well posed boundary value problems in daes having a sufficiently smooth solution the global error of the collocation scheme converges uniformly with the so called stage order due to the sin gularity superconvergence at the mesh points does not hold in general the theoretical results are supported by numerical experiments

the convergence of shooting methods for singular boundary value problems we investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems particular attention is paid to the well posedness of the process it is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available moreover convergence results for a perturbed newton iteration are discussed

long time dynamics for nonlinear porous thermoelasticity with second sound and delay this paper is concerned with the long time behavior of a damped porous thermoelastic problem it has been studied by many authors and most of the known results are concerned with decay issues under different boundary conditions and damping terms there are just a few references on the long time dynamics of such systems the purpose of the present work is to complement and extend some early studies on porous thermoelastic systems by establishing new results on the existence of attractors and some of their properties motivated by this scenario we consider a one dimensional porous thermoelastic system with linear frictional damping nonlinear source terms and a time varying delay term in the internal feedback where the heat flux depends on cattaneo law under some suitable assumptions on the weights of feedback we establish the global well posedness of the system by using the c0 semigroup theory of linear operators then we show the existence of a global attractor for finite energy solutions and we prove its smoothness and finite fractal dimension furthermore the existence of a generalized fractal exponential attractor is also derived this paper is concerned with the long time behavior of a damped porous thermoelastic problem it has been studied by many authors and most of the known results are concerned with decay issues under different boundary conditions and damping terms there are just a few references on the long time dynamics of such systems the purpose of the present work is to complement and extend some early studies on porous thermoelastic systems by establishing new results on the existence of attractors and some of their properties motivated by this scenario we consider a one dimensional porous thermoelastic system with linear frictional damping nonlinear source terms and a time varying delay term in the internal feedback where the heat flux depends on cattaneo law under some suitable assumptions on the weights of feedback we establish the global well posedness of the system by using the c0 semigroup theory of linear operators then we show the existence of a global attractor for finite energy solutions and we pro

patterns and coherence resonance in the stochastic swift hohenberg equation with pyragas control the turing bifurcation case abstract we provide a multiple time scales analysis for the swift hohenberg equation with delayed feedback via pyragas control with and without additive noise an analysis of the pattern formation near onset indicates both the possibility of either standing waves rolls or traveling waves via turing or turing hopf bifurcations respectively depending on the product of the strength of the feedback and the length of the delay the remainder of the paper is focused on turing bifurcations where the delay can drive the appearance of an additional time scale intermediate to the usual slow and fast time scales observed in the modulation of rolls without delay in the deterministic case a ginzburg landau type modulation equation is derived that inherits pyragas control terms from the original equation the eckhaus stability criteria is obtained for the rolls with the intermediate time scale observed in the transients in the stochastic context slow modulation equations are derived for the amplitudes of the primary modes that are coupled to a fast ornstein uhlenbeck type equation with delay for the zero mode driven by the additive noise by deriving an averaging approximation for the amplitude of the primary mode we show how the interaction of noise and delay influences the existence and stability range for the noisy roll type patterns furthermore approximations for the spectral densities of the primary and zero modes show that oscillations on the intermediate times scale are sustained through the phenomenon of coherence resonance these dynamics on the intermediate time scale are sustained through the interaction of noise and delay in contrast to the deterministic context where dynamics on the intermediate times scale are transient

modifications of real code genetic algorithm for global optimization a series of modifications to the real coded genetic algorithm for the task of locating the global minimum of multidimensional functions are presented here these modifications include a new stopping rule a novel mutation mechanism and a periodically application of a local search procedure the proposed modifications are tested on a series of optimization problems and the results are reported

oscillon type structures and their interaction in a swift hohenberg model abstract motivated by the observation of localized circular excitations oscillons in vertically vibrated granular layers p umbanhowar f melo h swinney nature 382 1996 793 we numerically investigate an extension of a swift hohenberg model that exhibits a subcritical transition to square patterns for sufficiently strong damping of the basic state stable oscillon structures are found the localization mechanism is quite general and is due to non adiabatic effects bound structures of oscillons of equal and opposite polarity are found with bound states of like polarity being less robust much of the phenomena are consistent with the experimental observations and suggest that oscillons are not specific to patterns in granular media or to parametrically driven systems experimental tests are suggested that would determine whether this minimal framework is sufficient to describe the phenomena

