recovery of a time dependent source from a surface measurement in maxwell s equations we study an inverse source problem arising in computations of electromagnetic fields governed by maxwell s equations the unknown source is solely time dependent and it will be recovered from a surface measurement the existence and uniqueness of a solution to this inverse problem is shown moreover a constructive algorithm for approximations is designed and its convergence is established

a source identification problem in a time fractional wave equation with a dynamical boundary condition abstract we focus on a inverse source problem in a partial differential equation containing a fractional derivative in time of the order 1 2 the equation is accompanied with a non standard boundary condition which consists of the classic flux term and the dynamical time fractional derivative term on the one part of the boundary to determine both the solution of the equation and the source term a measurement in a form of a integral over space domain is considered using a time discretization and rothe s method we prove an existence of the strong solution in the suitable functional spaces and the error estimate is established moreover the uniqueness is addressed the results are supported by some numerical experiments

identifying the diffusion coefficient by optimization from the final observation this paper deals with the inverse problem of determining a pair a u in the reaction diffusion equationu t au x x f x t with initial and boundary conditionsu x 0 email protected x u x x 0 u x x 1 0 from the final measurement data u x t z x which has important application in a large fields of applied science based on the optimal control framework the existence uniqueness and stability of the minimizer for the cost functional are established a necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced the gradient iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples

an averaging theorem for hamiltonian dynamical systems in the thermodynamic limit it is shown how to perform some steps of perturbation theory if one assumes a measure theoretic point of view i e if one renounces to control the evolution of the single trajectories and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase space for a system of coupled rotators estimates uniform in n for finite specific energy can be obtained in quite a direct way this is achieved by making reference not to the sup norm but rather following koopman and von neumann to the much weaker l2 norm

time scales to equipartition in the fermi pasta ulam problem finite size effects and thermodynamic limit we investigate numerically the common and the pure fpu models as well as some higher order generalizations we consider initial conditions in which only low frequency normal modes are excited and perform a very accurate systematic study of the equilibrium time as a function of the number n of particles the specific energy e and the parameters and while at any fixed n the equilibrium time is found to be a stretched exponential in 1 e in the thermodynamic limit i e for n at fixed e we observe a crossover to a power law concerning the usually disregarded dependence of teq on and we find it is nontrivial and propose and test a general law a central role is played by the comparison of the fpu models with the toda model

determination of a time dependent heat source under nonlocal boundary and integral overdetermination conditions abstract this paper investigates the inverse problem of finding a time dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions under some natural regularity and consistency conditions on the input data the existence uniqueness and continuously dependence upon the data of the solution are shown by using the generalized fourier method numerical tests using the crank nicolson finite difference scheme combined with an iterative method are presented and discussed

existence results for inverse problems associated with a nonlinear parabolic equation this paper deals with the determination of a pair p u in the nonlinear parabolic equation ut u p x f u 0 with initial and boundary conditions u x 0 0 u 0 t g from the overspecified data u t q the problem is reduced to a nonlinear operator equation and solved by taking advantage of a fixed point theory in ordered banach spaces similarly another form of overspecified data is considered

a series expansion for the time autocorrelation of dynamical variables we present here a general iterative formula which gives a formal series expansion for the time autocorrelation of smooth dynamical variables for all hamiltonian systems endowed with an invariant measure we add some criteria theoretical in nature which enable one to decide whether the decay of the correlations is exponentially fast or not one of these criteria is implemented numerically for the case of the fermi pasta ulam system and we find indications which might suggest a sub exponential decay of the time autocorrelation of a relevant dynamical variable

on the problem of energy equipartition for large systems of the fermi pasta ulam type analytical and numerical estimates abstract we report on some analytical and numerical results on the exchanges of energy in systems of the fermi pasta ulam type in the light of nekhoroshev s theorem with particular attention to the dependence of the estimates on the number n of degrees of freedom for the ordinary fpu problem we look for a control of the single normal mode energies and we find both the analytical and numerical estimates to agree in predicting that the energy exchanges of the single modes cannot be controlled in the thermodynamic limit we consider then a modified fpu model with alternating light and heavy particles which appears as composed of two subsystems of low acoustic frequency and of high optical frequency respectively we try to control the exchange of the total energy of the high frequency modes up to times increasing exponentially with the frequency in this case the numerical estimates are stronger than the available analytical ones and give indications for nonequipartition with constants essentially independent of the number n of degrees of freedom

extensive adiabatic invariants for nonlinear chains we look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit considering the quadratic part of the klein gordon hamiltonian by a linear change of variables we transform it into a sum of two parts in involution at variance with the usual method of introducing normal modes our constructive procedure allows us to exploit the complete resonance while keeping the extensive nature of the system next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear schrodinger model the fluctuations of this quantity are controlled via gibbs measure estimates independent of the system size for a large set of initial data at low specific energy finally by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory with fluctuation much smaller than expected according to standard statistical estimates

on metastability in fpu we present an analytical study of the fermi pasta ulam fpu model with periodic boundary conditions we analyze the dynamics corresponding to initial data with one low frequency fourier mode excited we show that correspondingly a pair of kdv equations constitute the resonant normal form of the system we also use such a normal form in order to prove the existence of a metastability phenomenon more precisely we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes subsequently the distribution remains unchanged up to the time scales of validity of our approximation the phenomenon is controlled by the specific energy

relaxation times for hamiltonian systems usually the relaxation times of a gas are estimated in the frame of the boltzmann equation in this paper instead we deal with the relaxation problem in the frame of the dynamical theory of hamiltonian systems in which the definition itself of a relaxation time is an open question we introduce a lower bound for the relaxation time and give a general theorem for estimating it then we give an application to a concrete model of an interacting gas in which the lower bound turns out to be of the order of magnitude of the relaxation times observed in dilute gases

