implicit finite difference approximation for time fractional diffusion equations time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical brownian motion model in this paper we develop an implicit unconditionally stable numerical method to solve the one dimensional linear time fractional diffusion equation formulated with caputo s fractional derivative on a finite slab several numerical examples of interest are also included

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

spectral multipliers for sub laplacians on amenable lie groups with exponential volume growth let g be a lie group and a left invariant sub laplacian on g if l g denotes the space of square integrable functions with respect to the right invariant haar measure on g then is a selfadjoint operator on l g therefore every bounded borel function f on r induces a continuous operator f on l g it is now natural to ask under which additional conditions on f the operator f is necessarily bounded on l g p 6 2 in this case we call f an l multiplier for for more background information and various multiplier theorems we refer to 1 3 2 5 10 8 and the literature mentioned therein here we focus our attention on amenable groups with exponential volume growth and continuous functions f with compact support our aim is to show for a reasonably large class of lie groups and sub laplacians that a certain degree of differentiability of f is sufficient for f to extend to a bounded operator on l g i e that has differentiable l functional calculus that this is not true on any group with exponential growth in contrast to the situation on lie groups with polynomial growth cf 1 was shown by m christ and d muller in 2 they gave examples of sub laplacians on solvable lie groups which are for any p 6 2 of holomorphic l type i e there exists some non isolated point in the l spectrum of and an open complex neighbourhood u of in c such that every continuous lmultiplier which vanishes at infinity extends holomorphically to u more recent articles dealing with this topic are 8 and 7

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

weighted average finite difference methods for fractional diffusion equations a class of finite difference methods for solving fractional diffusion equations is considered these methods are an extension of the weighted average methods for ordinary non fractional diffusion equations their accuracy is of order x 2 and t except for the fractional version of the crank nicholson method where the accuracy with respect to the timestep is of order t 2 if a second order approximation to the fractional time derivative is used their stability is analyzed by means of a recently proposed procedure akin to the standard von neumann stability analysis a simple and accurate stability criterion valid for different discretization schemes of the fractional derivative arbitrary weight factor and arbitrary order of the fractional derivative is found and checked numerically some examples are provided in which the new methods numerical solutions are obtained and compared against exact solutions

time fractional ihcp with caputo fractional derivatives the numerical solution of the time fractional inverse heat conduction problem tfihcp on a finite slab is investigated in the presence of measured noisy data when the time fractional derivative is interpreted in the sense of caputo a finite difference space marching scheme with adaptive regularization using mollification techniques is introduced error estimates are derived for the numerical solution of the mollified problem and several numerical examples of interest are provided

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

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

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

spectral multipliers on a class of na groups with rank two we consider a family of real na groups with rank two and we prove that these groups have sub laplacians with differentiable lp functional calculus for all p 1

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

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

the measurement of plane wave reflection coefficients by using two dimensional spatial transforms recently tamura has proposed a method for measuring plane wave reflection coefficients at arbitrary angles of incidence m tamura proceedings of inter noise 89 1085 1088 1989 the procedure is based on measuring the complex pressure distribution on two parallel planes located between a compact source and the reflecting surface by spatially transforming the measured data into the wave number domain the sound field may be decomposed into upward and downward going plane wave components the reflection coefficient is then simply the ratio of the complex amplitudes of the upward and downward going plane wave components at a particular angle of incidence in this presentation the fundamentals of the technique will be reviewed followed by the presentation of new results that illustrate the effect of source type and location on measurement accuracy and practicality in addition it will be shown that acceptable accuracy may be achieved using reasonable spatial aperture sizes and sample rates only if the spatial data are transformed using a hankel transform based on a cylindrical geometry rather than a two dimensional fourier transform based on a rectangular geometry finally it will be shown that the use of a generalized hankel transform can eliminate the usual requirement that the source radiate a cylindrically symmetric sound field both simulated and model scale experiments will be used to show that the technique yields accurate results when the source type and transform geometry are chosen appropriately