weak and strong solutions of a nonlinear subsonic flow structure interaction semigroup approach abstract we consider a non rotational subsonic flow structure interaction describing the flow of gas above a flexible plate a perturbed wave equation describes the flow and a second order nonlinear plate equation describes the plate s displacement it is shown that the linearization of the model generates a strongly continuous semigroup with respect to the topology generated by finite energy considerations an interesting feature of the problem is that linear perturbed flow structure interaction is not monotone with respect to the standard norm describing the finite energy space the main tool used in overcoming this difficulty is the construction of a suitable inner product on the finite energy space which allows the application of maximal monotone operator theory the obtained result allows us to employ suitable perturbation theory in order to discuss well posedness of weak and strong solutions corresponding to several classes of nonlinear dynamics including the full flow structure interaction with von karman berger s and other semilinear plate equations

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

stabilization of wave and plate like equations with nonlinear dissipation on the boundary abstract in this paper we consider second order in time equations of petrovsky type wave and plate like with nonlinear dissipative boundary conditions under certain geometric conditions imposed on the domain the results on asymptotic behavior of the solutions are established

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

a study of moving mesh pde methods for numerical simulation of blowup in reaction diffusion equations a new concept called the dominance of equidistribution is introduced for analyzing moving mesh partial differential equations for numerical simulation of blowup in reaction diffusion equations theoretical and numerical results show that a moving mesh method works successfully when the employed moving mesh equation has the dominance of equidistribution the property can be verified using dimensional analysis in several aspects the current work generalizes previous work where a moving mesh equation is shown to have this dominance of equidistribution if it preserves the scaling invariance of the underlying physical partial differential equation and uses a small constant value for t a parameter used for adjusting response time of the mesh movement to the change in the physical solution also cases with both constant and variable t are considered here

parallel genetic algorithms applied to the traveling salesman problem genetic algorithms are adaptive search algorithms that have been shown to be robust optimization algorithms for multimodal real valued functions and a variety of combinatorial optimization problems in contrast to more standard search algorithms genetic algorithms base their progress on the performance of a population of candidate solutions rather than on a single candidate solution the authors will concentrate on the application of genetic algorithms to the traveling salesman problem for this problem there exist several such algorithms ranging from pure genetic algorithms to genetic algorithms that incorporate heuristic information these algorithms will be reviewed and their performance contrasted a serious drawback of genetic algorithms is their inefficiency when implemented on a sequential machine however due to their inherent parallel properties they can be successfully implemented on parallel machines resulting in considerable speedup parallel genetic algorithms will be reviewed and their

smooth attractors of finite dimension for von karman evolutions with nonlinear frictional damping localized in a boundary layer abstract in this paper dynamic von karman equations with localized interior damping supported in a boundary collar are considered hadamard well posedness for von karman plates with various types of nonlinear damping are well known and the long time behavior of nonlinear plates has been a topic of recent interest since the von karman plate system is of hyperbolic type with critical nonlinearity noncompact with respect to the phase space this latter topic is particularly challenging in the case of geometrically constrained nonlinear damping in this paper we first show the existence of a compact global attractor for finite energy solutions and we then prove that the attractor is both smooth and finite dimensional thus the hyperbolic like flow is stabilized asymptotically to a smooth and finite dimensional set

large time behavior via the method of l trajectories abstract the method of l trajectories is presented in a general setting as an alternative approach to the study of the large time behavior of nonlinear evolutionary systems it can be successfully applied to the problems where solutions suffer from lack of regularity or when the leading elliptic operator is nonlinear here we concentrate on systems of a parabolic type and apply the method to an abstract nonlinear dissipative equation of the first order and to a class of equations pertinent to nonlinear fluid mechanics in both cases we prove the existence of a finite dimensional global attractor and the existence of an exponential attractor

analysis and numerical simulation of positive and dead core solutions of singular two point boundary value problems we investigate the solvability of the dirichlet boundary value problem u t lg u t l 0 u 0 1 u 1 1 where l is a nonnegative parameter we discuss the existence of multiple positive solutions and show that for certain values of l there also exist solutions that vanish on a subinterval r 1 r 0 1 the so called dead core solutions in order to illustrate the theoretical findings we present computational results for g u 1 u computed using the collocation method implemented in bvpsuite a new version of the standard matlab code sbvp1 0

