a new approach for the application of adomian decomposition method for the solution of fractional space diffusion equation with insulated ends abstract this paper presents the new approach for the solution of the space fractional diffusion equation defined in finite domain by adomian decomposition method adm a typical example of special interest with fractional space derivative of order 1 2 is considered in the present analysis and solved by adm after expressing the initial condition as fourier series the explicit solution of the space fractional diffusion equation has been presented in the closed form and then the numerical solution has been represented graphically the behaviour of adomian solutions and the effects of different values of are shown graphically

the asymptotic stability of steady solutions of reaction convection diffusion equations on montre comment une combinaison de l analyse mathematique et de l analyse asymptotique permet de donner des conditions suffisantes simples pour la stabilite asymptotique des solutions stationnaires du probleme aux valeurs limites et initiales u t f x u u x g x u eu xx x t dans 0 1 0 u x o e x x dans 0 1 u o t e au x o t e u 1 t e bu x 1 t e t dans 0

almost hermitian geometries that characterize hamilton jacobi distributions the author considers horizontal lagrangian sub bundles of a cotangent bundle t n that contain the dynamical vector field of a hamiltonian system it is shown that if the hamiltonian is quadratic on the fibres of t n then all such sub bundles can be characterized as solutions to a system of algebro partial differential equations that are defined solely in terms of vertical quantities this system of equations has a universal solution the universal solution is applied to the study of the geometry of the lorentz force law it is shown that the dynamical flow of the lorentz force law on t n is a geodesic vector field for a metric connection on t n with torsion the torsion of this connection is given by an extension of cartan s first structure equation to non linear connections

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 new approach to the converse of noether s theorem the concepts of vector fields and forms along a map are used to establish a condition characterising symmetries of the hamiltonian system associated with a regular lagrangian this condition does not mention any second order differential equation field but is expressed in terms of the geometry of the second order tangent bundle this result is also generalised to the case of lagrangian functions depending on higher order derivatives

asymptotic stability in the presence of convection t in 0 1 x 0 00 put simply theorem 2 1 asserts that bounds on the initial and boundary data which satisfy the appropriate differential inequalities in the domain under consideration are in fact bounds on the solution inside the domain as well thus this result is very much in the spirit of the more familiar maximum principle as an illustration of the applicability of theorem 2 1 suppose that in problem 2 1 a

internal deformation of lie algebroids and symplectic realizations given a lie algebroid and a bundle over its base which is endowed with a localizable poisson structure and a flat connection we construct an extended bundle whose dual is endowed with an almost poisson structure that is a quadratic poisson structure when a certain compatibility property is satisfied this new formalism on lie algebroids describes systems with internal degrees of freedom

causality stokes wave equation and acoustic pulse propagation in a viscous fluid stokes acoustic wave equation is solved for the impulse response of an isotropic viscous fluid two exact integral forms of solution are derived both of which are causal predicting a zero response before the source is activated at time t 0 moreover both integral solutions satisfy a stronger causality condition the pressure pulse is maximally flat with all its time derivatives identically zero at t 0 signifying that there is no instantaneous response to the source anywhere in the fluid a closed form approximation for each of the two integrals is derived with distinctly different properties in the two cases even though the original integrals are equivalent in that they predict identical pulse shapes one of these approximations reminiscent of transient solutions that have appeared previously in the literature is noncausal due to the incorrect representation of high frequency components in the propagating pulse in the second approximation all frequency components are treated correctly leading to an impulse response that satisfies the strong causality condition also satisfied by the original integrals whereby the predicted pressure pulse is zero when t 0 and maximally flat everywhere in the fluid immediately after t 0

on the differential geometry of the euler lagrange equations and the inverse problem of lagrangian dynamics the conditions for a system of second order differential equations to be derivable from a lagrangian the conditions of self adjointness in the terminology of santilli 1978 and others are related in the time independent case to the differential geometry of the tangent bundle of configuration space these conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second order differential equations necessary and sufficient conditions for such a vector field to be derivable from a lagrangian may be stated as the existence of a two form with certain properties it is interesting that it is a deduction not an assumption that this two form is closed and thus defines a symplectic structure some other differential geometric properties of euler lagrange second order differential equations are described

