fractional laplacian time space models for linear and nonlinear lossy media exhibiting arbitrary frequency power law dependency frequency dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2 the standard time domain partial differential equation models can describe merely two extreme cases of frequency independent and frequency squared dependent attenuations the otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation in this study a linear integro differential equation wave model was developed for the anomalous attenuation by using the space fractional laplacian operation and the strategy is then extended to the nonlinear burgers equation a new definition of the fractional laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional laplacian under the szabo s smallness approximation where attenuation is assumed to be much smaller than the wave number the linear model is found consistent with arbitrary frequency power law dependency

neumann problems with time singularities in this paper we study the existence and uniqueness of solutions to a nonlinear neumann problem for a scalar second order ordinary differential equation u atu f t u u where a 0 and f t x y satisfies the local caratheodory conditions on 0 t xrxr

new exact solutions of differential equations derived by fractional calculus abstract fractional calculus generalizes the derivative and antiderivative operations d n d z n of differential and integral calculus from integer orders n to the entire complex plane methods are presented for using this generalized calculus with laplace transforms of complex order derivatives to solve analytically many differential equations in physics facilitate numerical computations and generate new infinite series representations of functions as examples new exact analytic solutions of differential equations including new generalized bessel equations with complex power law variable coefficients are derived

periodic solutions for damped differential equations with a weak repulsive singularity abstract this paper deals with the existence of positive t periodic solutions for the following damped differential equation x p t x q t x f t x c t where p q c l 1 r are t periodic functions and f car r r r is t periodic in the first variable we will prove that a weak repulsive singularity enables the achievement of new existence criteria through a basic application of schauder s fixed point theorem

fractal ladder models and power law wave equations extensive measurement of the ultrasonic attenuation coefficient in human and mammalian tissue has revealed a power law dependence on frequency to describe this power law behavior in tissue a hierarchical fractal network model is proposed the viscoelastic and self similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite ladder topology involving alternating springs and dashpots in the low frequency limit this ladder network yields a stress strain constitutive equation with a time fractional derivative by combining this constitutive equation with linearized conservation principles and an adiabatic equation of state the fractional partial differential equations previously proposed by m caputo linear models of dissipation whose q is almost frequency independent ii geophys j r astron soc 13 529 539 1967 and mg wismer finite element analysis of broadband acoustic pulses through inhomogeneous media with power law attenuation j acoust soc am 120 3493 3502 2006 are derived the resulting attenuation coefficient is a power law with exponents ranging between 1 and 2 while the phase velocity is in agreement with the kramers kronig relations the power law exponent is interpreted in terms of the mechanical structure and related to the spectral dimension of the underlying fractal geometry

derivative computations for the log likelihood function this note improves on a method for computing derivatives of the log likelihood function used in the identification of discrete linear time invariant gaussian systems a formula for the derivative is obtained in terms of the derivatives of the matrices used to define the system under identification

layered incompressible fluid flow equations in the limit of low mach number and strong stratification abstract we investigate the low mach number regimes of the complete navier stokes fourier system which describes an evolution of a viscous heat conducting gas with a heat source term interpreted as a radiation cooling in models of atmospheric flows this investigation is performed in the context of weak solutions on an arbitrary large time interval and for the ill prepared initial data the physically expected limiting equations as the mach and froude numbers tend to zero at the same rate are layered incompressible flow equations with strong stratification we give a rigorous mathematical proof of this result

modified szabo s wave equation models for lossy media obeying frequency power law szabo s models of acoustic attenuation szabo j acoust soc am 96 1 491 500 1994 comply well with the empirical frequency power law involving noninteger and odd integer exponent coefficients while guaranteeing causality but nevertheless encounter the troublesome issues of hypersingular improper integral and obscurity in implementing initial conditions the purpose of this paper is to ease or remove these drawbacks of the szabo s models via the caputo fractional derivative concept the positive time fractional derivative is also introduced to include the positivity of the attenuation processes

numerical solutions of linear mixed boundary value problems using stochastic representations the application of ito s formula induces some probabilistic representations of solutions of deterministic linear problems with boundary conditions of dirichlet neumann fourier and mixed types these representations are used to establish some easily implementable algorithms which compute an approximate solution by means of simulation of reflected random walks they do not require selected configurations at the neighbourhood of the domain boundary nor a discretization mesh the associated simulation methods are obtained and applied to problems for each class of boundary conditions numerical experiments with distributed source in two or three dimensional geometries and computational results with estimation of error are reported 1997 by john wiley sons ltd