global attractors for a third order in time nonlinear dynamics abstract long time behavior of a third order in time nonlinear pde equation is considered this type of equations arises in the context of nonlinear acoustics 12 20 22 24 where modeling accounts for a finite speed of propagation paradox the latter results in hyperbolic nature of the dynamics it will be proved that the underlying pde generates a well posed dynamical system which admits a global and finite dimensional attractor the main difficulty associated with the problem studied is the lack of lyapunov function along with the lack of compactness of trajectories which fact prevents applicability of standard tools in the area of dynamical systems

a numerical study for multiple solutions of a singular boundary value problem arising from laminar flow in a porous pipe with moving wall abstract this paper is concerned with multiple solutions of a singular nonlinear boundary value problem bvp on the interval 0 1 which arises in a study of the laminar flow in a porous pipe with an expanding or contracting wall for the singular nonlinear bvp the correct boundary conditions are derived to guarantee that its linearization has a unique smooth solution then a numerical technique is proposed to find all possible multiple solutions for the suction driven pipe flow with the expanding wall e g 2 we find a new solution numerically and classify it as a type vi solution the computed results agree well with what can be obtained by the bifurcation package auto in addition we also construct asymptotic solutions for a few cases of parameters which agree well with numerical solutions these serve as validations of our numerical results thus we believe that the numerical technique designed in the paper is reliable and may be further applied to solve a variety of nonlinear equations that arise from other flow problems

convergence to steady states in asymptotically autonomous semilinear evolution equations abstract we study the convergence to equilibrium of bounded solutions of the nonautonomous first order problem u m u g t t r and of the second order problem u u m u g t t r applications to diffusion wave cahn hilliard and kirchhoff carrier equations are described

long time dynamics of von karman semi flows with non linear boundary interior damping abstract we study von karman evolution equations with non linear dissipation and with partially clamped and partially free boundary conditions two distinctive mechanisms of dissipation are considered i internal dissipation generated by non linear operator and ii boundary dissipation generated by shear forces friction acting on a free part of the boundary the main emphasis is given to the effects of boundary dissipation under suitable hypotheses we prove existence of a compact global attractor and finiteness of its fractal dimension we also show that any solution is stabilized to an equilibrium and estimate the rate of the convergence which in turn depends on the behaviour at the origin of the functions describing the dissipation

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

global attractors for von karman evolutions with a nonlinear boundary dissipation abstract dynamic von karman equations with a nonlinear boundary dissipation are considered questions related to long time behaviour existence and structure of global attractors are studied it is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global compact attractor for all weak finite energy solutions this result has been known in the case of full interior dissipation but it is new in the case when the boundary damping is the main dissipative mechanism in the system in addition we prove that fractal dimension of the attractor is finite the proofs depend critically on the infinite speed of propagation associated with the von karman model considered

mesh adaptation on the sphere using optimal transport and the numerical solution of a monge ampere type equation abstract an equation of monge ampere type has for the first time been solved numerically on the surface of the sphere in order to generate optimally transported ot meshes equidistributed with respect to a monitor function optimal transport generates meshes that keep the same connectivity as the original mesh making them suitable for r adaptive simulations in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers the semi implicit solution of the monge ampere type equation involves a new linearisation of the hessian term and exponential maps are used to map from old to new meshes on the sphere the determinant of the hessian is evaluated as the change in volume between old and new mesh cells rather than using numerical approximations to the gradients ot meshes are generated to compare with centroidal voronoi tessellations on the sphere and are found to have advantages and disadvantages ot equidistribution is more accurate the number of iterations to convergence is independent of the mesh size face skewness is reduced and the connectivity does not change however anisotropy is higher and the ot meshes are non orthogonal it is shown that optimal transport on the sphere leads to meshes that do not tangle however tangling can be introduced by numerical errors in calculating the gradient of the mesh potential methods for alleviating this problem are explored finally ot meshes are generated using observed precipitation as a monitor function in order to demonstrate the potential power of the technique

the application of runge kutta schemes to singular initial value problems a theory for explicit runge kutta schemes applied to the initial value problem for a first order system of differential equations with a singularity of the first kind is developed it is shown that in general the order of convergence is at most two but that the usual order up to a logarithmic term can be obtained for level three and four schemes applied to specific problems 1 introduction in this paper we examine the application of explicit runge kutta schemes to the numerical solution of the problem

pattern selection and localized structures in reaction diffusion systems abstract we present some theoretical concepts that have been used in the study of chemical disspative structures together with a brief description of recent experimental work on turing patterns and their interaction with travelling waves

