stable numerical solution of a fractional diffusion inverse heat conduction problem the ill posed problem of attempting to recover the boundary temperature and the heat flux functions from one measured transient data temperature at some interior point of a one dimensional semi infinite conductor when the governing linear diffusion equation is of fractional type is discussed a simple algorithm based on space marching mollification techniques is introduced for the numerical solution of the discrete problem stability bounds error estimates and numerical examples of interest are also presented

analytical and numerical solutions of a one dimensional fractional in space diffusion equation in a composite medium in this paper a one dimensional space fractional diffusion equation in a composite medium consisting of two layers in contact is studied both analytically and numerically since domain decomposition is the only approach available to solve this problem we at first investigate analytical and numerical strategies for a composite medium with the same fractal dimension in each layer to ascertain which domain decomposition approach is the most accurate and consistent with a global solution methodology which is available in this case we utilise a matrix representation of the fractional in space operator to generate a system of linear odes with the matrix raised to the same fractional exponent we show that the global and domain decomposition numerical strategies for this problem produce simulation results that are in good agreement with their analytic counterparts and conclude that the domain decomposition that imposes the neumann condition at the interface produces the most consistent results finally we carry this finding to study the composite problem with different fractal dimensions where we again favourably compare analytic and numerical solutions the resulting method can be naturally extended to space fractional diffusion in a composite medium consisting of more than two layers

fast finite difference approximation for identifying parameters in a two dimensional space fractional nonlocal model with variable diffusivity coefficients in this paper we consider an inverse problem for identifying the fractional derivative indices in a two dimensional space fractional nonlocal model based on a generalization of the two sided riemann liouville formulation with variable diffusivity coefficients first we derive an implicit difference method idm for the direct problem and the stability and convergence of the idm are discussed second for the implementation of the idm we develop a fast bi conjugate gradient stabilized method fbi cgstab that is superior in computational performance to gaussian elimination and attains the same accuracy third we utilize the levenberg marquardt l m regularization technique combined with the armijo rule the popular inexact line search condition to solve the modified nonlinear least squares model associated with the parameter identification finally we carry out numerical tests to verify the accuracy and efficiency of the idm numerical investigations are performed with both accurate data and noisy

an efficient implicit fem scheme for fractional in space reaction diffusion equations fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity however the presence of a fractional differential operator causes memory time fractional or nonlocality space fractional issues that impose a number of computational constraints in this paper we develop efficient scalable techniques for solving fractional in space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector our approach is show cased by solving the fractional fisher and fractional allen cahn reaction diffusion equations in two and three spatial dimensions and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying laplacian operator

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

a boundary collocation method for anomalous heat conduction analysis in functionally graded materials abstract this paper applies a semi analytical boundary collocation method the singular boundary method sbm in conjunction with the dual reciprocity method drm and laplace transformation technique to solve anomalous heat conduction problems under functionally graded materials fgms in this study transient heat conduction equation with caputo time fractional derivative is considered to describe anomalous heat conduction phenomena in the present numerical implementation laplace transformation and numerical inverse laplace transformation are used to deal with the caputo time fractional derivative which avoid the effect of time step on the computational efficiency of the time fractional derivation approximation the sbm in conjunction with the drm is employed to obtain the high accurate results in the solution of laplace transformed time independent nonhomogeneous problems to demonstrate the effectiveness of the proposed method for anomalous heat conduction analysis under functionally graded materials three numerical examples are considered and the present results are compared with known analytical solutions and comsol simulation

a fast finite difference method for distributed order space fractional partial differential equations on convex domains abstract fractional partial differential equations pdes provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long range spatial interactions which cannot be modeled accurately by classical second order diffusion equations however numerical methods for space fractional pdes usually generate dense or full stiffness matrices for which a direct solver requires o n 3 computations per time step and o n 2 memory where n is the number of unknowns the significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three dimensional space fractional diffusion equations computationally intractable fast numerical methods were previously developed for space fractional pdes on multidimensional rectangular domains without resorting to lossy compression but rather via the exploration of the tensor product form of the toeplitz like decompositions of the stiffness matrices in this paper we develop a fast finite difference method for distributed order space fractional pdes on a general convex domain in multiple space dimensions the fast method has an optimal order storage requirement and almost linear computational complexity without any lossy compression numerical experiments show the utility of the method