a more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators l a corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented the fundamental approach is based on fractional formulations of the stress strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in prieur and holm nonlinear acoustic wave equations with fractional loss operators j acoust soc am 130 3 1125 1132 2011 the loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged

the inverse problem for six dimensional codimension two nilradical lie algebras ado s theorem asserts that every real lie algebra g of dimension n has a faithful representation as a subalgebra of gl p r for some p the theorem offers no practical information about the size of p in relation to n and in principle p may be very large compared to n this article is concerned with finding representations for a certain class of six dimensional lie algebras specifically real indecomposable algebras that have a codimension two nilradical these algebras were classified by turkowski and comprise of 40 cases some of which contain up to four parameters linear representations are found for each algebra in these classes more precisely a matrix lie group is given whose lie algebra corresponds to each algebra in turkowski s list and can be found by differentiating and evaluating at the identity element of the group in addition a basis for the right invariant vector fields that are dual to the maurer cartan forms are given thereby providing an effective realization of lie s third theorem t

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

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

numerical methods and analysis for a class of fractional advection dispersion models in this paper a class of fractional advection dispersion models fadms is considered these models include five fractional advection dispersion models i e the time fadm the mobile immobile time fadm with a time caputo fractional derivative 0 c 1 the space fadm with two sides riemann liouville derivatives the time space fadm and the time fractional advection diffusion wave model with damping with index 1 c 2 these equations can be used to simulate the regional scale anomalous dispersion with heavy tails we propose computationally effective implicit numerical methods for these fadms the stability and convergence of the implicit numerical methods are analysed and compared systematically finally some results are given to demonstrate the effectiveness of theoretical analysis

analysis of four parameter fractional derivative model of real solid materials the introduction of fractional derivatives into the constitutive equation of the differential operator type of linear solid materials has led to the development of the so called fractional derivative models one of these models characterized by four parameters has been found usable for describing the variation of dynamics elastic and damping properties in a wide frequency range provided that there is only one loss peak in this paper this four parameter model is theoretically analyzed the effect of the parameters on the frequency curves is demonstrated and it is shown that there is a strict relation between the dispersion of the dynamic modulus the loss peak and the slope of the frequency curves the half value bandwidth of the loss modulus frequency curve is investigated and conditions are developed to establish the applicability of the model for a class of materials moreover it is shown that the model can be used to predict the frequency dependences of dynamic properties for a wide range even if measurements are made in only a narrow frequency range around the loss peak

new approach to find exact solutions of time fractional kuramoto sivashinsky equation abstract in the present paper we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics viz time fractional kuramoto sivashinsky equation by a new proposed method via fractional complex transform as a result new types of exact analytical solutions are obtained here the fractional derivative is described in jumarie s modified riemann liouville sense

a fractional order darcy s law abstract by using spatial averaging methods in this work we derive a darcy s type law from a fractional newton s law of viscosity which is intended to describe shear stress phenomena in non homogeneous porous media as a prerequisite towards this end we derive an extension of the spatial averaging theorem for fractional order gradients the usage of this tool for averaging continuity and momentum equations yields a darcy s law with three contributions i similar to the classical darcy s law a term depending on macroscopic pressure gradients and gravitational forces ii a fractional convective term induced by spatial porosity gradients and iii a fractional brinkman type correction in the three cases the corresponding permeability tensors should be computed from a fractional boundary value problem within a representative cell consistency of the resulting darcy s type law is demonstrated by showing that it is reduced to the classical one in the case of integer order velocity gradients and homogeneous porous media

extended tangent bundle formalism for time dependent lagrangian systems the lagrangian formalism for time dependent systems is developed using vector fields defined in the extended tangent bundle t q r the definition of the time independent extended lagrangian function l associated to a time dependent lagrangian l is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determining the euler lagrange vector field the relations between the time independent and the time dependent helmholtz conditions are analyzed from a geometric perspective and finally the noether theorem for the extended lagrangian l is considered

lagrangian systems with constraints a geometric approach to the method of lagrange multipliers a geometric approach to the method of lagrange multipliers is presented using the framework of the tangent bundle geometry the nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then and using this study of the nonholonomic mechanical systems as a previous result the holonomic systems are considered the lagrangian inverse problem is also analysed and finally the theory is illustrated with several examples