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

a class of trust region methods for parallel optimization we present a new class of optimization methods that incorporates a parallel direct search pds method within a trust region newton framework this approach combines the inherent parallelism of pds with the rapid and robust convergence properties of newton methods numerical tests have yielded favorable results for both standard test problems and engineering applications in addition the new method appears to be more robust in the presence of noisy functions which are inherent in many engineering simulations

global attractors for von karman equations with nonlinear interior dissipation abstract in this paper we study the asymptotic behavior of weak solutions for von karman equations with nonlinear interior dissipation we prove the existence of a global attractor in the space w 2 2 l 2

global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain abstract in this paper we study the asymptotic behavior of solutions for the plate equation with a localized damping and a critical exponent we prove the existence regularity and finite dimensionality of a global attractor in w 2 2 r n l 2 r n

finite dimensional attractors for the quasi linear strongly damped wave equation abstract we present a new method of investigating the so called quasi linear strongly damped wave equations t 2 u t x u x u f u x x u g in bounded 3d domains this method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non linearity is less than 6 and f may have arbitrary polynomial growth rate moreover the existence of a finite dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established in a particular case 0 which corresponds to the so called semi linear strongly damped wave equation our result allows to remove the long standing growth restriction f u c 1 u 5

convergence and decay estimates for a class of second order dissipative equations involving a non negative potential energy abstract we estimate the rate of decay of the difference between a solution and its limiting equilibrium for the following abstract second order problem u t g u t m u t 0 t r where m is the gradient operator of a non negative functional and g is a non linear damping operator under some conditions relating the ojasiewicz exponent of the functional and the growth of the damping around the origin the main result is applied to non linear wave or plate equations in some cases direct constructive proofs of the ojasiewicz gradient inequality are given applicable to some non analytic functionals in presence of multiple critical points at the end similar results are obtained when a fast decaying source term is added in the right hand side

trace regularity of the solutions of the wave equation with homogeneous neumann boundary conditions and data supported away from the boundary let rn be a smooth domain with boundary let u be the solution of a second order hyperbolic scalar equation with homogeneous neumann boundary conditions defined on due to initial conditions u0 u1 and to a forcing term the present paper studies the regularity of the map u0 u1 u where u0 u1 l2 q h1 l2 q 0 t and u is the dirichlet trace of the solution in particular the case of compactly supported data u0 u1 is considered and contrasted with corresponding results where the data do not have compact support in this latter case an optimal regularity result is given for the wave equation on the half space of dimension 2 it states contrary to expectations from elliptic or parabolic theory that u h34 but u h34 e e 0 where 0 t

differential algebraic equations with properly stated leading terms in this paper we study differential algebraic equations daes of the form a g t d g t b g t 0 with in some sense well matched matrix functions a g t and d g t d g g t as they arise e g in circuit simulation we characterize index 1 daes in this context after analyzing those index 1 equations themselves we apply runge kutta methods and bdfs provide stability inequalities and show convergence the cases of the image space of d g t or the nullspace of a g t remaining constant are pointed out to be essentially favourable for the qualitative behaviour of the approximations on long intervals hence when modelling with daes one should try for those constant subspaces relations to quasilinear daes in standard formulation e g t g g t 0 are considered too

collocation for singular boundary value problems of second order the collocation methods for solving linear boundary value problems with a singularity of the first kind are investigated the stability and convergence results are established particular attention is paid to the superconvergence properties of the methods which are different as in the classical case when the singularity is not present the results are extended to the nonlinear case and illustrated by numerical examples

the łojasiewicz gradient inequality in the infinite dimensional hilbert space framework abstract we provide a reasonably optimal answer to the natural question of the conditions under which an analytic function on an infinite dimensional hilbert space satisfies the ojasiewicz gradient inequality