an analog architecture on parameter estimation of arma models an analog architecture that is suitable for parameter estimation of autoregressive moving average arma models is proposed the convergence theorem that connects this architecture with arma parameter estimation is presented simulation results indicate that its convergence takes only a few microseconds hence this architecture can lead to online implementations

fractional biharmonic operator equation model for arbitrary frequency dependent scattering attenuation in acoustic wave propagation this paper proposes a fractional biharmonic operator equation model in the time space domain to describe scattering attenuation of acoustic waves in heterogeneous media compared with the existing models the proposed fractional model is able to describe arbitrary frequency dependent scattering attenuation which typically obeys an empirical power law with an exponent ranging from 0 to 4 in stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature little has been reported on frequency dependent scattering attenuation this is largely because the order of the fractional laplacian is from 0 to 2 and is infeasible for scattering attenuation in this study the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power law dependency an

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

functional calculus for semigroup generators via transference abstract in this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators in particular it is proved that if a generates a c 0 semigroup on a hilbert space then for each 0 the operator a has a bounded calculus for the closed ideal of bounded holomorphic functions on a sufficiently large right half plane that satisfy f z o e re z as z the bound of this calculus grows at most logarithmically as 0 as a consequence f a is a bounded operator for each holomorphic function f on a right half plane with polynomial decay at then we show that each semigroup generator has a so called strong m bounded calculus for all m n and that this property characterizes semigroup generators similar results are obtained if the underlying banach space is a umd space upon restriction to so called bounded semigroups the hilbert space results actually hold in general banach spaces

on the relation between filter maps and correction factors in likelihood ratios the robust form of the likelihood ratio for a signal in the presence of white noise has an additional term in the exponent called the correction factor which corresponds to the trace of the conditional covariance of the signal given the observations the authors show that this correction term is nothing but the trace of the symmetrized frechet derivative of the nonlinear filter map and hence the likelihood ratio can be completely represented in terms of the observations and the filter map

on radon nikodym derivatives of finitely additive measures induced by nonlinear transformations on hilbert space the study of finitely additive probability measures arises in problems of modelling and estimation of stochastic signals with bounded variation paths in l o t this is because physical signals are of bounded variation and of finite energy for which l o t is the natural setting and results obtained via ito theory hold only on a set of wiener measure 1 it is well known that integrals of paths in jo t lie in a set of wiener measure zero hence there is a need to construct a theory of white noise to model large bandwidth noise arising in signal analysis with the usual properties associated with it in a series of papers l 3 in which he advocated the use of such a framework balakrishnan showed that it could indeed be done and thus laid the basis for a such a theory he considered white noise to be the identity map on a hilbert space h with standard gauss measure thereon it is well known that such a measure is only finitely additive on the algebra of cylinder sets and cannot be extended to the bore1 sets of h this line of work culminated in the excellent treatise by kallianpur and karandikar 4 devoted to filtering and smoothing problems the key assumption in the development of the theory in the nonlinear context so far has been that the signal process is assumed to be defined on a countably additive probability space with paths in h while the measurement noise process is defined on a cylindrical probability space associated with the gauss measure the formulation does not allow for signal noise dependence and one is forced to work with a quasi cylindrical probability space a product space despite the fact that the mathematical difficulties are daunting there nevertheless arises the need to develop a complete theory of white noise in order to study modelling issues as well as signal noise dependence since these issues arise quite naturally in physical problems this paper presents one step in such a direction and is motivated by the issue of likelihood ratio evaluation for signals arising in differential systems driven by white noise the specific problem addressed in this paper is the study of radon nikodym derivatives and their evaluation of cylindrical or finitely additive measures induced by nonlinear transformations on h with standard gauss measure thereon in the linear case this problem was solved by balakrishnan 5 balakrishnan 2 also obtained some results in the nonlinear case when the transformation is given by i k where k defines a homogeneous finite volterra polynomial by exploiting the connection with the pioneering work of cameron and martin 6

bolza problems with discontinuous lagrangians and lipschitz continuity of the value function we study the local lipschitz continuity of the value function v associated with a bolza problem in the presence of a lagrangian l x q convex and uniformly superlinear in q but only borel measurable in x under these assumptions the associated integral functional is not lower semicontinuous with respect to the suitable topology which ensures the existence of minimizers so all results known in the literature fail to apply yet the lipschitz regularity of v does not depend on the existence of minimizers in fact it is enough to control the derivatives of quasi minimal curves but the problem is nontrivial due to the general growth conditions assumed here on l x cdot we propose a new approach based on suitable reparameterization arguments to obtain suitable a priori estimates on the lipschitz constants of quasi minimizers as a consequence of our analysis we derive the lipschitz continuity of v and a compactness result for value functions associated with sequences of locally equibounded discontinuous lagrangians

one dimensional quaternion homogeneous polynomial differential equations in this paper we study the dynamics of real four dimensional polynomial differential equations of the forms q aqkq m or q qkaqm where q a are quaternions q is the conjugate of q and k and m are non negative integers in many parts of the work the quaternion structure is used along the proofs

nonlinear second order equations with functional implicit impulses and nonlinear functional boundary conditions in this paper we deal with nonlinear second order boundary value problems with impulses the impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear in both cases functional dependence on the solution is allowed the existence results follow from the existence of a pair of well ordered lower and upper solutions

symmetry properties and solutions of nonlinear dispersive thin film equations in three dimensions the lie point symmetries are obtained for each one of two nonlinear dispersive thin film equations in 2 1 variables by means of the invariants of the one and two dimensional subgroups all reductions to ordinary differential equations and to partial differential equations in two variables are obtained solutions are provided for a number of the reduced equations it is also ascertained that neither one of the two considered thin film equations is integrable in some cases solitary and periodic wave solutions having curved wavecrests are obtained

representation of the absorption of nonlinear waves by fractional derivatives fractional derivatives of arbitrary order are incorporated into a model nonlinear equation in order to simulate the absorption mechanism in a finite frequency domain that makes it possible to take into account a power dependence of the absorption coefficient on the frequency at any real degree the model gives a mathematical description of a large set of well known absorption mechanisms the method is applied to calculate weakly nonlinear oscillations in a nonrigid tube as a result the value of the absorption coefficient and its dependence on the frequency are found based on experimental data

an improved log likelihood gradient for continuous time stochastic systems with deterministic input using a covariance operator approach we derive an explicit expression for the log likelihood ratio gradient for system parameter estimation for continuous time stochastic systems with deterministic inputs the gradient formula includes the smoother estimates and derivatives of system matrices with no derivatives of estimates or covariance matrices a deterministic input is also permitted and the state noise covariance is not required to be nonsingular stable numerical techniques to calculate the gradient are also discussed

propagation of quasiplane nonlinear waves in tubes and the approximate solutions of the generalized burgers equation this paper deals with using the generalized burgers equation for description of nonlinear waves in circular ducts two new approximate solutions of the generalized burgers equation gbe are presented these solutions take into account the boundary layer effects the first solution is valid for the preshock region and gives more precise results than the fubini solution whereas the second one is valid for the postshock sawtooth region and provides better results than the fay solution the approximate solutions are compared with numerical results of the gbe furthermore the limits of validity of the used model equation are discussed with respect to boundary conditions and radius of a circular duct

on powers of gaussian white noise classical gaussian white noise in communications and signal processing is viewed as the limit of zero mean second order gaussian processes with a compactly supported flat spectral density as the support goes to infinity the difficulty of developing a theory to deal with nonlinear transformations of white noise has been to interpret the corresponding limits in this paper we show that a renormalization and centering of powers of band limited gaussian processes is gaussian white noise and as a consequence homogeneous polynomials under suitable renormalization remain white noises

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

on complementary models and fixed interval smoothing a new algorithm is derived for the standard fixed interval linear smoothing problem in which the signal is generated by a state model the structure of this new algorithm allows the smoothed estimate to be easily updated in response to a change in the initial state covariance matrix pi 0 since unlike in existing algorifiuns the relevant riccati equation is entirely independent of pi 0 the derivation of the algorithm is based on properties of complementary models

a new method for evaluating the log likelihood gradient score of linear dynamic systems a novel method is presented that is based on the optimal smoothing equations the result can be used for efficient calculations and approximations of gradient search algorithms for maximum likelihood estimation of the unknown system parameters the method is applied to the continuous discrete case

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

an approximate solution of a nonlinear fractional differential equation by adomian decomposition method the aim of the present analysis is to apply adomian decomposition method for the solution of a nonlinear fractional differential equation finally the solution obtained by the decomposition method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method a good agreement of the results is observed

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

multiple solutions for nonlocal system of p x q x kirchhoff type abstract in this paper we study the existence of solutions for a neumann elliptic system with variable exponents under some suitable conditions and by applying an equivalent variational approach to a recent ricceri s three critical points theorem we established the existence of at least three weak solutions

a new formula for the log likelihood gradient for continuous time stochastic systems using a finitely additive white noise approach we obtain an explicit expression for the gradient of the log likelihood ratio for system parameter estimation for continuous time linear stochastic systems with noisy observations our gradient formula includes the smoother estimates of the state vector and derivatives of only the system matrices and not the estimates or error covariances a scheme to calculate the log likelihood gradient without solving a riccati equation is described when only one matrix and the initial covariance depend on the unknown parameter

operators of fractional integration and their applications the main purpose of this paper is to present a systematic and historical account of the investigations carried out by various authors in the field of fractional calculus and its applications several interesting results relevant to the present investigation are also considered

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

complete sets of invariants for dynamical systems that admit a separation of variables consider a classical hamiltonian h in n dimensions consisting of a kinetic energy term plus a potential if the associated hamilton jacobi equation admits an orthogonal separation of variables then it is possible to generate algorithmically a canonical basis q p where p1 h p2 pn are the other second order constants of the motion associated with the separable coordinates and qi qj pi pj 0 qi pj ij the 2n 1 functions q2 qn p1 pn form a basis for the invariants we show how to determine for exactly which spaces and potentials the invariant qj is a polynomial in the original momenta we shed light on the general question of exactly when the hamiltonian admits a constant of the motion that is polynomial in the momenta for n 2 we go further and consider all cases where the hamilton jacobi equation admits a second order constant of the motion not necessarily associated with orthogonal separable coordinates or even separable coordinates at all in each of these cases we construct an addition

planar nonautonomous polynomial equations the riccati equation abstract we give a few sufficient conditions for the existence of two periodic solutions of the riccati ordinary differential equation in the plane we give also examples of the equation without periodic solutions

a boundary element numerical scheme for the two dimensional convection diffusion equation in this study the theoretical and numerical fundamentals of biem techniques for the two dimensional convection diffusion problem are presented after an extended presentation of the basic integral formulation the discretizing and iterative processes for its resolution are introduced interesting remarks on general expressions versus previously published particularized results are worth mentioning a numerical solution scheme is provided which has been completely developed and designed to the physical problem posed a novel scheme based in the simultaneous solving of the potential and the gradient of the potential boundary integral equations is included a diversity of problems is tested to prove the possibilities of the method copyright 2008 john wiley sons ltd

the 16th hilbert problem on algebraic limit cycles abstract for real planar polynomial differential systems there appeared a simple version of the 16th hilbert problem on algebraic limit cycles is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m in j llibre r ramirez n sadovskaia on the 16th hilbert problem for algebraic limit cycles j differential equations 248 2010 1401 1409 llibre ramirez and sadovskaia solved the problem providing an exact upper bound in the case of invariant algebraic curves generic for the vector fields and they posed the following conjecture is 1 m 1 m 2 2 the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have in this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves this result includes the llibre et al s as a special one for the polynomial vector fields having only non dicritical invariant algebraic curves we answer the simple version of the 16th hilbert problem

nonlinear hamiltonian equations with fractional damping existence and uniqueness are proved for a class of hamiltonian nonlinear differential equations of second order with a caputo fractional derivative of order 0 2 for noninteger the rate of convergence of solutions to a critical point is algebraic for 0 1 a non negative dissipation function is constructed for 2 the equation is hamiltonian the solution is an analytic function of for 0 1 and for 1 2 numerical studies of nonlinear pendulum and the duffing equation with a fractional derivative term replacing the usual damping show striking differences between ordinary differential and fractional differential equations

gradient of the log likelihood ratio for infinite dimensional stochastic systems using a covariance operator approach we derive an expression for the log likelihood ratio gradient for system parameter estimation for continuous time infinite dimensional stochastic systems the gradient formula includes the smoother estimates and derivatives of system operators with no derivatives of estimates or covariance operators the unbounded operators typically found in partial differential equations limit how much the gradient formula can be simplified a random heat equation is considered

on an instability exhibited by the ballistic diffusive heat conduction model of xu and hu we show that the constitutive relation for the thermal flux proposed by xu hu 2011 admits an unconditional instability we also highlight the difference between mathematical models containing delay and those that include relaxation effects

duality and uniqueness of convex solutions to stationary hamilton jacobi equations value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality hamilton jacobi characterizations and the conjugacy of primal and dual value functions are of main interest close ties between the uniqueness of convex solutions to a hamilton jacobi equation the uniqueness of such solutions to a dual hamilton jacobi equation and the conjugacy of primal and dual value functions are displayed simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon for which the value functions are conjugate leads to sufficient conditions on the conjugacy of the infinite time horizon value functions consequently uniqueness results for the hamilton jacobi equation are established little regularity is assumed on the cost functions in the control problems correspondingly the hamiltonians need not display any strict convexity and may have several saddle points

computation of lie transfer maps for perturbed hamiltonian systems time evolution of a hamiltonian system can be viewed as a canonical transformation therefore perturbations giving rise to near identity deviations from an unperturbed solution can be represented by products of lie transformations or together with the unperturbed solution lie transfer maps in this paper i broaden the applicability to all perturbed hamiltonian systems the method of dragt and finn and subsequent co workers who developed a representation using a product of lie transformations factored by phase space variable order in the present paper perturbation parameters are no longer necessarily associated with the phase space variables this method treats both internal and external perturbations on an equal footing and a rank is assigned to each variable to reflect the degree of perturbation it represents with the companion program pglt analytic development of the lie transfer maps is relatively easy for many systems

spatiotemporal accessible solitons in fractional dimensions we report solutions for solitons of the accessible type in globally nonlocal nonlinear media of fractional dimension fd viz for self trapped modes in the space of effective dimension 2 d 3 with harmonic oscillator potential whose strength is proportional to the total power of the wave field the solutions are categorized by a combination of radial orbital and azimuthal quantum numbers n l m they feature coaxial sets of vortical and necklace shaped rings of different orders and can be exactly written in terms of special functions that include gegenbauer polynomials associated laguerre polynomials and associated legendre functions the validity of these solutions is verified by direct simulations the model can be realized in various physical settings emulated by fd spaces in particular it applies to excitons trapped in quantum wells

renormalized squares of white noise and other non gaussian noises as levy processes on real lie algebras it is shown how the relations of the renormalized squared white noise defined by accardi lu and volovich alv99 can be realized as factorizable current representations or levy processes on the real lie algebra 2 this allows to obtain its ito table which turns out to be infinite dimensional the linear white noise without or with number operator is shown to be a levy process on the heisenberg weyl lie algebra or the oscillator lie algebra furthermore a joint realization of the linear and quadratic white noise relations is constructed but it is proved that no such realizations exist with a vacuum that is an eigenvector of the central element and the annihilator classical levy processes are shown to arise as components of levy processes on real lie algebras and their distributions are characterized in particular the square of white noise analogue of the quantum poisson process is shown to have a 2 probability density and the analogue of the field operators to have a density proportional to where is the usual function and m 0 a real parameter

extended hamilton jacobi characterization of value functions in optimal control this paper examines generalized solutions to the hamilton jacobi equation the hamiltonian h t x p is assumed convex in p but is not constrained to have linear growth in this variable this corresponds to a certain class of generalized bolza problems related to optimal control lower semicontinuous solutions are considered and it is shown that there is a unique solution the so called value function of the underlying bolza problem in proving the main result we use recent improved necessary optimality conditions viability is also used in a new way in connection to differential inclusions with unbounded images

two dimensional dynamical system associated with abel s nonlinear differential equation a two dimensional dynamical system is proposed that is described by a pair of nonlinear ordinary differential equations odes with a complex parameter it reduces to abel s nonlinear ode of the first kind by an appropriate transformation using this fact the properties of solutions are investigated in detail with the aid of numerical computations it is found that various types of bifurcation phenomena occur depending on the values of the parameter in particular the solution is shown to blow up in finite time under certain conditions in order to visualize the behaviors of dynamical motions the trajectories of solutions are depicted in the plane finally a discussion is made on some generalizations of the proposed system