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

spectral regularization method for a cauchy problem of the time fractional advection dispersion equation in this paper a cauchy problem for the time fractional advection dispersion equation tfade is investigated such a problem is obtained from the classical advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order a 0 a 1 we show that the cauchy problem of tfade is severely ill posed and further apply a spectral regularization method to solve it based on the solution given by the fourier method the convergence estimate is obtained under a priori bound assumptions for the exact solution numerical examples are given to show the effectiveness of the proposed numerical method

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 novel approach of high accuracy analysis of anisotropic bilinear finite element for time fractional diffusion equations with variable coefficient abstract in this paper by using bilinear finite element and l 1 approximation a fully discrete scheme is established for time fractional diffusion equation with variable coefficient on anisotropic meshes unconditionally stable analysis of the proposed scheme are presented in both l 2 norm and h 1 norm moreover convergence superclose and superconvergence results are derived by combining interpolation with projection which is the key technique for the numerical analysis specifically by defining a novel projection operator the error estimate between the projection and the exact solution is obtained on anisotropic meshes furthermore high accuracy analysis on interpolation of bilinear finite element and projection is gained by means of some known results about the interpolation and mean value technique based on the related results about projection and interpolation the optimal error estimate in l 2 norm and superclose of interpolation in h 1 norm are deduced by skillfully dealing with fractional derivative at the same time the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

some uniqueness and existence results for the initial boundary value problems for the generalized time fractional diffusion equation in this paper some uniqueness and existence results for the solutions of the initial boundary value problems for the generalized time fractional diffusion equation over an open bounded domain gx 0 t email protected r n are given to establish the uniqueness of the solution a maximum principle for the generalized time fractional diffusion equation is used in turn the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is considered in the paper too another important consequence of the maximum principle is the continuous dependence of the solution on the problem data to show the existence of the solution the fourier method of the variable separation is used to construct a formal solution under certain conditions the formal solution is shown to be a generalized solution of the initial boundary value problem for the generalized time fractional diffusion equation that turns out to be a classical solution under some additional conditions

the fractional tikhonov regularization methods for identifying the initial value problem for a time fractional diffusion equation abstract in this paper we identify the initial value for a time fractional diffusion equation on a columnar axis symmetric domain this problem is ill posed i e the solution if it exists does not depend continuously on the domain two different kinds of fractional tikhonov methods are used to solve this problem under the a priori and the a posteriori regularization parameter choice rules the error estimates between the regularization solutions and the exact solution are obtained respectively different numerical examples are presented to illustrate the validity and effectiveness of our methods

separation of nonstationary sound fields in the time wavenumber domain a number of sound field separation techniques have been proposed for different purposes however these techniques just consider the separation of sound fields in the space domain and are restricted to stationary sound fields when the sound fields are nonstationary it is also necessary to perform the separation in the time domain therefore on the basis of the propagation principle of sound pressure in the time wavenumber domain a nonstationary sound field separation technique with two closely spaced parallel measurement surfaces is proposed it can separate the nonstationary signals generated by the primary sources in both time and space domains when the disturbing sources exist on the other side of the measurement plane the signals in time and space domains are separated by using the spatial fourier transform method and the time domain deconvolution method a simulation involving two monopoles driven by nonstationary signals demonstrates that the method proposed can remove the influence of disturbing sources in both time and space domains the feasibility of this method is also demonstrated by an experiment with two loudspeakers located on two sides of measurement planes additionally to comment more objectively on the separation results some indicators are computed in both the simulation and experiment

block preconditioning strategies for time space fractional diffusion equations abstract we present a comparison of four block preconditioning strategies for linear systems arising in the numerical discretization of time space fractional diffusion equations in contrast to the traditional time marching procedure the discretization via finite difference is considered in a fully coupled time space framework the resulting fully coupled discretized linear system is a summation of two kronecker products the four preconditioning methods are based on block diagonal banded block triangular and kronecker product splittings of the coefficient matrix all preconditioning approaches use structure preserving methods to approximate blocks of matrix formed from the spatial fractional diffusion operator numerical experiments show the efficiency of the four block preconditioners and in particular of the banded block triangular preconditioner that usually outperforms the other three when the order of the time fractional derivative is close to one