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

a posteriori error estimation for hp adaptivity for fourth order equations a posteriori error estimates developed to drive hp adaptivity for second order reaction diffusion equations are extended to fourth order equations a c 1 hierarchical finite element basis is constructed from hermite lobatto polynomials a priori estimates of the error in several norms for both the interpolant and finite element solution are derived in the latter case this requires a generalization of the well known aubin nitsche technique to time dependent fourth order equations we show that the finite element solution and corresponding hermite lobatto interpolant are asymptotically equivalent a posteriori error estimators based on this equivalence for solutions at two orders are presented both are shown to be asymptotically exact on grids of uniform order these estimators can be used to control various adaptive strategies computational results for linear steady state and time dependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings

symmetric collocation methods for linear differential algebraic boundary value problems we present symmetric collocation methods for linear differential algebraic boundary value problems without restrictions on the index or the structure of the differential algebraic equation in particular we do not require a separation into differential and algebraic solution components instead we use the splitting into differential and algebraic equations which arises naturally by index reduction techniques and apply gaus type for the differential part and lobatto type for the algebraic part collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points under standard assumptions we show solvability and stability of the discrete problem and determine its order of convergence moreover we show superconvergence when using the combination of gaus and lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations finally we present some numerical comparisons to show the reliability and efficiency of the new methods

asymptotically correct error estimation for collocation methods applied to singular boundary value problems we discuss an a posteriori error estimate for collocation methods applied to boundary value problems in ordinary differential equations with a singularity of the first kind as an extension of previous results we show the asymptotical correctness of our error estimate for the most general class of singular problems where the coefficient matrix is allowed to have eigenvalues with positive real parts this requires a new representation of the global error for the numerical solution obtained by piecewise polynomial collocation when applied to our problem class

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

asymptotics of cellular buckling close to the maxwell load we study the deformation of an elastic strut on a nonlinear winkler foundation subjected to an axial compressive load p using multi scale analysis and numerical methods we describe the localized cellular post buckled state of the system when p is removed from the critical load p 2 the solutions and their modulation frequencies differ significantly from those predicted by weakly nonlinear analysis very close to p 2 in particular when p approaches the maxwell load pm the localized solutions approach a large amplitude heteroclinic connection between an unbuckled solution and a periodic solution an asymptotic description of pm in terms of the system parameters is given the agreement between the numerical calculations and the asymptotic approximations is striking

analytical and numerical treatment of a singular initial value problem in avalanche modeling we discuss a leading edge model used in the computation of the run out length of dry flowing avalanches the model has the form of a singular initial value problem for a scalar ordinary differential equation describing the avalanche dynamics existence uniqueness and smoothness properties of the analytical solution are shown we also prove the existence of a unique root of the solution moreover we present a fortran 90 code for the numerical computation of the run out length the code is based on a solver for singular initial value problems which is an implementation of the acceleration technique known as iterated defect correction based on the implicit euler method

a nonhyperbolic cauchy problem for b c and its applications to elasticity theory on etudie l unicite et la stabilite d une fonction u satisfaisant les conditions suivantes b c u a 3 u f sur t t g n j u g j sur t t j 3 ou b u bu tt u a k est un operateur differentiel d ordre k g r n une partie de sa frontiere et n la normale a

abundance of stable stationary localized solutions to the generalized 1d swift hohenberg equation abstract a number of stationary localized solutions to the well known pattern forming gradient system mentioned in the title have been found their search is based on the theory of homoclinic orbits to a saddle focus equilibrium and some results of linear symmetric differential operators with decaying coefficients along with computer simulations

on a singular boundary value problem arising in the theory of shallow membrane caps abstract we investigate the following singular boundary value problem which originates from the theory of shallow membrane caps t 3 u t t 3 1 8 u 2 t a 0 u t b 0 t 2 4 0 lim t 0 t 3 u t 0 u 1 0 where a 0 b 0 and are given constants we show the existence of a positive solution to the above problem by means of a generalized lower and upper functions method involving limiting processes we illustrate the theory by numerical experiments in which we used the new version of the matlab code sbvp based on polynomial collocation to approximate the solution of the membrane problem