the heat flux identification problem for a nonlinear parabolic equation in 2d we consider the heat flux identification problem hfip based on the boundary measurements for a nonlinear parabolic equation in 2 dimensional space the standard linearization algorithm is applied to the nonlinear direct problem the method of conjugate gradient algorithm based on the gradient formula for the cost functional is then proposed for numerical solution of the inverse heat flux problem numerical analysis of the algorithm applied to the inverse problem in typical classes of flux functions is presented computational results obtained for random noisy output data indicate how the iteration number of the conjugate gradient algorithm can be estimated numerical results illustrate bounds of applicability of proposed algorithm as well as its efficiency and accuracy

birkhoff coordinates for the toda lattice in the limit of infinitely many particles with an application to fpu abstract in this paper we study the birkhoff coordinates cartesian action angle coordinates of the toda lattice with periodic boundary condition in the limit where the number n of the particles tends to infinity we prove that the transformation introducing such coordinates maps analytically a complex ball of radius r n in discrete sobolev analytic norms into a ball of radius r n with r r 0 independent of n if and only if 2 then we consider the problem of equipartition of energy in the spirit of fermi pasta ulam we deduce that corresponding to initial data of size r n 2 0 r 1 and with only the first fourier mode excited the energy remains forever in a packet of fourier modes exponentially decreasing with the wave number finally we consider the original fpu model and prove that energy remains localized in a similar packet of fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet the proof of the theorem on birkhoff coordinates is based on a new quantitative version of a vey type theorem by kuksin and perelman which could be interesting in itself

realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory part ii the so called problem of the realization of the holonomic constraints of classical mechanics is here revisited in the light of nekhoroshev like classical perturbation theory precisely if constraints are physically represented by very steep potential wells with associated high frequency transversal vibrations then one shows that within suitable assumptions the vibrational energy and the energy associated to the constrained motion are separately almost constant for a very long time scale growing exponentially with the frequency i e with the rigidity of the constraint one aims to realize this result can also be applied to microscopic physics providing a possible entirely classical mechanism for the freezing of the high frequency degrees of freedom in terms of non equilibrium statistical mechanics according to some ideas expressed by boltzmann and jeans at the turn of the century in this part i we introduce the problem and prove a first theorem concerning the realization of a single constraint within a system of any number of degrees of freedom the problem of the realization of many constraints will be considered in a forthcoming part ii

an averaging theorem for fpu in the thermodynamic limit consider an fpu chain composed of n gg 1 particles and endow the phase space with the gibbs measure corresponding to a small temperature beta 1 given a fixed k we construct k packets of normal modes whose energies are adiabatic invariants i e are approximately constant for times of order beta 1 a a 0 for initial data in a set of large measure furthermore the time autocorrelation function of the energy of each packet does not decay significantly for times of order beta the restrictions on the shape of the packets are very mild all estimates are uniform in the number n of particles and thus hold in the thermodynamic limit n rightarrow infty beta 0

determination of a solely time dependent source in a semilinear parabolic problem by means of boundary measurements a semilinear parabolic problem of second order with an unknown solely time dependent source term is studied the missing source is recovered from an additional integral measurement over the boundary the global in time existence uniqueness as well as the regularity of a solution are addressed a new numerical scheme based on rothe s method is designed and convergence of iterates towards the exact solution is shown

simultaneous determination of source terms in a linear parabolic problem from the final overdetermination weak solution approach the problem of determining the pair w f x t t0 t of source terms in the parabolic equation ut k x ux x f x t and robin boundary condition k l ux l t v u l t t0 t from the measured final data t x u x t is formulated it is proved that both components of the frechet gradient of the cost functional j w t x u x t w 02 can be found via the same solution of the adjoint parabolic problem lipschitz continuity of the gradient is derived the obtained results permit one to prove existence of a quasi solution of the considered inverse problem as well as to construct a monotone iteration scheme based on a gradient method

analytical approximations and pade approximants for volterra s population model in this paper an analytic approximation for volterra s model for population growth of a species in a closed system is presented the nonlinear integro differential model includes an integral term that characterizes accumulated toxicity on the species in addition to the terms of the logistic equation the series solution method and the decomposition method are implemented independently to the model and to a related ode the pade approximants that often show superior performance over series approximations are effectively used in the analysis to capture the essential behavior of the population u t of identical individuals

asymptotic localization of energy in nondisordered oscillator chains we study two popular one dimensional chains of classical anharmonic oscillators the rotor chain and a version of the discrete nonlinear schrodinger chain we assume that the interaction between neighboring oscillators controlled by the parameter 0 is small we rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant and we provide strong evidence that it decays faster than any power law in as 0 the weak coupling regime also translates into a high temperature regime suggesting that the conductivity vanishes faster than any power of the inverse temperature to our knowledge it is the first time that a clear connection has been established between kam like phenomena and thermal conductivity 2015 wiley periodicals inc

identification of a source in a fractional wave equation from a boundary measurement abstract we deal with an inverse source problem in a time fractional wave equation the time dependent source term is reconstructed using the additional non invasive measurement in the form of integral over a part of the boundary we look for the solution and the source term obeying the variational formulation and the equation obtained from applying the measurement on the equation using the rothe method the existence of the solution is proven the uniqueness is shown in appropriate spaces and some numerical experiments are presented

a parabolic inverse source problem with a dynamical boundary condition an inverse source problem for the heat equation is studied in a bounded domain a dynamical nonlinear boundary condition containing the time derivative of a solution is prescribed on one part of the boundary this models a non perfect contact on the boundary the missing purely time dependent source is recovered from an additional integral measurement the global in time existence and uniqueness of a solution in corresponding function spaces is addressed using the backward euler method for the time discretization error estimates for time discrete approximations are derived