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

simulation of thermal disturbances with finite wave speeds using a high order method the classical heat diffusion theory based on the fourier s model breaks down when considering transient heat flow for short times extreme thermal gradients or at low temperatures the hyperbolic heat conduction equation based on the cattaneo model for the heat flux incorporates a relaxation mechanism in order to gradually adjust to a change in the temperature gradient a spectral element method is applied for solving the hyperbolic system treating the heat flux as an independent variable in addition to temperature the numerical solution is based on the time space least squares spectral method numerical examples are included for discussing the effects of the thermal waves

connecting the grain shearing mechanism of wave propagation in marine sediments to fractional order wave equations the characteristic time dependent viscosity of the intergranular pore fluid in buckingham s grain shearing gs model buckingham j acoust soc am 108 2796 2815 2000 is identified as the property of rheopecty the property corresponds to a rare type of a non newtonian fluid in rheology which has largely remained unexplored the material impulse response function from the gs model is found to be similar to the power law memory kernel which is inherent in the framework of fractional calculus the compressional wave equation and the shear wave equation derived from the gs model are shown to take the form of the kelvin voigt fractional derivative wave equation and the fractional diffusion wave equation respectively therefore an analogy is drawn between the dispersion relations obtained from the fractional framework and those from the gs model to establish the equivalence of the respective wave equations further a physical interpretation of the characteristic fractional order present in the wave e

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

acoustics in a two deck viscothermal boundary layer over an impedance surface the acoustics of a mean flow boundary layer over an impedance surface or acoustic lining is considered by considering a thick mean flow boundary layer possibly due to turbulence the boundary la

new exact solutions of nonlinear fractional acoustic wave equations in ultrasound in this paper new exact solutions of fractional nonlinear acoustic wave equations have been devised the travelling periodic wave solutions of fractional burgers hopf equation and khokhlov zabolotskaya kuznetsov kzk equation have obtained by first integral method nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible taking nonlinear effects into account the ultrasound beam analysis makes more accurate in these applications the burgers hopf equation is one of the extensively studied models in mathematical physics in addition the kzk parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3d diffraction sound beams in dissipative media in the present analysis these nonlinear equations have solved by first integral method as a result new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations the obtained results are presented graphically to demonstrate the efficiency of this proposed method

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

bi invariant and noninvariant metrics on lie groups a restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied specifically for solvable lie algebras of dimension up to and including six all algebras for which there is a compatible pseudo riemannian metric on the corresponding linear lie group are found of the 19 such metrics four are bi invariant the lie algebras are taken from tables compiled originally by mubarakzyanov izv vyssh uchebn zaved mat 4 104 116 1963 and morozov izv vyssh uchebn zaved mat 4 161 171 1958

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

extended covariance for the lagrange equations of motion a geometric analysis the infinitesimal transformations which leave the lagrangian structure of the equations of motion unchanged are intrinsically characterized a new condition is given with which to obtain alternative lagrangians

large time behavior for conservation laws with source in a bounded domain here u u x t u r precise assumptions will be given later we are in particular interested in the asymptotic behavior of solutions to problem 1 3 equation of the form 1 is called reaction convection equation or balance law the correct setting to deal with for such equations is the one given by entropy solutions in the sense of kruz kov in 15 in such class it is well known that the cauchy problem has a unique solution continuously depending on initial data in l norm in this article we consider boundary conditions and we analyze the influence of such conditions on the whole solution as time goes on such conditions for equations of the form 1 were considered in 1 it turns out that in order to have existence uniqueness and continuous dependence conditions 3 have to be interpreted in a non classical fashion following 1 we look for entropy solutions of 1 such that their boundary values belong to appropriate sets details will be given later on article id jdeq 1999 3669 available online at http www idealibrary com on