stability estimate and the modified regularization method for a cauchy problem of the fractional diffusion equation in this paper we investigate a non characteristic cauchy problem for a fractional diffusion equation using the fourier transformation technique we give a conditional stability estimate on the solution since the problem is highly ill posed in the hadamard sense a modified version of the tikhonov regularization technique is devised for stable numerical reconstruction of the solution an error bound with optimal order is proven for illustration several numerical experiments are constructed to demonstrate the feasibility and efficiency of the proposed method

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

identifying an unknown source in time fractional diffusion equation by a truncation method in this paper an inverse source problem for the time fractional diffusion equation is investigated we prove a conditional stability for this problem a truncation method is presented to deal with the ill posedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter numerical examples show that the proposed method is effective and stable

a new regularization method for solving a time fractional inverse diffusion problem in this paper we consider an inverse problem for a time fractional diffusion equation in a one dimensional semi infinite domain the temperature and heat flux are sought from a measured temperature history at a fixed location inside the body we show that such problem is severely ill posed and further apply a new regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under the a priori bound assumptions for the exact solution finally numerical examples are given to show that the proposed numerical method is effective

on the stable numerical evaluation of caputo fractional derivatives the computation of caputo s fractional derivatives in the presence of measured data is considered as an ill posed problem and treated by mollification techniques it is shown that with the appropriate choice of the radius of mollification the method is a regularizing algorithm and the order of convergence is derived error estimates are included together with numerical examples of interest

unsteady flow of a generalized maxwell fluid with fractional derivative due to a constantly accelerating plate the velocity field and the adequate shear stress corresponding to the unsteady flow of a generalized maxwell fluid are determined using fourier sine and laplace transforms they are presented as a sum of the newtonian solutions and the corresponding non newtonian contributions the similar solutions for maxwell and newtonian fluids performing the same motion are obtained as limiting cases of our general results graphical illustrations show that the velocity profiles corresponding to a generalized maxwell fluid are going to that for an ordinary maxwell fluid if a 1

the space fractional diffusion equation with feller s operator abstract this paper investigates the space fractional diffusion equation with fractional feller s operator the green s function is obtained by using fourier transform and the analytical solutions of some space fractional diffusion equations with initial or initial and boundary condition are obtained in terms of green s function in addition numerical simulations are discussed the results indicate that the effect range of skewness parameter has more effect on probability density than that of parameter the results also explain the property of the skewness and long tail in the asymmetry diffusion process

thermal wave model of bioheat transfer with modified riemann liouville fractional derivative in this paper a new fractional thermal wave model of the bioheat transfer ftwmbt caused by spatial heating is built using taylor s series expansion of modified riemann liouville fractional derivatives a one dimensional analytical solution of the ftwmbt in a finite medium is obtained the ftwmbt in the case 1 interpolates the standard thermal wave model of bioheat transfer and the well known pennes bioheat equation 0 finally numerical results are presented graphically for various values of different parameters this study demonstrates that fractional models can provide a unified approach to examine the heat transfer in biological tissue

a fractional tikhonov regularization method for identifying a space dependent source in the time fractional diffusion equation abstract in this paper we are concerned with an inverse source problem for the time fractional diffusion equation with variable coefficients in a general bounded domain the problem is mildly ill posed a new fractional tikhonov method is proposed we discuss the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule and prove the corresponding convergence estimates numerical experiments are conducted for illustrating effectiveness of the proposed method

an inverse problem for fractional diffusion equation in 2 dimensional case stability analysis and regularization abstract in this paper we investigate an inverse problem for a time fractional diffusion equation which is highly ill posed in the two dimensional setting based on an a priori assumption we give a conditional stability result some new regularization methods are constructed for solving the inverse problem and the corresponding error estimates are proved for numerical illustration several examples are constructed to demonstrate the feasibility and efficiency of the proposed methods

the fractional finite hankel transform and its applications in fractal space in the present work a generalized finite hankel transform is derived which is useful in solving equations in fractal dimension df and involving a fractal diffusion coefficient d0r the corresponding inversion formula is established and some properties are given then the transform is successfully used to solve a class of time fractional diffusion equations in fractional spatial dimension with an absorbent term and schrodinger equation in fractional dimensional space green s functions and exact wave function of the above problems are found

a posteriori regularization parameter choice rule for a modified kernel method for a time fractional inverse diffusion problem abstract in this paper we consider a time fractional inverse diffusion problem tfidp where the measured data is given at x 1 and the solution is required in the interval 0 x 1 we show that tfidp is severely ill posed and further apply a modified kernel method to deal with this problem based on the solution in the frequency domain the convergence estimate is obtained by using a posteriori parameter choice rule numerical examples show that the proposed method works effectively

parameters estimation for a new anomalous thermal diffusion model in layered media in this paper we study an inverse problem of parameters estimation for a new time fractional heat conduction model in multilayered medium in the anomalous thermal diffusion model we consider the fractional derivative boundary conditions and the conduction obeys modified fourier law with riemannliouville fractional operator of different order in each layer for the direct problem we construct an effective finite difference scheme by using the balance method to deal with the discontinuity interface for the inverse problem we apply the nonlinear conjugate gradient ncg method with different conjugated coefficients to simultaneously identify the fractional exponent in each layer finally we use experimental data to verify the effectiveness of the proposed technique in which the jacobian matrix is achieved by a derivative free approach we analyze the sensitivity coefficients and the convergence behaviors of the ncg algorithm the simulation results confirm that the fractional heat conduction model with estimated parameters gives a more accurate fitting than the classical counterpart and the ncg method is a feasible and effective technique for the inverse problem of parameters estimation in fractional model

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

a new regularization method for the time fractional inverse advection dispersion problem in this paper we consider the time fractional inverse advection dispersion problem tfiadp in a quarter plane the solute concentration and dispersion flux are sought from a measured concentration history at a fixed location inside the body such a problem is obtained from the classical advection dispersion equation by replacing the first order time derivative with the caputo fractional derivative of order alpha 0 alpha 1 we show that the tfiadp is severely ill posed and we further apply a new regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under a priori bound assumptions for the exact solution finally one numerical example is given to show that the proposed numerical method is effective

novel techniques in parameter estimation for fractional dynamical models arising from biological systems in recent years both parameter estimation and fractional calculus have attracted a considerable interest parameter estimation of the fractional dynamical models is a new topic in this paper we consider novel techniques for parameter estimation of fractional nonlinear dynamical models in systems biology first a computationally effective fractional predictor corrector method is proposed for simulating fractional complex dynamical models second we convert the parameter estimation of fractional complex dynamical models into a minimization problem of the unknown parameters third a modified hybrid simplex search mhss and a particle swarm optimization pso is proposed finally these techniques are applied to a dynamical model of competence induction in a cell with measurement error and noisy data some numerical results are given that demonstrate the effectiveness of the theoretical analysis

initial value boundary value problems for fractional diffusion wave equations and applications to some inverse problems abstract we consider initial value boundary value problems for fractional diffusion wave equation t u x t l u x t where 0 2 where l is a symmetric uniformly elliptic operator with t independent smooth coefficients first we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to and the proofs are based on the eigenfunction expansions second for 0 1 we apply the eigenfunction expansions and prove i stability in the backward problem in time ii the uniqueness in determining an initial value and iii the uniqueness of solution by the decay rate as t iv stability in an inverse source problem of determining t dependent factor in the source by observation at one point over 0 t

structure from motion with impaired local speed and global motion field computations humans can recover the three dimensional structure of moving objects from their changing two dimensional retinal image in the absence of other cues to three dimensional structure wallach and o connell 1953 braunstein 1976 in this paper we describe a patient a f with bilateral lesions involving the visual cortex who is severely impaired on computing local speed and global motion fields but who can recover structure from motion the data suggest that although possibly useful global motion fields are not necessary for deriving structure from motion we discuss these results from the perspective of theoretical models for this computation

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

spectral regularization method for solving a time fractional inverse diffusion problem abstract in this paper we consider an inverse problem for a time fractional diffusion equation with one dimensional semi infinite domain the temperature and heat flux are sought from a measured temperature history at a fixed location inside the body we show that such problem is severely ill posed and further apply a spectral regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under a priori bound assumptions for the exact solution finally numerical examples are given to show that the proposed numerical method is effective

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 schrodinger equation with noninteger dimensions the spatial and time dependent solutions of the schrodinger equation incorporating the fractional time derivative of distributed order and extending the spatial operator to noninteger dimensions are investigated they are obtained by using the green function approach in two situations the free case and in the presence of a harmonic potential the results obtained show an anomalous spreading of the wave packet which may be related to an anomalous diffusion process

analytical solutions to time fractional partial differential equations in a two dimensional multilayer annulus abstract in this paper analytical solutions to time fractional partial differential equations in a multi layer annulus are presented the final solutions are obtained in terms of mittag leffler function by using the finite integral transform technique and laplace transform technique in addition the classical diffusion equation 1 the helmholtz equation 0 and the wave equation 2 are discussed as special cases finally an illustrative example problem for the three layer semi circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative

effective medium equations for fractional fick s law in porous media abstract this paper studies reaction diffusion phenomena in disordered porous media with non fickian diffusion effects the aim is to obtain an effective medium equation of the concentration dynamics having a fractional fick s law description for the particles flux since the methodology is based on a volume averaging approach a fractional spatial averaging theorem is developed to interchange averaging integration and fractional differentiation model structure simplifications are made on the basis of an order of magnitude analysis from physical insights the closure problem associated with the effective diffusivity definition is also developed showing that the macroscale diffusion parameter is affected by i the scaling from mesoscales to macroscales and ii by the disordered structure of the porous medium

the levinson theorem the levinson theorem is a fundamental theorem in quantum scattering theory which shows the relation between the number of bound states and the phase shift at zero momentum for the schrodinger equation the levinson theorem was established and developed mainly with the jost function with the green function and with the sturm liouville theorem in this review we compare three methods of proof study the conditions of the potential for the levinson theorem and generalize it to the dirac equation the method with the sturm liouville theorem is explained in some detail references to development and application of the levinson theorem are introduced

exact solutions of fractional schrodinger like equation with a nonlocal term we study the time space fractional schrodinger equation with a nonlocal potential by the method of fourier transform and laplace transform the green function and hence the wave function is expressed in terms of h functions graphical analysis demonstrates that the influence of both the space fractal parameter and the nonlocal parameter on the fractional quantum system is strong indeed the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative

existence and uniform stability analysis of fractional order complex valued neural networks with time delays this paper deals with the problem of existence and uniform stability analysis of fractional order complex valued neural networks with constant time delays complex valued recurrent neural networks is an extension of real valued recurrent neural networks that includes complex valued states connection weights or activation functions this paper explains sufficient condition for the existence and uniform stability analysis of such networks three numerical simulations are delineated to substantiate the effectiveness of the theoretical results

the time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems abstract in this paper a time fractional fourier law is obtained from fractional calculus according to the fractional fourier law a fractional heat conduction equation with a time fractional derivative in the general orthogonal curvilinear coordinate system is built the fractional heat conduction equations in other orthogonal coordinate systems are readily obtainable as special cases in addition we obtain the solution of the fractional heat conduction equation in the cylindrical coordinate system in terms of the generalized h function using integral transformation methods the fractional heat conduction equation in the case 0 1 interpolates the standard heat conduction equation 1 and the localized heat conduction equation 0 finally numerical results are presented graphically for various values of order of fractional derivative

the fractional fick s law for non local transport processes fick s law is extensively adopted as a model for standard diffusion processes however requiring separation of scales it is not suitable for describing non local transport processes we discuss a generalized non local fick s law derived from the space fractional diffusion equation generating the levy feller statistics this means that the fundamental solutions can be interpreted as levy stable probability densities in the feller parameterization with index 1 2 and skewness 2 we explore the possibility of defining an equivalent local diffusivity by displaying a few numerical case studies concerning the relevant quantities flux and gradient it turns out that the presence of asymmetry 0 plays a fundamental role it produces shift of the maximum location of the probability density function and gives raise to phenomena of counter gradient transport

scattering theory for one dimensional systems with dx v x 0 abstract low energy scattering for schrodinger operators of the type h v in l2 r with rdx v x 0 is considered the possibility of zero energy eigenstates of h is taken into account explicitly in particular a laurent expansion for the transition operator and recursion relations for its coefficients are provided and the leading behaviour of the scattering operator is given in all possible cases

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

a fast finite difference method for three dimensional time dependent space fractional diffusion equations and its efficient implementation fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the non local property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved with gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory to store where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of these methods makes a numerical simulation of three dimensional space fractional diffusion equations computationally prohibitively expensive in this paper we develop an efficient and faithful solution method for the implicit finite difference discretization of time dependent space fractional diffusion equations in three space dimensions by carefully analyzing the structure of the coefficient matrix of the finite difference method and delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices the fast method has a computational work count of o nlogn per iteration and a memory requirement of o n while retaining the same accuracy as the underlying finite difference method solved with gaussian elimination numerical experiments of a three dimensional space fractional diffusion equation show the utility of the fast method

a fast finite difference method for two dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate complicated dense or full coefficient matrices consequently these numerical methods were traditionally solved by gaussian elimination which requires computational work of o n 3 per time step and o n 2 of memory where n is the number of spatial grid points in the discretization the significant computational work and memory requirement of the numerical methods impose a serious challenge for the numerical simulation of two and especially three dimensional space fractional diffusion equations we develop a fast and yet accurate solution method for the implicit finite difference discretization of space fractional diffusion equations in two space dimensions by carefully analyzing the structure of the coefficient ma

a fourier method for the fractional diffusion equation describing sub diffusion in this paper a fractional partial differential equation fpde describing sub diffusion is considered an implicit difference approximation scheme idas for solving a fpde is presented we propose a fourier method for analyzing the stability and convergence of the idas derive the global accuracy of the idas and discuss the solvability finally numerical examples are given to compare with the exact solution for the order of convergence and simulate the fractional dynamical systems

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

fourth order compact scheme for space fractional advection diffusion reaction equations with variable coefficients abstract in this work a new fourth order compact approximation is derived for riemann liouville space fractional derivatives modified wave numbers are obtained for various approximations of fractional derivatives considering these modified wave numbers fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations a novel fourth order compact scheme is developed for space fractional advection diffusion reaction equations with variable coefficients in one and two dimensions in the proposed compact scheme the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations this splitting of second derivative approximations allows us to obtain a similar system of linear equations as in zhao and tian 2017 which presents only second order accurate finite difference method for advection diffusion reaction equations in one dimension the stability of the proposed compact scheme is demonstrated numerically furthermore the proposed compact scheme is employed to space fractional black scholes equation for pricing european options as an application of fractional derivatives in mathematical finance numerical illustrations are presented to validate the theoretical claims

parameter estimation for the generalized fractional element network zener model based on the bayesian method abstract in the present study an inverse problem to estimate parameters in the zener model of viscoelasticity based on the generalized fractional element gfe network is studied the bayesian method is proposed to obtain the optimal estimation of the viscoelastic parameters three examples are performed to certify the validity of the method all numerical results lead to an excellent fitting between the calculative results and experimental data it is shown that the bayesian method is feasible in the inverse problem of parameter estimation for the fractional constitutive equation and the gfe network zener model is efficient in the modeling of the viscoelastic behavior

numerical algorithms to estimate relaxation parameters and caputo fractional derivative for a fractional thermal wave model in spherical composite medium in this paper we formulate a fractional thermal wave model for a bi layered spherical tissue implicit finite difference method is employed to obtain the solution of the direct problem the inverse analysis for simultaneously estimating the caputo fractional derivative and the relaxation time parameters is implemented by means of the levenberg marquardt method compared with the experimental data we can obviously find out that the estimated temperature increase values are excellently consistent with the measured temperature increase values in the experiment we have also discussed the effect of the fractional derivative the relaxation time parameters the initial guess as well as the sensitivity problem all the results show that the proposed fractional thermal wave model is efficient and accurate in modeling the heat transfer in the hyperthermia experiment and the proposed numerical method for simultaneously estimating multiple parameters for the fractional thermal wave model in a spherical composite medium is effective

solution of a modified fractional diffusion equation abstract recently a modified fractional diffusion equation has been proposed i sokolov j klafter from diffusion to anomalous diffusion a century after einstein s brownian motion chaos 15 2005 026103 a v chechkin r gorenflo i m sokolov v yu gonchar distributed order time fractional diffusion equation frac calc appl anal 6 3 2003 259279 i m sokolov a v checkin j klafter distributed order fractional kinetics acta phys pol b 35 2004 1323 for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term in this letter we give the solution of the modified equation on an infinite domain in contrast to the solution of the traditional fractional diffusion equation the solution of the modified equation requires an infinite series of fox functions instead of a single fox function

levinson s theorem for schrodinger operators with point interaction a topological approach in this note levinson theorems for schrodinger operators in r n with one point interaction at 0 are derived using the concept of winding numbers these results are based on new expressions for the associated wave operators

existence result for fractional neutral stochastic integro differential equations with infinite delay this paper is concerned with the existence of mild solutions for a class of fractional neutral stochastic integro differential equations with infinite delay in hilbert spaces a sufficient condition for the existence is obtained under non lipschitz conditions by means of sadovskii s fixed point theorem an example is given to illustrate the theory

the use of finite difference element approaches for solving the time fractional subdiffusion equation in this paper two finite difference element approaches for the time fractional subdiffusion equation with dirichlet boundary conditions are developed in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method the two methods are unconditionally stable and convergent of order o tau q h r 1 in the l 2 norm where q 2 beta or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth beta 0 beta 1 is the order of the fractional derivative tau and h are the step sizes in time and space respectively and r is the degree of the polynomial space the corresponding schemes for the subdiffusion equation with neumann boundary conditions are presented as well where the stability and convergence are shown numerical examples are provided to verify the theoretical analysis comparisons between the algorithms derived in this paper and the existing algorithms are given

parameter estimation for the fractional fractal diffusion model based on its numerical solution in this paper we mainly consider a problem of parameter estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media the bayesian method is proposed to estimate parameters for the fractional fractal diffusion model based on its numerical solution firstly the center box difference method is employed to solve the fractional fractal diffusion equation subject to the initial boundary conditions then we apply the bayesian method to estimate three parameters for the model simultaneously that is the fractional order the fractal dimension d f and the structure parameter to testify the validity of the results for the model s parameter estimation experimental data from the fast desorption process of methane in coal are used it is shown that the numerical results modeled with parameters given by the bayesian method coincide well with the desorption experimental data indicating that the bayesian method is efficient and valid in identifying multi parameter for the fractional fractal diffusion model besides the fractional fractal diffusion model is demonstrated to be more capable than the classical fick s model to describe the anomalous diffusion behavior of gas in coal to clarify the parameters effect on the relative diffusion amount of methane for the fractional fractal diffusion model parameter sensitivity analysis is also carried out this paper provides a specific and efficient parameter estimation method for the fractional fractal diffusion model

two dimensional axisymmetric stresses exerted by instantaneous pulses and sources of diffusion in an infinite space in a case of time fractional diffusion equation the theory of diffusive stresses based on the time fractional diffusion equation is formulated the source problem is discussed as well as the cauchy problem the stresses are found in axially symmetric cases for plane deformation the numerical results for the concentration and stress distributions are presented graphically for various values of order of fractional derivative

does levinson s theorem count complex eigenvalues by considering a quantum mechanical system with complex eigenvalues we show that indeed levinson s theorem extends to the non self adjoint setting the perturbed system corresponds to a realization of the schrodinger operator with inverse square potential on the half line while the dirichlet laplacian on the half line is chosen for the reference system the resulting relation is an equality between the number of eigenvalues of the perturbed system and the winding number of the scattering system together with additional operators living at 0 energy and at infinite energy

the topological meaning of levinson s theorem half bound states included we propose to interpret levinson s theorem as an index theorem this exhibits its topological nature it furthermore leads to a more coherent explanation of the corrections due to resonances at thresholds

efficient kansa type mfs algorithm for time fractional inverse diffusion problems in this study we propose an efficient kansa type method of fundamental solutions mfs k for the numerical solution of time fractional inverse diffusion problems by approximating the time fractional derivative through a finite difference scheme the time fractional inverse diffusion problem is transformed into a sequence of cauchy problems associated with inhomogeneous elliptic type equations which can be conveniently solved using the mfs k since the matrix arising from the mfs k discretization is highly ill conditioned a regularized solution is obtained by employing the tikhonov regularization method while the choice of the regularization parameter is based on the generalized cross validation criterion numerical results are presented for several examples with smooth and piecewise smooth boundaries the stability of the method with respect to the noise in the data is investigated

regularization for a fractional sideways heat equation we consider a sideways problem for a fractional heat equation which is highly ill posed this study gives a new dynamic method for choosing a regularization parameter by using the spectral methods some convergence rates on the temperature and heat flow are given for illustration several numerical examples are constructed to show the feasibility and efficiency of the proposed methods comparing with the traditional stationary methods for choosing regularization parameter the proposal methods are more accurate and effective

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

surface interpolation in three dimensional structure from motion perception although it is appreciated that humans can use a number of visual cues to perceive the three dimensional 3 d shape of an object for example luminance orientation binocular disparity and motion the exact mechanisms employed are not known de yoe and van essen 1988 an important approach to understanding the computations performed by the visual system is to develop algorithms marr 1982 or neural network models lehky and sejnowski 1988 siegel 1987 that are capable of computing shape from specific cues in the visual image in this study we investigated the ability of observers to see the 3 d shape of an object using motion cues so called structure from motion sfm we measured human performance in a two alternative forced choice task using novel dynamic random dot stimuli with limited point lifetimes we show that the human visual system integrates motion information spatially and temporally across several point lifetimes as part of the process for computing sfm we conclude that sfm algorithms must include surface interpolation to account for human performance our experiments also provide evidence that local velocity information and not position information derived from discrete views of the image as proposed by some algorithms is used to solve the sfm problem by the human visual system

efficient solution of two sided nonlinear space fractional diffusion equations using fast poisson preconditioners we develop a fast poisson preconditioner for the efficient numerical solution of a class of two sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines using the shifted gr unwald finite difference formulas to approximate the two sided i e the left and right riemann liouville fractional derivatives the resulting semi discrete nonlinear systems have dense jacobian matrices owing to the non local property of fractional derivatives we employ a modern initial value problem solver utilising backward differentiation formulas and jacobian free newton krylov methods to solve these systems for efficient performance of the jacobianfree newton krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver the key contribution of our work is to generalise the fast poisson preconditioner widely used for integer order diffusion equations so that it applies to the two sided space fractional diffusion equation a number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy

an inverse problem for a fractional diffusion equation in this paper we consider an inverse problem for a fractional diffusion equation which is highly ill posed such a problem is obtained from the classical diffusion equation by replacing the first order time derivative by the caputo fractional derivative of order a 0 a 1 we show that the problem is severely ill posed and further apply an optimal regularization method to solve it based on the solution in the frequency domain we can prove the optimal convergence estimate which shows that the regularized solution depends continuously on the data and is a good approximation to the exact solution numerical examples show that the proposed method works well

some results for a fractional diffusion equation with radial symmetry in a confined region abstract we investigate an n dimensional fractional diffusion equation with radial symmetry by taking a spatial and time dependent diffusion coefficient into account i e d r t d t r with d t d t d t the equation is considered in a confined region and subjected to time dependent boundary conditions which may be related to inhomogeneous characteristics of the surfaces confining the system the results show an anomalous spreading of the solutions and an unusual behavior of the survival probability

levinson s theorem and higher degree traces for aharonov bohm operators we study levinson type theorems for the family of aharonov bohm models from different perspectives the first one is purely analytical involving the explicit calculation of the wave operators and allowing to determine precisely the various contributions to the left hand side of levinson s theorem namely those due to the scattering operator the terms at 0 energy and at energy the second one is based on non commutative topology revealing the topological nature of levinson s theorem we then include the parameters of the family into the topological description obtaining a new type of levinson s theorem a higher degree levinson s theorem in this context the chern number of a bundle defined by a family of projections on bound states is explicitly computed and related to the result of a 3 trace applied on the scattering part of the model