a robust numerical method for a fractional differential equation abstract this paper is devoted to giving a rigorous numerical analysis for a fractional differential equation with order 0 1 first the fractional differential equation is transformed into an equivalent volterra integral equation of the second kind with a weakly singular kernel based on the a priori information about the exact solution an integral discretization scheme on an a priori chosen adapted mesh is proposed by applying the truncation error estimate techniques and a discrete analogue of gronwall s inequality it is proved that the numerical method is first order convergent in the discrete maximum norm numerical results indicate that this method is more accurate and robust than finite difference methods when is close to 0

particle tracking for fractional diffusion with two time scales abstract previous work y zhang m m meerschaert b baeumer particle tracking for time fractional diffusion phys rev e 78 2008 036705 showed how to solve time fractional diffusion equations by particle tracking this paper extends the method to the case where the order of the fractional time derivative is greater than one a subordination approach treats the fractional time derivative as a random time change of the corresponding cauchy problem with a first derivative in time one novel feature of the time fractional case of order greater than one is the appearance of clustering in the operational time subordinator which is non markovian solutions to the time fractional equation are probability densities of the underlying stochastic process the process models movement of individual particles the evolution of an individual particle in both space and time is captured in a pair of stochastic differential equations or langevin equations monte carlo simulation yields particle location and the ensemble density approximates the solution to the variable coefficient time fractional diffusion equation in one or several spatial dimensions the particle tracking code is validated against inverse transform solutions in the simplest cases further applications solve model equations for fracture flow and upscaling flow in complex heterogeneous porous media these variable coefficient time fractional partial differential equations in several dimensions are not amenable to solution by any alternative method so that the grid free particle tracking approach presented here is uniquely appropriate

wigner ville spectral analysis of nonstationary processes the wigner ville spectrum has been recently introduced as the unique generalized spectrum for time varying spectral analysis its properties are revised with emphasis on its central role in the analysis of second order properties of nonstationary random signals we propose here a general class of spectral estimators of the wigner ville spectrum this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time frequency representations of deterministic signals with finite energy classical estimators like short time periodograms and the recently introduced pseudo wigner estimators are shown to be special cases of the general class the generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo wigner estimators the effective numerical implementation by an n point fft of pseudo wigner estimators of 2n points is indicated and various examples are given

variable rate vector quantization for medical image compression three techniques for variable rate vector quantizer design are applied to medical images the first two are extensions of an algorithm for optimal pruning in tree structured classification and regression due to breiman et al the code design algorithms find subtrees of a given tree structured vector quantizer tsvq each one optimal in that it has the lowest average distortion of all subtrees of the tsvq with the same or lesser average rate since the resulting subtrees have variable depth natural variable rate coders result the third technique is a joint optimization of a vector quantizer and a noiseless variable rate code this technique is relatively complex but it has the potential to yield the highest performance of all three techniques

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

non linear boundary value problems involving caputo derivatives of complex fractional order abstract we study approximate solutions of c d t y t f t y t separately for 0 1 and 1 2 with different boundary data where c d t is the caputo fractional derivative of complex order for this purpose we use the expansion formula for such fractional derivatives and prove the existence and the uniqueness of approximate solutions under certain conditions and their convergence to the original solutions

alternatives to the discrete cosine transform for irreversible tomographic image compression full frame irreversible compression of medical images is currently being performed using the discrete cosine transform dct although the dct is the optimum fast transform for video compression applications the authors show here that it is outperformed by the discrete fourier transform dft and discrete hartley transform dht for images obtained using positron emission tomography pet and magnetic resonance imaging mri and possibly for certain types of digitized radiographs this difference occurs because pet and mri images are characterized by a roughly circular region d of non zero intensity bounded by a region r in which the image intensity is essentially zero clipping r to its minimum extent can reduce the number of low intensity pixels but the practical requirement that images be stored on a rectangular grid means that a significant region of zero intensity must remain an integral part of the image to be compressed with this constraint imposed the dct loses its advantage over the dft because neither transform introduces significant artificial discontinuities the dft and dht have the further important advantage of requiring less computation time than the dct

a combined transform coding ctc scheme for medical images a combined transform coding ctc scheme is proposed to reduce the blocking artifact of conventional block transform coding and hence to improve the subjective performance the proposed ctc scheme is described and its information theoretic properties are investigated computer simulation results for a class of chest x ray images are presented a comparison between the ctc scheme and the conventional discrete cosine transform dct and discrete walsh hadamard transform dwht demonstrates the performance improvement of the proposed scheme in addition combined coding can also be used in noiseless coding yielding a slight improvement in the compression performance if it is used properly

piecewise polynomial collocation for linear boundary value problems of fractional differential equations we consider a class of boundary value problems for linear multi term fractional differential equations which involve caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties the numerical solution of boundary value problems by piecewise polynomial collocation methods is discussed in particular we study the attainable order of convergence of proposed algorithms and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by two numerical examples

a nonpolynomial collocation method for fractional terminal value problems in this paper we propose a nonpolynomial collocation method for solving a class of terminal or boundary value problems for differential equations of fractional order 0 1 the approach used is based on the equivalence between a problem of this type and a fredholm integral equation of a particular form taking into account the asymptotic behaviour of the solution of this problem we propose a nonpolynomial collocation method on a uniform mesh we study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained in order to illustrate the theoretical results and the performance of the method we present several numerical examples

coupled continuous time random walks in finance continuous time random walks ctrws are used in physics to model anomalous diffusion by incorporating a random waiting time between particle jumps in finance the particle jumps are log returns and the waiting times measure delay between transactions these two random variables log return and waiting time are typically not independent for these coupled ctrw models we can now compute the limiting stochastic process just like brownian motion is the limit of a simple random walk even in the case of heavy tailed power law price jumps and or waiting times the probability density functions for this limit process solve fractional partial differential equations in some cases these equations can be explicitly solved to yield descriptions of long term price changes based on a high resolution model of individual trades that includes the statistical dependence between waiting times and the subsequent log returns in the heavy tailed case this involves operator stable space time random vectors that generalize the familiar stable models in this paper we will review the fundamental theory and present two applications with tick by tick stock and futures data

a mixed transform approach for efficient compression of medical images a novel technique is presented to compress medical data employing two or more mutually nonorthogonal transforms both lossy and lossless compression implementations are considered the signal is first resolved into subsignals such that each subsignal is compactly represented in a particular transform domain an efficient lossy representation of the signal is achieved by superimposing the dominant coefficients corresponding to each subsignal the residual error which is the difference between the original signal and the reconstructed signal is properly formulated adaptive algorithms in conjunction with an optimization strategy are developed to minimize this error both two dimensional 2 d and three dimensional 3 d approaches for the technique are developed it is shown that for a given number of retained coefficients the discrete cosine transform dct walsh mixed transform representation yields a more compact representation than using dct or walsh alone this lossy technique is further extended for the lossless case the coefficients are quantized and the signal is reconstructed the resulting reconstructed signal samples are rounded to the nearest integer and the modified residual error is computed this error is transmitted employing a lossless technique such as the huffman coding it is shown that for a given number of retained coefficients the mixed transforms again produces the smaller rms modified residual error the first order entropy of the error is also smaller for the mixed transforms technique than for the dct thus resulting in smaller length huffman codes

numerical solutions of nonlinear fractional differential equations by alternative legendre polynomials abstract in this paper numerical techniques are presented for solving initial value problems of nonlinear fractional differential equations the method is implemented by applying alternative legendre polynomials the operational matrix of fractional integration and the product for the alternative legendre polynomials are derived in order to transform the nonlinear equations into a system of algebraic equations the study of the error analysis of the obtained method is also considered furthermore numerical examples demonstrate that this method is applicable and accurate

a differential index assignment scheme for tree structured vector quantization a differential index di assignment scheme is proposed for the image encoding system in which a variable length tree structured vector quantizer vltsvq is adopted each source vector is quantized into a terminal node of vltsvq and each terminal node is represented as a unique binary vector the proposed index assignment scheme utilizes the correlation between interblocks of the image to increase the compression ratio with the image quality maintained simulation results show that the proposed scheme achieves a much higher compression ratio than the conventional one does and that the amount of the bit rate reduction of the proposed scheme becomes large as the correlation of the image becomes large the proposed encoding scheme can be effectively used to encode mr images whose pixel values are in general highly correlated with those of the neighbor pixels

similarity and other spectral relations for symmetric cones abstract the similarity relations that are derived in this paper reduce to well known results in the special case of symmetric matrices in particular for two positive definite matrices x and y the square of the spectral geometric mean is known to be similar to the matrix product xy it is shown in this paper that this property carries over to symmetric cones more elementary similarity relations such as xy 2 x yx 2 y are generalized as well we also extend the result that the eigenvalues of a matrix product xy are less dispersed than the eigenvalues of the jordan product xy yx 2 the paper further contains a number of inequalities on norms and spectral values this type of inequality is often used in the analysis of interior point methods in optimization we also derive an extension of stein s theorem to symmetric cones

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

central difference approximation of convection in caputo fractional derivative two point boundary value problems a rigorous numerical analysis is given for a fractional derivative two point boundary value problem the highest order term of the differential operator is a caputo fractional derivative of order 1 2 the second and higher order derivatives of the solution of the problem are in general unbounded at one end of the interval which creates difficulties for the analysis the problem is discretized on an equidistant mesh of diameter h using a standard central difference approximation of the convection term and nodal convergence of order o h 1 is proved provided that h satisfies a stability condition that depends on and the convective coefficient this condition may be restrictive when is near 1 when it is violated the solution may exhibit small oscillations but these can be removed by a simple and inexpensive postprocessing technique numerical results are given to display the performance of the method

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results