efficient two dimensional simulations of the fractional szabo equation with different time stepping schemes the modified szabo wave equation is one of the various models that have been developed to model the power law frequency dependent attenuation phenomena in lossy media the purpose of this study is to develop two different efficient numerical methods for the two dimensional szabo equation and to compare the relative merits of each method in both methods we employ the adi scheme to split directions however we use different time discretization specifically in the first adi method adi i we include a third order correction term to achieve second order convergence for smooth solutions hence extending the work of sun and wu 2006 in the second adi method adi ii we employ the scheme in zeng et al submitted for publication to two dimensional fractional wave equation using multiple correction terms to enhance accuracy for non smooth solutions our simulation results show that both methods are computationally efficient for the fractional wave equation but have different advantages in terms of accuracy specifically adi ii seems to produce more accurate results than adi i for non smooth solutions however for smooth solutions and fractional order close to two adi i seems to outperform adi ii

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

relaxation matched modeling of propagation through porous media including fractal pore structure the viscous and thermal dissipation of an acoustic wave propagating through a porous medium is shown to be characteristic of a relaxation process based on this interpretation a new model for the complex density dynamic permeability and bulk modulus which describe the viscous and thermal processes respectively is proposed the model is based on approximating the relaxational characteristic as opposed to previous models based on matching low and high frequency asymptotic behavior an advantage of the relaxation model is that one fewer parameter is required the relaxation model is also simpler than the commonly used zwikker kosten attenborough or biot allard models in the sense that it contains no bessel or kelvin functions and is not a modified form of the solution for uniform circular pores and unlike the delany bazley empirical equations the relaxation model is physically realistic for all frequencies extension of the relaxation model to fractal pore surfaces is also discussed

justification for power laws and fractional models in previous chapters it has been shown that the standard linear viscoelastic models can be generalized to their fractional counterparts it has also been demonstrated that the fractional zener kelvin voigt maxwell and newtonian models all satisfy physical criteria such as passivity and causality these models have temporal responses that are characterized by power laws either exactly or asymptotically and this corresponds to observed temporal characteristics in many media the character of the fractional derivative operator is also such that it describes temporal memory in the media

on the existence of local and global lagrangians for ordinary differential equations necessary and sufficient conditions for the existence of local and global lagrangians for ordinary differential equations of arbitrary order are described in terms of the geometry of higher order tangent bundles the results are applied to the study of gauge invariant differential equations and the second order differential equation defined by the 2 1 dimensional yang mills lagrangian with the chern simons term is discussed

time dependent lagrangian systems a geometric approach using semibasic forms a new presentation of the geometric formalism in r tq of the time dependent lagrangian systems is given the formalism is used to prove some properties of the equations determining the euler lagrange vector field xl for relating the theory with the time independent case and for studying the time dependent lagrangian inverse problem

a geometric approach to a generalized virial theorem the virial theorem introduced by clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics is studied by making use of symplectic formalism as an approach in the case of both the hamiltonian and lagrangian systems the possibility of establishing virial like theorems from one parameter groups of non strictly canonical transformations is analysed the case of systems with a position dependent mass is also discussed using the modern symplectic approach to quantum mechanics we arrive at the quantum virial theorem in full analogy with the classical case

five parameter fractional derivative model for polymeric damping materials fractional derivative models offer a powerful tool to describe the dynamic behaviour of real viscoelastic materials a version of the fractional derivative models characterized by five parameters is presented and investigated in this paper in order to describe asymmetrical loss factor peak and the high frequency behaviour of polymeric damping materials the speculative derivation of the model constitutive equation containing time derivatives of stress and strain of different orders is given the model behaviour is investigated in the frequency domain the physical meaning of the model parameters is defined and constraints on the parameter values are made it is shown that the asymmetry of loss peak and the high frequency behaviour of the model are governed by the difference between the order of time derivatives of stress and strain moreover it is shown that this difference is related to the high frequency limit value of the loss factor the model is fitted to experimental data on some polymeric damping materials to verify its behaviour

restrictions on wave equations for passive media most derivations of acoustic wave equations involve ensuring that causality is satisfied here the consequences of also requiring that the medium should be passive are explored this is a stricter criterion than causality for a linear system and implies that there are restrictions on the relaxation modulus and its first few derivatives the viscous and relaxation models of acoustics satisfy passivity and have restrictions on not only a few but all derivatives of the relaxation modulus these models are described as a system of springs and dampers with positive parameters and belong to the important class of completely monotone systems it is shown here that the attenuation as a function of frequency for such media has to increase slower than a linear function likewise the phase velocity has to increase monotonically this gives criteria on which one may judge whether a proposed wave equation is passive or not as illustrated by comparing two different versions of the viscous wave equation

a linear connection for higher order ordinary differential equations a linear connection is defined on the space of n jets of sections of where b is one dimensional this is a first step toward classifying n 1 th order time dependent ordinary differential equations the module of vector fields on splits into coordinate invariant components n 1 of them isomorphic and one component generated by the differential equation field the connection is shown to uniquely determine the differential equation field to which it is associated

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

hierarchical analogues to fractional relaxation equations recently fractional calculus has become an important tool in the analysis of slow relaxation phenomena such as stress strain relationships in polymeric materials in the rheological constitutive equations this implies the replacement of the first order derivatives by fractional derivatives here we show that such procedures have hierarchical mechanical analogues we focus on the generalized dashpot and the generalized maxwell model and display the corresponding arrangements our models allow a transparent interpretation of the parameters which enter the fractional equations and reveal that the internal dynamics are hierarchically constrained

on haar wavelet operational matrix of general order and its application for the numerical solution of fractional bagley torvik equation abstract in the present analysis the motion of an immersed plate in a newtonian fluid described by torvik and bagley s fractional differential equation 1 has been considered this bagley torvik equation has been solved by operational matrix of haar wavelet method the obtained result is compared with analytical solution suggested by podlubny 2 haar wavelet method is used because its computation is simple as it converts the problem into algebraic matrix equation

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

origin of the lagrangian constraints and their relation with the hamiltonian formulation the theory of presymplectic systems is used for the study of mechanical systems described by singular lagrangians in order to clarify the geometric meaning of the euler lagrange equations for such systems the two different types of primary constraint functions arising in the lagrangian formulation are analyzed by means of the relation between the image under the vertical endomorphism of the kernel of the presymplectic form l and its vertical part the connection with the hamiltonian dirac theory is also studied and the theory is illustrated with several examples

the cattaneo type time fractional heat conduction equation for laser heating in this paper the laser short pulse heating of a solid surface is considered the time fractional cattaneo model is used as the heat conduction model and the corresponding fractional heat conduction equation with a volumetric heat source is built the analytical solution for the temperature distribution is obtained using the laplace transformation method finally the numerical results are presented graphically for various values of model parameters and their effects on the speed of heat conduction propagation are discussed

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

stability switches and hopf bifurcations of an isolated population model with delay dependent parameters the dynamical behaviors of an isolated population model involving delay dependent parameters are investigated it is shown that the positive equilibrium switches from being stable to unstable and then back to stable as the delay increases and the hopf bifurcation occurs finite times between the two critical values of stability changes which can be analytically determined moreover the bifurcating periodic solutions are expressed analytically in an approximate form by the perturbation approach and floquet technique the direction and stability of the bifurcating periodic solutions are also determined finally the validity of the results is shown by the consistency with the numerical simulations

singular lagrangians affine in velocities the properties of lagrangians affine in velocities are analysed in a geometric way these systems are necessarily singular and exhibit in general gauge invariance the analysis of constraint functions and gauge symmetry leads us to a complete classification of such lagrangians

fractional differential models for anomalous diffusion abstract in this study we investigate three kinds of fractional differential models distributed order model variable order model and random order model for characterization of anomalous diffusion the characteristics physical advantages and potential applications of each model are highlighted the numerical simulations also validate our analytical and comparison results furthermore a generalized distributed variable order model and a more generalized distributed variable random order model are proposed to combine the advantages of distributed order model variable order model and random order model

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

a fourth order scheme for space fractional diffusion equations abstract a weighted and shifted difference formula is constructed based on the lubich operators which gives a forth order and unconditionally stable difference scheme for the cauchy problem of space fractional diffusion equations the novelty of the proposed method here is that only four weighted parameters are required compared to eight parameters used in the previous work to achieve the fourth order accuracy and to ensure the stability at the same time to verify the efficiency of the proposed scheme several numerical experiments for both one dimensional and two dimensional fractional diffusion problems are provided

fitted reproducing kernel hilbert space method for the solutions of some certain classes of time fractional partial differential equations subject to initial and neumann boundary conditions latterly many problems arising in different fields of science and engineering can be reduced by applying some appropriate discretization to a series of time fractional partial differential equations unlike the normal case derivative the differential order in such equations is with a fractional order which will lead to new challenges for numerical simulation the purpose of this analysis is to introduce the reproducing kernel hilbert space method for treating classes of time fractional partial differential equations subject to neumann boundary conditions with parameters derivative arising in fluid mechanics chemical reactions elasticity anomalous diffusion and population growth models the method provides appropriate representation of the solutions in convergent series formula with accurately computable components numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n term of the exact solutions finally the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time fractional neumann problems

time fractional heat equations and negative absolute temperatures the classical parabolic heat equation based on fourier s law implies infinite heat propagation speed to remedy this physical flaw the hyperbolic heat equation is used but it may instead predict temperatures less than absolute zero in recent years fractional heat equations have been proposed as generalizations of heat equations of integer order by simulating a 1d model problem of size on the order of a thermal energy carrier s mean free path length we have done a study of four fractional generalized cattaneo equations from compte and metzler 1997 called gce gce i gce ii and gce iii and also a fractional version of the parabolic heat equation we have observed that when the fractional order is large enough these equations give temperatures less than absolute zero but if the fractional order is small enough gce i does not have this problem when the domain length is comparable to the mean free path length with larger size gce i and gce iii also give non oscillating solutions for both small and large values of the fractional order

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

extended legendre transformation approach to the time dependent hamiltonian formalism the hamiltonian formalism for time dependent systems is developed using vector fields defined in the extended cotangent bundle t q r the extended hamiltonian h is obtained using the extended legendre transformation associated to an extended lagrangian l previously defined in t q r and then the techniques of symplectic mechanics are employed in t q r to study the properties of the extended hamiltonian system finally this approach is related with the multisymplectic formalism and other extended approaches

waves in fractional zener type viscoelastic media classical wave equation is generalized for the case of viscoelastic materials obeying fractional zener model instead of hooke s law cauchy problem for such an equation is studied existence and uniqueness of the fundamental solution is proven and solution is calculated

diffusive approximation of a time fractional burger s equation in nonlinear acoustics a fractional time derivative is introduced into burger s equation to model losses of nonlinear waves this term amounts to a time convolution product which greatly penalizes the numerical modeling a diffusive representation of the fractional derivative is adopted here replacing this nonlocal operator by a continuum of memory variables that satisfy local in time ordinary differential equations then a quadrature formula yields a system of local partial differential equations well suited to numerical integration the determination of the quadrature coefficients is crucial to ensure both the well posedness of the system and the computational efficiency of the diffusive approximation for this purpose optimization with constraint is shown to be a very efficient strategy strang splitting is used to solve successively the hyperbolic part by a shock capturing scheme and the diffusive part exactly numerical experiments are proposed to assess the efficiency of the numerical modeling and to illustrate the effect of the fractional attenuation on the wave propagation

the first integral method to study the burgers korteweg de vries equation in this paper applying the theory of commutative algebra we propose a new approach which we currently call the first integral method to study the burgers korteweg de vries equation

improved fractional sub equation method for 3 1 dimensional generalized fractional kdv zakharov kuznetsov equations in the present paper we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics namely the time fractional kdv zakharov kuznetsov kdv zk and space time fractional modified kdv zakharov kuznetsov mkdv zk equations by using improved fractional sub equation method as a result new types of exact analytical solutions are obtained the obtained results are shown graphically here the fractional derivative is described in the jumarie s modified riemann liouville sense

quasi coordinates from the point of view of lie algebroid structures a geometrical description of the lagrangian dynamics in quasi coordinates on the tangent bundle using the lie algebroid framework is given linear nonholonomic systems on lie algebroids are solved in local coordinates adapted to the constraints through lagrangian multipliers and gibbs appell generalized methods

