the eigenvalues of second order spectral differentiation matrices the eigenvalues of the pseudospectral second derivative matrix with homogeneous dirichlet boundary conditions are important in many applications of spectral methods this paper investigates some of their properties numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately but the errors in the remaining ones are large it is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid in particular it is proved that 7r points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a chebyshev distribution of nodes and six points per wavelength for a uniform distribution these results are in agreement with the observed fractions of accurate eigenvalues by using the characteristic polynomial a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2 or better the effect of filtering on the eigenvalues is studied numerically

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

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

causal theories and data for acoustic attenuation obeying a frequency power law this study compares causal theories based on kramers kronig relations fractional calculus and on those derived from new time domain causal relationships to diverse data all these theories are based on the assumptions that the functional form of the attenuation persists beyond the measurement range and that attenuation is much smaller than the wave number the data for lossy media with attenuation having a power law frequency dependence with an exponent y include cases for both liquids and solids ranging from acoustic to ultrasound frequencies data are in closer correspondence with the new theory which predicts decreasing dispersion as the power exponent y approaches zero or an even integer experimental results and supporting evidence show that the classical case of frequency squared attenuation is dispersionless an approximate nearly local kramers kronig theory is in agreement with the time causal theory when the exponent is close to one but deviates for other values the comprehensive time causal theory is shown to be equivalent to two other theories derived from exact kramers kronig relations and from fractional calculus and it covers the y odd integer cases which are missing or incomplete in these approaches attenuation dispersion relations are presented in two forms one for a general frequency range and another for a finite range it is demonstrated that complete velocity dispersion within a signal bandwidth can be predicted from knowledge of the attenuation data and velocity at a single frequency including the velocity at either zero frequency y 1 and or at a high frequency limit 0 y 1 in addition attenuation data can be predicted from velocity data

the fourier pseudospectral method with a restrain operator for the rlw equation in this paper we develop a new fourier pseudospectral method with a restrain operator for the modified korteweg de vries mkdv equation we prove the generalized stability of the scheme from which the convergence follows

fractional model equation for anomalous diffusion abstract in recent years the phenomenon of anomalous diffusion has attracted more and more attention one of the main impulses was initiated by de gennes idea of the ant in the labyrinth several authors presented asymptotic probability density functions for the location of a random walker on a fractal object as this density function and the time dependence of its second moment are now well established a modified diffusion equation providing the correct result is formulated the parameters of this fractional partial differential equation are uniquely determined by the fractal hausdorff dimension of the underlying object and the anomalous diffusion exponent the presented equation reduces exactly to the ordinary isotropic diffusion equation by appropriate choice of the parameters a closed form solution is given in terms of fox s h function in the asymptotic case a halved diffusion equation can be established furthermore the differences to equations considered previously are discussed

fractional diffusion equation on fractals one dimensional case and asymptotic behaviour a fractional differential equation is studied and its application for describing diffusion on random fractal structures is considered it represents the simplest generalization of the fractional diffusion equation valid in euclidean systems the solution of the fractional equation in one dimension is discussed and compared with exact results for the fractional brownian motion and the one dimensional version of the standard diffusion equation on fractals in higher dimensions it correctly describes the asymptotic scaling behaviour of the probability density function on random fractals as obtained recently by using scaling arguments and exact enumeration calculations for the infinite percolation cluster at criticality

pseudospectral solution of near singular problems using numerical coordinate transformations based on adaptivity the work presented here describes a method of coordinate transformation that enables spectral methods to be applied efficiently to differential problems with steep solutions the approach makes use of the adaptive finite difference method presented by huang and sloan siam j sci comput 15 1994 pp 776 797 this method is applied on a coarse grid to obtain a rough approximation of the solution and a suitable adapted mesh the adaptive finite difference solution permits the construction of a smooth coordinate transformation that relates the computational space to the physical space the map between the spaces is based on chebyshev polynomial interpolation finally the standard pseudospectral ps method is applied to the transformed differential problem to obtain highly accurate nonoscillatory numerical solutions numerical results are presented for steady problems in one and two space dimensions

on discontinuous solutions of hyperbolic equations abstract a modified fourier method which is applicable to numerical simulations of discontinuous solutions of hyperbolic equations is presented truncated fourier series representations are de truncated at every timestep by utilising step functions in the reconstruction of the discontinuous solutions the de truncated expansions are used to eliminate the numerical dispersion and diffusion which normally ruin the discontinuous solutions in a few timesteps when traditional fourier methods are applied

generalized viscoelastic models their fractional equations with solutions recently fractional calculus fc has encountered much success in the description of complex dynamics in particular fc has proved to be a valuable tool to handle viscoelastic aspects in this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models especially the maxwell the kelvin voigt the zener and the poynting thomson model to this end we introduce a fractional element in addition to the standard purely elastic and purely viscous elements as we proceed to show many of the fractional differential equations which we obtain by this construction method admit closed form analytical solutions in terms of fox h functions of the minag leffler type

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

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

fox function representation of non debye relaxation processes applying the liouville riemann fractional calculus we derive and solve a fractional operator relaxation equation we demonstrate how the exponent of the asymptotic power law decay t relates to the order of the fractional operatordv dtv 0 1 continuous time random walk ctrw models offer a physical interpretation of fractional order equations and thus we point out a connection between a special type of ctrw and our fractional relaxation model exact analytical solutions of the fractional relaxation equation are obtained in terms of fox functions by using laplace and mellin transforms apart from fractional relaxation fox functions are further used to calculate fourier integrals of kohlrausch williams watts type relaxation functions because of its close connection to integral transforms the rich class of fox functions forms a suitable framework for discussing slow relaxation phenomena

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

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

singularity categories and singular equivalences for resolving subcategories let mathcal x be a resolving subcategory of an abelian category in this paper we investigate the singularity category mathsf d sg underline mathcal x mathsf d b mathsf mod underline mathcal x mathsf k b mathsf proj mathsf mod underline mathcal x of the stable category underline mathcal x of mathcal x we consider when the singularity category is triangle equivalent to the stable category of gorenstein projective objects and when the stable categories of two resolving subcategories have triangle equivalent singularity categories applying this to the module category of a gorenstein ring we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type mathsf a 1 we also generalize several results of yoshino on totally reflexive modules

on the gottlieb turkel time filter for chebyshev spectral methods abstract highly accurate solutions to hyperbolic boundary value problems may be obtained using chebyshev spectral methods however with explicit time differencing the increased resolution of the chebyshev series near the boundaries necessitates using very small time steps for stability thus compromising efficiency recently a method of time filtering was proposed which is claimed to make explicit time integrations unconditionally stable so that time steps may be chosen by accuracy requirements alone an analysis of this method shows that the filtering can in fact lead to absolute instability for any time step and that it does not relax the stability condition of the unfiltered method in a useful manner

frequency dependent ultrasonic differentiation of normal and diffusely diseased liver the attenuation coefficient in 38 pathologically graded in vitro liver specimens was measured over a frequency range from 1 25 8 mhz and fitted to the power law model the attenuation in the normal group n 17 exhibited a frequency dependence of the form 0 399f1 139 in the mild disease group n 13 it exhibited a dependence of the form 0 395f1 212 and in the moderate severe disease group n 8 it exhibited a dependence of the form 0 391f1 325 using a student s t test it is shown that due to these differences in the frequency dependence the statistical significance level at which the null hypothesis regarding the difference between the mean attenuation slopes of any two of these categories is rejected is a strong function of frequency in the range of 1 4 mhz the significance level relating to the difference between the normal and moderate severe disease group is more than one order of magnitude better than the other categories in all cases no substantial improvement occurs beyond 4 mhz it is

combined finite element and pseudospectral method for the two dimensional evolutionary navier stokes equations a combined finite element and pseudospectral scheme is proposed for solving the two dimensional evolutionary navier stokes equations the artificial compressibility and filtering techniques are used it is also shown that the inf sup condition holds naturally for this combined method the generalized stability and optimal convergence rate in l 2 norm are proved some numerical results are presented

a 3 d discontinuous spectral element time domain method for maxwell s equations a discontinuous spectral element time domain method is proposed to analyze transient electromagnetic properties of general 3 d structures this method is advantageous in that its mass matrices are block diagonal due to the gauss lobatto legendre polynomials and it allows different orders of basis functions for each subdomain the riemann solver is employed in the boundary integral terms to communicate fields between adjacent subdomains perfectly matched layers are utilized to truncate the computational domain galerkin method is used for spatial discretization and a fourth order runge kutta scheme is employed for the time integration the validity of the proposed approach is demonstrated through several numerical examples of initial value problems and scattering problems

the effect of filtering on the pseudospectral solution of evolutionary partial differential equations abstract this paper examines the effect of filtering on the solution of time dependent partial differential equations by the pseudospectral method it is shown that if spatial discretisation is effected using the fourier pseudospectral method the computed solution will be an approximation to the solution of a modified differential equation the changes in dispersive and dissipative properties induced by the modification are examined and numerical results are presented which illustrate these changes for both linear and nonlinear equations numerical results are also presented which show the effect of filtering on chebyshev pseudospectral solutions of time dependent equations