on localised hotspots of an urban crime model abstract we investigate stationary spatially localised crime hotspots on the real line and the plane of an urban crime model of short et al m short m dorsogna a statistical model of criminal behavior mathematical models and methods in applied sciences 18 2008 1249 1267 extending the weakly nonlinear analysis of short et al we show in one dimension that localised hotspots should bifurcate off the background spatially homogeneous state at a turing instability provided the bifurcation is subcritical using path following techniques we continue these hotspots and show that the bifurcating pulses can undergo the process of homoclinic snaking near the singular limit we analyse the singular limit to explain the existence of spike solutions and compare the analytical results with the numerical computations in two dimensions we show that localised radial spots should also bifurcate off the spatially homogeneous background state localised planar hexagon fronts and hexagon patches are found and depending on the proximity to the singular limit these solutions either undergo homoclinic snaking or act like multi spot solutions finally we discuss applications of these localised patterns in the urban crime context and the full agent based model

front motion metastability and subcritical bifurcations in hydrodynamics metastability occurs whenever under given external conditions as defined by the reynolds number in hydrodynamics more than one linearly stable solution may exist in variational problems the preferred state is usually claimed to be the one with the lowest energy but in hydrodynamics and except for very special conditions no energy functional exists from an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations as turbulent and laminar in pipe flows or boundary layers moves with a constant mean velocity depending on the control parameter the sign of this velocity allows to decide which state is stable and which is metastable possible physical implications of these ideas for various problems of hydrodynamics are discussed

localized periodic patterns for the non symmetric generalized swift hohenberg equation abstract a new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized swift hohenberg equation an analysis of this envelope equation results in parametric conditions for localized patterns furthermore it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case the analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing hamiltonian and lagrangian for periodic solutions one striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an o 1 distance from the bifurcation thus providing a novel approach for studying these localized patterns

adaptive solution to two dimensional partial differential equations in curved domains using the monge ampere equation we present a finite difference method for mesh generation and adaptation with application to the solution of partial differential equations in curved domains for mesh generation we construct a ma

efficient mesh selection for collocation methods applied to singular bvps we describe a mesh selection strategy for the numerical solution of boundary value problems for singular ordinary differential equations this mesh adaptation procedure is implemented in our matlab code sbvp which is based on polynomial collocation we prove that under realistic assumptions our mesh selection strategy serves to approximately equidistribute the global error of the collocation solution thus enabling to reach prescribed tolerances efficiently moreover we demonstrate that this strategy yields a favorable performance of the code and compare its computational effort with other implementations of polynomial collocation

parabolic monge ampere methods for blow up problems in several spatial dimensions this paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular pdes in one or more spatial dimensions the scheme is based on computing a legendre transformation from a regular to a spatially non uniform mesh via the solution of a relaxed form of the monge ampere equation the method is shown to preserve the inherent scaling properties of the pde and to identify natural computational coordinates numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach

exact semi internal control of an euler bernoulli equation this paper proves exact semi internal controllability of an euler bernoulli equation with variable coefficients the basic principle of the proof is the hilbert uniqueness method for this a multiplier technique and a unique continuation property have been used the proof of the unique continuation property is the main feature of this work

adaptive geometric integrators for hamiltonian problems with approximate scale invariance we consider adaptive geometric integrators for the numerical integration of hamiltonian systems with greatly varying time scales a time regularization is considered using either the sundman or the poincare transformation in the latter case this gives a new hamiltonian which is usually separable but with one of the parts not always exactly solvable this system can be numerically integrated with a splitting scheme where each part can be computed using a symplectic implicit or explicit method preserving the qualitative properties of the exact solution in this case a backward error analysis for the numerical integration is presented for a one dimensional near singular problem this analysis reveals a strong dependence of the performance of the method with the choice of the monitor function g which is also observed when using other symmetric nonsymplectic integrators we also show how this dependence greatly increases with the order of the numerical integrator used the optimal choice corresponds to the function g which nearly preserves the scaling invariance of the system numerical examples supporting this result are presented in some cases a canonical transformation can also be considered making the system more regular or easy to compute and this is also illustrated with some examples

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude