what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

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

a petrov galerkin finite element method for variable coefficient fractional diffusion equations abstract fractional diffusion equations have found increasingly more applications in recent years but introduce new mathematical and numerical difficulties galerkin formulation which was proved to be coercive and well posed for fractional diffusion equations with a constant diffusivity coefficient may lose its coercivity for variable coefficient problems the corresponding finite element method fails to converge we utilize the discontinuous petrov galerkin dpg framework to develop a petrov galerkin finite element method for variable coefficient fractional diffusion equations we prove the well posedness and optimal order convergence of the petrov galerkin finite element method numerical examples are presented to verify the theoretical results

order conditions for canonical runge kutta schemes when numerically integrating hamiltonian systems of differential equations it is often advantageous to use canonical methods i e methods that preserve the symplectic structure of the phase space thus reproducing an important feature of the hamiltonian flow an s stage runge kutta rk method without redundant stages is canonical if and only if with a standard notation b i a ij b j a ji b i b j 0 1 leqq i j leqq s it is shown that for canonical rk methods there are many redundancies in the standard order conditions for a canonical method to have order p it is sufficient that the b i s a ij s satisfy a system of algebraicequations that has roughly speaking an equation per nonrooted tree of order leqq p furthermore a new methodology is presented for the investigation of the order of canonical integration methods not necessarily rk methods when applied to hamiltonian systems in the new approach consistency is studied by taylor expanding a suitable scalar function in ter

subordination approach to multi term time fractional diffusion wave equations abstract this paper is concerned with the fractional evolution equation with a discrete distribution of caputo time derivatives such that the largest and the smallest orders and m satisfy the conditions 1 2 and m 1 first based on a study of the related propagation function the nonnegativity of the fundamental solutions to the spatially one dimensional cauchy and signaling problems is proven and propagation speed of a disturbance is discussed next we study the equation with a general linear spatial differential operator defined in a banach space and suppose it generates a cosine family a subordination principle is established which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function explicit representation of the probability density function is derived the subordination principle is applied for obtaining regularity results the analytical findings are supported by numerical work

thermoacoustic tomography for an integro differential wave equation modeling attenuation abstract in this article we study the inverse problem of thermoacoustic tomography tat on a medium with attenuation represented by a time convolution or memory term and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters under the assumption of being able to measure data on the whole boundary we prove uniqueness and stability and propose a convergent reconstruction method for a class of smooth variable sound speeds by a suitable modification of the time reversal technique we obtain a neumann series reconstruction formula

analytical time domain green s functions for power law media frequency dependent loss and dispersion are typically modeled with a power law attenuation coefficient where the power law exponent ranges from 0 to 2 to facilitate analytical solution a fractional partial differential equation is derived that exactly describes power law attenuation and the szabo wave equation time domain wave equations for lossy media obeying a frequency power law j acoust soc am 96 491 500 1994 is an approximation to this equation this paper derives analytical time domain green s functions in power law media for exponents in this range to construct solutions stable law probability distributions are utilized for exponents equal to 0 1 3 1 2 2 3 3 2 and 2 the green s function is expressed in terms of dirac delta exponential airy hypergeometric and gaussian functions for exponents strictly less than 1 the green s functions are expressed as fox functions and are causal for exponents greater than or equal than 1 the green s functions are expressed as fox and

partitioned runge kutta methods for separable hamiltonian problems separable hamiltonian systems of differential equations have the form dp dt dh dq dq dt dh dp with a hamiltonian function h that satisfies h t p k q t and v are respectively the kinetic and potential energies we study the integration of these systems by means of partitioned runge kutta methods i e by means of methods where different runge kutta tableaux are used for the p and q equations we derive a suffi cient and almost necessary condition for a partitioned runge kutta method to be canonical i e to conserve the symplectic structure of phase space thereby reproducing the qualitative properties of the hamiltonian dynamics we show that the requirement of canonicity operates as a simplifying assumption for the study of the order conditions of the method

a new integrable equation valued on a cayley dickson algebra we introduce a new integrable equation valued on a cayley dickson c d algebra in the particular case in which the algebra reduces to a complex one the new interacting term in the equation cancels and the equation becomes the known korteweg de vries kdv equation for each c d algebra the equation has an infinite sequence of local conserved quantities we obtain a backlund transformation in the sense of walhquist estabrook for the equation for any cayley dickson algebra and relate it to a generalized gardner equation from this the infinite sequence of conserved quantities follows directly we give the explicit expression for the first few from the backlund transformation we get the lax pair and the one soliton and two soliton solutions generalizing the known solutions for the quaternion valued kdv equation from the gardner equation we obtain the generalized modified kdv equation which also has an infinite sequence of conserved quantities the new integrable equation is preserved under a subgroup of the automorphisms of the c d algebra in the particular case of the algebra of octonions the equation is invariant under

tree graphs and the solution to the hamilton jacobi equation a combinatorial method is used to construct solutions of the hamilton jacobi equation an exact expression for hamilton s principal function s is obtained for classical systems of finitely many particles interacting via a certain class of time dependent potentials if x p and t are the position momentum and time variables for n point particles of mass m it is shown that hamiltonians of the form h x p t 1 2m p2 v x t have complete integrals s that are analytic functions of the inverse mass parameter m 1 in a punctured disk about the origin if v x t is bounded c in the x variable and has controlled x derivative growth then the coefficients of the laurent expansion of s about m 1 0 may be expressed in terms of gradient structures associated with tree graphs this series expansion for s x t y t0 converges absolutely and uniformly for all x y for time displacements t t0 t 2k 1 m eu 1 2 where k and u are bounds associated with the space derivatives of the potential for t t0 t the classi

hamiltonian and super hamiltonian systems of a hierarchy of soliton equations abstract by considering an isospectral eigenvalue problem a hierarchy of soliton equations are derived two types of extensions are presented by enlarging the associated spectral problem with the aid of generalized trace identity and the super trace identity the hamiltonian and super hamiltonian structures for the integrable extensions are established

locally solving fractional laplacian viscoacoustic wave equation using hermite distributed approximating functional method abstractaccurate seismic modeling in realistic media serves as the basis of seismic full waveform inversion and imaging recently viscoacoustic seismic modeling incorporating attenuation effects has been performed by solving a fractional laplacian viscoacoustic wave equation in this equation attenuation being spatially heterogeneous is represented partially by the spatially varying power of the fractional laplacian operator previously approximated by the global fourier method we have developed a local spectral approach based on the hermite distributed approximating functional hdaf method to implement the fractional laplacian in the viscoacoustic wave equation our approach combines local methods simplicity and global methods accuracy several numerical examples are developed to evaluate the feasibility and accuracy of using the hdaf method for the frequency independent q fractional laplacian wave equation

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

a symplectic integration algorithm for separable hamiltonian functions abstract we derive an algorithm to numerically integrate differential equations derivable from a separable hamiltonian function this symplectic algorithm is accurate to fourth order in the time step and preserves exactly the poincare cartan integral invariants associated with the topology of the phase flow we compare the efficiency and accuracy of this method to that of existing integrators both symplectic and non symplectic by integrating the equations of motion corresponding to a nonlinear pendulum a particle in the field of a standing wave and a harmonic oscillator perturbed by a plane wave

infinite energy solutions for critical wave equation with fractional damping in unbounded domains abstract this work is devoted to infinite energy solutions of semi linear wave equations in unbounded smooth domains of r 3 with fractional damping of the form x 1 1 2 t u the work extends previously known results for bounded domains in finite energy case furthermore well posedness and existence of locally compact smooth attractors for the critical quintic non linearity are obtained under less restrictive assumptions on non linearity relaxing some artificial technical conditions used before this is achieved by virtue of new type lyapunov functional that allows to establish extra space time regularity of solutions of strichartz type

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

a geometrical formulation of fermionic integrable systems a fermionic hamiltonian system is formulated on a supermanifold it is shown that if the system possesses a bi hamiltonian structure one can naturally define a lax equation associated with a 1 1 tensor on this supermanifold and this allows one to construct a set of conserved quantities furthermore if the corresponding nijenhuis tensor vanishes it is shown that all these conserved quantities would be in involution which is a sufficient condition for integrability of the system the fermionic extension of the kdv equation with a bi hamiltonian structure within this geometrical approach is studied

on the nature of the gardner transformation it is shown that every higher korteveg de vries equation can be included in a one parameter family of integrable equations

factorisation of energy dependent schrodinger operators miura maps and modified systems we consider the energy dependent schrodinger operator mathbb l sum limits i 0 n lambda i varepsilon i partial 2 u i which we have previously shown to be associated with multi hamiltonian structures 2 in this paper we use an unusual form of the lax approach to derive by asingle construction the time evolutions of the eigenfunctions of mathbb l the associated hamiltonian operators and the hamiltonian functionals we then generalise the well known factorisation of standard lax operators to the case of energy dependent operators the simple product of linear factors is replaced by a dependent quadratic form we thus generalise the resulting construction of miura maps and modified equations we show that for some of our systems there exists a sequence ofn such modifications therth modification possessing n r 1 hamiltonian structures

conservation of integrals and symplectic structure in the integration of differential equations by multistep methods we consider the question of whether multistep methods inherit in some sense quadratic first integrals possessed by the differential system being integrated we also investigate whether in the integration of hamiltonian systems multistep methods conserve the symplectic structure of the phase space

on the integrability of the super kdv equation supersymmetric analogues of the gelfand dikii polynomials are derived an alternative proof of the super kdv equation is given using identities obeyed by the polynomials they also allow the recursive generation of the infinite number of conserved quantities for the super kdv flow

the bi hamiltonian structure of fully supersymmetric korteweg de vries systems the bi hamiltonian structure of integrable supersymmetric extensions of the korteweg de vries kdv equation related to then 1 and then 2 superconformal algebras is found it turns out that some of these extensions admit inverse hamiltonian formulations in terms of presymplectic operators rather than in terms of poisson tensors for one extension related to then 2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical kdv they can be explained by a factorization of the corresponding lax operator all the bi hamiltonian formulations are derived in a systematic way from the lax operators

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

symplectic runge kutta and related methods recent results symplectic algorithms are numerical integrators for hamiltonian systems that preserve the symplectic structure in phase space in long time integrations these algorithms tend to perform better than their nonsymplectic counterparts some symplectic algorithms are derived by explicitly finding a generating function other symplectic algorithms are members of standard families of methods such as runge kutta methods that just turn out to preserve the symplectic structure here we survey what is known about the second type of symplectic algorithms

symplectic phase flow approximation for the numerical integration of canonical systems new methods are presented for the numerical integration of ordinary differential equations of the important family of hamiltonian dynamical systems these methods preserve the poincare invariants and therefore mimic relevant qualitative properties of the exact solutions the methods are based on a runge kutta type ansatz for the generating function to realize the integration steps by canonical transformations a fourth order method is given and its implementation is discussed numerical results are presented for the henon heiles system which describes the motion of a star in an axisymmetric galaxy

a k space method for coupled first order acoustic propagation equations a k space method for large scale simulation of ultrasonic pulse propagation is presented the present method which solves the coupled first order differential equations for wave propagation in inhomogeneous media is derived in a simple form analogous to previous finite difference methods with staggered spatial and temporal grids like k space methods based on second order wave equations the present method is exact for homogeneous media unconditionally stable for slow c r c0 media and highly accurate for general weakly scattering media in addition unlike previous k space methods the form of the method allows straightforward inclusion of relaxation absorption and perfectly matched layer pml nonreflecting boundary conditions numerical examples illustrate the capabilities of the present k space method for weakly inhomogeneous media accurate results are obtained using coarser temporal and spatial steps than possible with comparable finite difference and pseudospectral methods the low dispers

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

integrable system constructed out of two interacting superconformal fields we consider an interaction between superconformal fields of the same gradation this entails the construction of a supersymmetric poisson tensor for these fields generating a new class of hamiltonian systems the lax representation is found for one of them by supersymmetrizing the lax operator for the hirota satsuma equation the supersymmetric equation is not reducible to the classical hirota satsuma case we show that our generalized system can be reduced to the the supersymmetric kdv equation a 4 surprisingly the integrals of motion are not reduced to integrals of motion of the supersymmetric kdv equation

on the bihamiltonian structure of the supersymmetric kdv hierarchies a lie superalgebraic approach we give a lie superalgebraic interpretation of the bihamiltonian structure of known supersymmetric kdv equations we show that the loop algebra of a lie superalgebra carries a natural poisson pencil and we subsequently deduce the bihamiltonian structure of the supersymmetric kdv hierarchies by applying to loop superalgebras an appropriate reduction technique this construction can be regarded as a superextension of the drinfeld sokolov method for building a kdv type hierarchy from a simple lie algebra

modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional laplacians we evaluated a time domain wave equation for modeling acoustic wave propagation in attenuating media the wave equation was derived from kjartansson s constant q constitutive stress strain relation in combination with the mass and momentum conservation equations our wave equation expressed by a second order temporal derivative and two fractional laplacian operators described very nearly constant q attenuation and dispersion effects the advantage of using our formulation of two fractional laplacians over the traditional fractional time derivative approach was the avoidance of time history memory variables and thus it offered more economic computations in numerical simulations we formulated the first order constitutive equations with the perfectly matched layer absorbing boundaries the temporal derivative was calculated with a staggered grid finite difference approach the fractional laplacians are calculated in the spatial frequency domain using a fourier pseudospectral implementation we validated our numerical results through comparisons with theoretical constant q attenuation and dispersion solutions field measurements from the pierre shale and results from 2d viscoacoustic analytical modeling for the homogeneous pierre shale we also evaluated different formulations to show separated amplitude loss and dispersion effects on wavefields furthermore we generalized our rigorous formulation for homogeneous media to an approximate equation for viscoacoustic waves in heterogeneous media we then investigated the accuracy of numerical modeling in attenuating media with different q values and its stability in largecontrast heterogeneous media finally we tested the applicability of our time domain formulation in a heterogeneous medium with high attenuation

gpu accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion the efficient computation of matrix function vector products has recently become an important area of research driven in particular by two important applications the numerical solution of fractional partial differential equations and the integration of large systems of ordinary differential equations in this work we consider a problem that combines these two applications in the form of a numerical solution algorithm for fractional reaction diffusion equations that after spatial discretization is advanced in time using the exponential euler method we focus on the efficient implementation of the algorithm on graphics processing units gpus as we wish to make use of the increased computational power available with this hardware we compute the matrix function vector products using the contour integration method in n hale n j higham and l n trefethen siam j numer anal 46 2008 pp 2505 2523 multiple levels of preconditioning are applied to reduce the gpu memory footprint and to furth

linking multiple relaxation power law attenuation and fractional wave equations the acoustic wave attenuation is described by an experimentally established frequency power law in a variety of complex media e g biological tissue polymers rocks and rubber recent papers present a variety of acoustical fractional derivative wave equations that have the ability to model power law attenuation on the other hand a multiple relaxation model is widely recognized as a physically based description of the acoustic loss mechanisms as developed by nachman et al j acoust soc am 88 1584 1595 1990 through assumption of a continuum of relaxation mechanisms each with an effective compressibility described by a distribution related to the mittag leffler function this paper shows that the wave equation corresponding to the multiple relaxation approach is identical to a given fractional derivative wave equation this work therefore provides a physically based motivation for use of fractional wave equations in acoustic modeling

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

boundary conditions for two sided fractional diffusion abstract this paper develops appropriate boundary conditions for the two sided fractional diffusion equation where the usual second derivative in space is replaced by a weighted average of positive left and negative right fractional derivatives mass preserving reflecting boundary conditions for two sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary stable consistent explicit and implicit euler methods are detailed and steady state solutions are derived steady state solutions for two sided fractional diffusion equations using both riemann liouville and caputo flux are computed for riemann liouville flux and reflecting boundary conditions the steady state solution is singular at one or both of the end points for caputo flux and reflecting boundary conditions the steady state solution is a constant function numerical experiments illustrate the convergence of these numerical methods finally the influence of the reflecting boundary on the steady state behavior subject to both the riemann liouville and caputo fluxes is discussed

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

riemann liouville integrals of fractional order and extended kp hierarchy an attempt to formulate the extensions of the kp hierarchy by introducing fractional order pseudo differential operators is given in the case of the extension with the half order pseudo differential operators a system analogous to the supersymmetric extensions of the kp hierarchy is obtained unlike the supersymmetric extensions no grassmannian variable appears in the hierarchy considered here more general hierarchies constructed by the 1 nth order pseudo differential operators their integrability and the reduction procedure are also investigated in addition to finding the new extensions of the kp hierarchy a brief introduction to the riemann liouville integral is provided to yield a candidate for the fractional order pseudo differential operators

full wave modeling of therapeutic ultrasound efficient time domain implementation of the frequency power law attenuation for the simulation of therapeutic ultrasound applications a method including frequency dependent attenuation effects directly in the time domain is highly desirable this paper describes an efficient numerical time domain implementation of the power law attenuation model presented by szabo szabo j acoust soc am 96 491 500 1994 simulations of therapeutic ultrasound applications are feasible in conjunction with a previously presented finite differences time domain fdtd algorithm for nonlinear ultrasound propagation ginter et al j acoust soc am 111 2049 2059 2002 szabo implemented the empirical frequency power law attenuation using a causal convolutional operator directly in the time domain equation though a variety of time domain models has been published in recent years no efficient numerical implementation has been presented so far for frequency power law attenuation models solving a convolutional integral with standard time domain techniques requires enormous computational ef

reprint of boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

on the applicability of kramers kronig relations for ultrasonic attenuation obeying a frequency power law in the recent literature concern has been raised regarding the validity of kramers kronig relations for media with ultrasonic attenuation obeying a frequency power law it is demonstrated however that the kramers kronig dispersion relations for application to these types of media are available the developed dispersion relations are compared with measurements on several liquids and agreement is found to better than 1 m s over the experimentally available bandwidth a discussion regarding the validity of these dispersion relations in particular how the dispersion relations relate to the so called paley wiener conditions forms the conclusion

fermionic covariant prolongation structure theory for multidimensional super nonlinear evolution equation the fermionic covariant prolongation structure theory is investigated we extend the fermionic covariant prolongation structure technique to the multidimensional super nonlinear evolution equation and present the fermionic covariant fundamental equations determining the prolongation structure furthermore we investigate a 2 1 dimensional super nonlinear schrodinger equation and analyze its integrability by means of this prolongation structure technique we derive its lax representation and backlund transformation moreover we present a solution of this multidimensional super integrable equation

an easily implementable fourth order method for the time integration of wave problems abstract we are concerned with the time integration of systems of ordinary differential equations arising from the space discretization of partial differential wave equations with smooth solutions a method is suggested that while being as easily implementable as the standard implicit mid point rule is fourth order accurate the new method is symplectic so that it is very well suited for long time integrations of problems with a hamiltonian structure numerical experiments are reported that refer to a fourth order galerkin space discretization of the korteweg de vries equation and to a pseudospectral space discretization of the same equation

an operator valued extension of the super korteweg de vries equations an extension of the super korteweg de vries kdv integrable system in terms of operator valued functions is obtained following the ideas of gardner a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented the approach is applied to the above described system and infinitely many conserved quantities are constructed in a particular case they reduce to the corresponding conserved quantities of super kdv

an explicit runge kutta nystro m method is canonical if and only if its adjoint is explicit canonical runge kutta nystrom methods for hamiltonian dynamical systems are considered these systems arise in various areas in the physical sciences canonical methods preserve certain properties of the system in this paper it is shown that an explicit runge kutta nystrom method is canonical if and only its adjoint is explicit the adjoint of a method is obtained by time reversal one application for this result and other results in this paper is that if an explicit canonical runge kutta nystrom method of odd order is concatenated with its adjoint the result is an explicit canonical method of one order higher

integrable fermionic extensions of the garnier system and the anharmonic oscillator we first derive a fermionic extension of the garnier system by applying the method of binary nonlinearization of spectral problem to the supersymmetric kdv equation of kupershmidt and then a fermionic extension of the anharmonic oscillator by a simple reduction the integrable properties of resulting systems such as lax representations corresponding r matrices and conversed integrals of motion are established

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

modeling power law absorption and dispersion in viscoelastic solids using a split field and the fractional laplacian the absorption of compressional and shear waves in many viscoelastic solids has been experimentally shown to follow a frequency power law it is now well established that this type of loss behavior can be modeled using fractional derivatives however previous fractional constitutive equations for viscoelastic media are based on temporal fractional derivatives these operators are non local in time which makes them difficult to compute in a memory efficient manner here a fractional kelvin voigt model is derived based on the fractional laplacian this is obtained by splitting the particle velocity into compressional and shear components using a dyadic wavenumber tensor this allows the temporal fractional derivatives in the kelvin voigt model to be replaced with spatial fractional derivatives using a lossless dispersion relation with the appropriate compressional or shear wave speed the model is discretized using the fourier collocation spectral method which allows the fractional operators to be efficiently computed the field splitting also allows the use of a k space corrected finite difference scheme for time integration to minimize numerical dispersion the absorption and dispersion behavior of the fractional laplacian model is analyzed for both high and low loss materials the accuracy and utility of the model is then demonstrated through several numerical experiments including the transmission of focused ultrasound waves through the skull