computational algorithm for solving fredholm time fractional partial integrodifferential equations of dirichlet functions type with error estimates abstract numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science which means increased attention to fractional calculus this paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in hilbert space within appropriate kernel functions the algorithm relies on the reproducing kernel hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the fourier coefficients of orthogonalization process the caputo fractional derivatives are introduced to address these issues moreover the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions the superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples the outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method

on the inverse problem of the calculus of variations in field theory the inverse problem of the calculus of variations is investigated in the case of field theory uniqueness of the action principle is demonstrated for the vector laplace equation in a non decomposable riemannian space as well as for the harmonic map equation

statistical origins of fractional derivatives in viscoelasticity many linear viscoelastic materials show constitutive behavior involving fractional order derivatives linear time invariant systems without memory have exponential decay in time but contradictorily not the power law decay associated with fractional derivatives the physics literature has noted that apparently non exponential decays can be observed when several simultaneously decaying processes have closely spaced exponential decay rates many engineer researchers interested in viscoelastic damping however seem unaware of these observations in this letter i give an unoriginal explanation but with a fresh engineering flavor for the appearance of these fractional order derivatives by this explanation fractional order damping can be expected from many materials with sufficiently disordered dissipation mechanisms

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

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

the first integral method for some time fractional differential equations abstract in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the first integral method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations the power of this manageable method is presented by applying it to several examples this approach can also be applied to other nonlinear fractional differential equations

a geometrical version of the helmholtz conditions in time dependent lagrangian dynamics appropriate geometrical machinery for the study of time dependent lagrangian dynamics is developed it is applied to the inverse problem of the calculus of variations and a set of necessary and sufficient conditions for the existence of a lagrangian are given in terms of the existence of a 2 form with suitable properties which are exactly equivalent to the helmholtz conditions

a generalization of the fourier pseudospectral method the fourier pseudospectral ps method is generalized to the case of derivatives of nonnatural order fractional derivatives and irrational powers of the differential operators the generalization is straightforward because the calculation of the spatial derivatives with the fast fourier transform is performed in the wavenumber domain where the operator is an irrational power of the wavenumber modeling constant q propagation with this approach is highly efficient because it does not require memory variables or additional spatial derivatives the classical acoustic wave equation is modified by including those with a space fractional laplacian which describes wave propagation with attenuation and velocity dispersion in particular the example considers three versions of the uniform density wave equation based on fractional powers of the laplacian and fractional spatial derivatives

traveling waves to a burgers korteweg de vries type equation with higher order nonlinearities abstract in this paper first we survey some recent advances in the study of traveling wave solutions to the burgers korteweg de vries equation and some comments are given then we study a burgers korteweg de vries type equation with higher order nonlinearities a qualitative analysis to a two dimensional autonomous system which is equivalent to the burgers kdv type equation is presented and indicates that under certain conditions the burgers korteweg de vries type equation has neither nontrivial bell profile solitary waves nor periodic waves finally a solitary wave solution is obtained by means of the first integral method which is based on the ring theory of commutative algebra

on the coefficient of nonlinearity β in nonlinear acoustics an analysis valid to second order in the acoustic variables is used to investigate the origin and form of the coefficient of nonlinearity two cases are considered first for progressive plane waves the well known formula is 1 b 2a where b 2a is the coefficient of the quadratic term in the isentropic pressure density relation it is shown that the component 1 which is associated with convection comes solely from nonlinear terms in the equation of continuity the second component comes from a quadratic term in the equation of state the momentum equation is linear for this type of wave motion and therefore contributes nothing to the expression for second for noncollinear interaction of two plane waves quasilinear analysis shows that the expression for appropriate for the sum and difference frequency waves has three components in this case the momentum equation does contribute to the formula for most practical problems the three component formula reduces to cos b 2a where is the angle of interaction the factor cos represents the combination of contributions cos2 2 and sin2 2 from the continuity and momentum equations respectively

jet fields connections and second order differential equations jet fields are defined on fibred manifolds and represented as type 1 1 tensor fields corresponding to cartan ehresmann connections when defined on the first jet bundle they may be used to generalise the second order differential equation fields used in theoretical mechanics and to extend several existing results to field theories