time domain wave equations for lossy media obeying a frequency power law for attenuation described by a slowly varying power law function of frequency 0 y classical lossy time domain wave equations exist only for the restricted cases where y 0 or y 2 for the frequently occurring practical situation in which attenuation is much smaller than the wave number a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence in order to obtain the corresponding time domain lossy wave equation time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions three forms of lossy wave equations are found depending on whether y is an even or odd integer or a noninteger a time domain expression of causality analogous in function to the kramers kronig relations in the frequency domain is used to derive the causal wave equations final causal versions of the time domain wave equations are obtained even for the cases where y 1 which according to the paley wiener th

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

studies in numerical nonlinear instability iii augmented hamiltonian systems leap frog explicit mid point discretizations of hamiltonian systems are considered it is proved that the discrete evolution preserves the symplectic structure of the phase space the kolmogorov arnold moser kam theory is applied to guarantee the boundedness of the computed points under suitable restrictions of the time step the general ideas introduced are employed to provide an extensive analysis of the leap frog discretization of the complex equation iz t z 2 z 0 which describes the nonlinear selfinteraction of a fourier mode the paper emphasizes the relations between numerical analysis and classical mechanics dynamical systems

development of a 3d staggered fdtd scheme for solving maxwell s equations in drude medium an explicit finite difference scheme is developed to solve the three dimensional maxwell s equations in drude medium our aim of developing this scheme in time domain is to compute solutions in staggered grids from the faraday s and ampere s equations the electric and magnetic field solutions are sought subject to the discrete zero divergence condition gauss s law the local conservation laws in ideal maxwell s equations are also numerically preserved all the time using the explicit second order accurate symplectic partitioned runge kutta temporal scheme the spatial derivative terms in the faraday s and ampere s equations are discretized to obtain fourth order accuracy using the developed scheme underlying the concept of minimizing the discrepancy between the exact and the derived numerical phase velocities dispersion and anisotropy errors have been much reduced through the procedure of minimizing phase velocity error in addition to performing the fundamental analysis on the proposed scheme the computational efficiency and long term accurate properties embedded in the proposed symplectic dispersion error reduction centered scheme are numerically demonstrated through several test problems investigated in ideal and drude media

symmetries of supersymmetric integrable hierarchies of kp type this article is devoted to the systematic study of additional non isospectral symmetries of constrained reduced supersymmetric integrable hierarchies of kp type the so called skp r mb mf models the latter are supersymmetric extensions of ordinary constrained kp hierarchies which contain as special cases basic integrable systems such as m kdv akns fordy kulish yajima oikawa etc as a first main result it is shown that any skp r mb mf hierarchy possesses two different mutually anti commuting types of superloop superalgebra additional symmetries corresponding to the positive and negative grade parts of certain superloop superalgebras the second main result is the systematic construction of the full algebra of additional virasoro symmetries of skp r mb mf hierarchies which requires nontrivial modifications of the virasoro flows known from the general case of unconstrained manin radul super kp hierarchies the latter flows do not define symmetries for constrained skp r mb mf hierarchies as

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

frequency domain wave equation and its time domain solutions in attenuating media our purpose in this paper is to describe the wave propagation in media whose attenuation obeys a frequency power law to achieve this a frequency domain wave equation was developed using previously derived causal dispersion relations an inverse space and time fourier transform of the solution to this algebraic equation results in a time domain solution it is shown that this solution satisfies the convolutional time domain wave equation proposed by szabo j acoust soc am 96 491 500 1994 the form of the convolutional loss operator contained in this wave equation is obtained solutions representing the propagation of both plane sinusoidal and transient waves propagating in media with specific power law attenuation coefficients are investigated as special cases of our solution using our solution comparisons are made for transient one dimensional propagation in a medium whose attenuation is proportional to frequency with recently obtained numerical solutions of szabo s equation these show good agreement

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

a supertrace identity and its applications to superintegrable systems a supertrace identity on lie superalgebras is established it provides a tool for constructing super hamiltonian structures of zero curvature equations associated with lie superalgebras applications in the case of the lie superalgebra b 0 1 present super hamiltonian structures of a super akns soliton hierarchy and a super dirac soliton hierarchy

supersymmetric extension of the scalar born infeld equation in this paper a supersymmetric extension of the scalar born infeld equation is constructed through a superspace formalism which involves the addition of one independent fermionic grassmann variable to the existing bosonic spacetime coordinates the bosonic scalar field is replaced by a fermionic superfield which is composed of two component fields one bosonic and one fermionic the resulting equation is invariant under a space supersymmetric transformation and in its most general form involves four arbitrary parameters for a certain specific case the lie superalgebra of symmetries was identified and the one dimensional subalgebras were systematically classified into splitting and non splitting conjugate classes a number of group invariant solutions were obtained including polynomial solutions solutions by radicals and solitary waves including bumps kinks and doubly periodic solutions

a causal and fractional all frequency wave equation for lossy media this work presents a lossy partial differential acoustic wave equation including fractional derivative terms it is derived from first principles of physics mass and momentum conservation and an equation of state given by the fractional zener stress strain constitutive relation for a derivative order in the fractional zener relation the resulting absorption k obeys frequency power laws as k 1 in a low frequency regime k 1 2 in an intermediate frequency regime and k 1 in a high frequency regime the value 1 corresponds to the case of a single relaxation process the wave equation is causal for all frequencies in addition the sound speed does not diverge as the frequency approaches infinity this is an improvement over a previously published wave equation building on the fractional kelvin voigt constitutive relation

deriving fractional acoustic wave equations from mechanical and thermal constitutive equations it is argued that fractional acoustic wave equations come in two kinds the first kind is constructed ad hoc to have loss operators that fit power law measurements the second kind is more fundamental as they in addition are based on underlying physical equations here that means constitutive equations these equations are the fractional kelvin voigt and the more general fractional zener stress strain relationships as well as a fractional version of the fourier heat law the properties of the wave equations are given in terms of attenuation and phase group velocities for low intermediate and high frequency regions in the most general case the attenuation exhibits power law behavior in all frequency ranges while the phase and group velocities increase sharply in the intermediate frequency range and converge to a constant finite value for high frequencies it is also shown that the fractional zener wave equation is equivalent to the multiple relaxation model for attenuation

a k space green s function solution for acoustic initial value problems in homogeneous media with power law absorption an efficient green s function solution for acoustic initial value problems in homogeneous media with power law absorption is derived the solution is based on the homogeneous wave equation for lossless media with two additional terms these terms are dependent on the fractional laplacian and separately account for power law absorption and dispersion given initial conditions for the pressure and its temporal derivative the solution allows the pressure field for any time t 0 to be calculated in a single step using the fourier transform and an exact k space time propagator for regularly spaced cartesian grids the former can be computed efficiently using the fast fourier transform because no time stepping is required the solution facilitates the efficient computation of the pressure field in one two or three dimensions without stability constraints several computational aspects of the solution are discussed including the effect of using a truncated fourier series to represent discrete initial condit

on the gelfand dickey algebra gd sln and the wn symmetry i the bosonic case the algebra of nonlinear local and nonlocal differential operators acting on the ring of analytic functions is studied it is shown in particular that this space splits into 3 2 special subalgebras jr j 0 1 r 1 each subalgebra is completely specified by quantum numbers s and p q describing the conformal spin and the lowest and the highest degrees respectively the algebra and its dual of local pure nonlocal differential operators is used to calculate the general expression of the gelfand dickey bracket and the wn symmetry poisson one in terms of a set of spin j canonical fields uj 2 j n and a nonlinear u cubic dependent differential operator d n i j z u the explicit form of this operator is worked out other remarkable features are also discussed

approximate analytical time domain green s functions for the caputo fractional wave equation the caputo fractional wave equation geophys j r astron soc 13 529 539 1967 models power law attenuation and dispersion for both viscoelastic and ultrasound wave propagation the caputo model can be derived from an underlying fractional constitutive equation and is causal in this study an approximate analytical time domain green s function is derived for the caputo equation in three dimensions 3d for power law exponents greater than one the green s function consists of a shifted and scaled maximally skewed stable distribution multiplied by a spherical spreading factor 1 4 r the approximate one dimensional 1d and two dimensional 2d green s functions are also computed in terms of stable distributions finally this green s function is decomposed into a loss component and a diffraction component revealing that the caputo wave equation may be approximated by a coupled lossless wave equation and a fractional diffusion equation

the zero curvature formulation of the skdv equations the fermionic extensions of the kdv equation are derived from the zero curvature condition associated with the superalgebra osp 2 1 this derivation clarifies why there are only two such extensions possible and why only one of them is supersymmetric a lenard type of derivation of these equations is also presented

a new n 2 supersymmetric korteweg de vries equation there are two known integrable n 2 space supersymmetric extensions of the kdv equation both can be written as hamiltonian systems with a common poisson structure which corresponds to the n 2 supersymmetric form of the second kdv structure by analyzing the conservation laws of the one parameter family of equations that share this hamiltonian structure one finds that a third system is singled out by the existence of nontrivial conservation laws it is conjectured to be integrable

bi hamiltonian structure of a generalized super korteweg de vries equation the bi hamiltonian structure is constructed for a super extension of the kdv equation the related miura map is recovered by a proper gauge transformation and a new spectral problem is presented for the modified system

time domain comparisons of power law attenuation in causal and noncausal time fractional wave equations the attenuation of ultrasound propagating in human tissue follows a power law with respect to frequency that is modeled by several different causal and noncausal fractional partial differential equations to demonstrate some of the similarities and differences that are observed in three related time fractional partial differential equations time domain green s functions are calculated numerically for the power law wave equation the szabo wave equation and for the caputo wave equation these green s functions are evaluated for water with a power law exponent of y 2 breast with a power law exponent of y 1 5 and liver with a power law exponent of y 1 139 simulation results show that the noncausal features of the numerically calculated time domain response are only evident very close to the source and that these causal and noncausal time domain green s functions converge to the same result away from the source when noncausal time domain green s functions are convolved with a short pulse no evidence of noncausal behavior remains in the time domain which suggests that these causal and noncausal time fractional models are equally effective for these numerical calculations

on the bi hamiltonian structures of the skdv the first and the second hamiltonian structures of the supersymmetric kdv equation are derived following the conventional approach of gel fand and dickey in particular it is shown that the nonlocal nature of the first bracket is a direct consequence of the constrained nature of the lax operator

viscoacoustic modeling and imaging using low rank approximation abstracta constant q wave equation involving fractional laplacians was recently introduced for viscoacoustic modeling and imaging this fractional wave equation has a convenient mixed domain space wavenumber formulation which involves the fractional laplacian operators with a spatially varying power we have applied the low rank approximation to the mixed domain symbol which enables a space variable attenuation specified by the variable fractional power of the laplacians using the proposed approximation scheme we formulated the framework of the q compensated reverse time migration q rtm for attenuation compensation numerical examples using synthetic data demonstrated the improved accuracy of using low rank wave extrapolation with a constant q fractional laplacian wave equation for seismic modeling and migration in attenuating media low rank q rtm applied to viscoacoustic data is capable of producing images comparable in quality with those produced by conventional rtm from acoustic data

integrable hierarchies related to the kuper ch spectral problem in this paper from a given kuper ch spectral problem we propose two kinds of super integrable hierarchies one is the kuper ch hierarchy the other is the generalized kuper harry dym hierarchy moreover we construct their zero curvature representations and super bi hamiltonian structures

a fourth order avf method for the numerical integration of sine gordon equation abstract in this paper a new scheme which has energy preserving property is proposed for solving the sine gordon equation with periodic boundary conditions it is obtained by the fourier pseudo spectral method and the fourth order average vector field method in numerical experiments the new high order energy preserving scheme is compared with a number of existing numerical schemes for the one dimensional sine gordon equation the new high order energy preserving scheme for the two dimensional sine gordon equation is also investigated numerical results are addressed to further illustrate the conservation of energy and the evolutional behaviors of solitons

defect controlled numerical methods and shadowing for chaotic differential equations abstract it is well known that chaotic ode s are in the classical sense unstable or ill conditioned it is not clear that variable step size codes based on local error control can solve such problems in a useful way in particular it is usually difficult to show that the apparent chaos in the resulting solution is not a numerical artifact i show here that a defect controlled method gives useful solutions for chaotic problems a pragmatically modified definition of what it means for a dynamical system to be chaotic is also presented