on adi like iteration method for fractional diffusion equations abstract the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of the identity matrix with two diagonal times toeplitz matrices in this paper the alternating direction implicit adi like method based on the classical adi iteration is proposed to solve the discretized linear system theoretical analyses show that the adi like iteration method is convergent each iteration of this method requires the solutions of two linear systems with hessenberg coefficient matrices these two systems can be solved effectively by hessenberg lu factorization numerical examples are presented to illustrate the effectiveness of the adi like iteration method

sound field separation technique based on equivalent source method and its application in nearfield acoustic holography a technique for separating sound fields using two closely spaced parallel measurement surfaces and based on equivalent source method is proposed the method can separate wave components crossing two measurement surfaces in opposite directions which makes nearfield acoustic holography nah applications in a field where there exist sources on the two sides of the hologram surface in a reverberant field or in a scattered field possible the method is flexible in applications simple in computation and very easy to implement the measurement surfaces can be arbitrarily shaped and they are not restricted to be regular as in the traditional field separation technique and because the method performs field separation calculations directly in the spatial domain not in the wave number domain it avoids the errors and limitations the window effects etc associated with the traditional field separation technique based on the spatial fourier transform method in the paper a theoretical description is first

recovery of the solute concentration and dispersion flux in an inhomogeneous time fractional diffusion equation abstract in this paper we consider recovery of solute concentration and dispersion flux in an inhomogeneous time fractional diffusion equation we prove that the considered problem is ill posed i e the solution does not depend continuously on the data in order to obtain a regularized solution we propose a truncation regularization method the convergence estimates are established under some priori bound assumptions for the exact solution we present three numerical examples to show efficiency of the method

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

the third kind chebyshev wavelets collocation method for solving the time fractional convection diffusion equations with variable coefficients in this paper a numerical method based on the third kind chebyshev wavelets is proposed for solving a class of time fractional convection diffusion equations with variable coefficients the third kind chebyshev wavelets operational matrices of the integer order integration and the fractional order integration are derived respectively they are utilized to reduce the problem to a system of linear algebraic equations by combining the collocation method the uniform convergence analysis and error estimation for the third kind chebyshev wavelets expansion are investigated illustrative examples are given and the numerical results are presented to demonstrate the efficiency and accuracy of the proposed method

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

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

spectral multipliers on exponential growth solvable lie groups let m be a measure space and let l be a positive definite operator on l2 m by the spectral theorem for any bounded borel measurable function f 0 7 c the operator f l f 0 f de f is bounded on l2 m we are interested in sufficient conditions on f for f l to be bounded on lp m p 6 2 we direct the reader to 1 3 4 8 9 10 12 and 13 for more background on various multiplier theorems in this paper we assume f is compactly supported and have some smoothness finite number of derivatives and we consider only the case p 1 our measure space g is semidirect product of stratified nilpotent lie group n and the real line the operator l is minus sublaplacian on g our group has exponential volume growth the earlier theory suggested that one needs holomorphic f for f l to be bounded on l1 however the recent results 5 6 7 showed that estimates on only a finite number of derivatives of f imply boundedness of f l on l1 on some solvable g of exponential growth in this case we say that g more precisely l has ckfunctional calculus on the other hand christ and muller give an example of a solvable lie group on which f must be holomorphic the problem is to find the condition on g and possibly l which decides whether g has a ck functional calculus or not here our condition is in terms of roots of adjoint representation of the lie algebra of g our groups are of rank one

sub laplacians of holomorphic lp type on rank one an groups and related solvable groups consider a right invariant sub laplacian l on an exponential solvable lie group g endowed with a left invariant haar measure depending on the structure of g and possibly also that of l l may admit differentiable lp functional calculi or may be of holomorphic lp type for a given p 2 as recent studies of specific classes of groups g and sub laplacians l have revealed by holomorphic lp type we mean that every lp spectral multiplier for l is necessarily holomorphic in a complex neighborhood of some point in the l2 spectrum of l this can only arise if the group algebra l1 g is non symmetric in this article we prove that for large classes of exponential groups including all rank one an groups a certain lie algebraic condition which characterizes the non symmetry of l1 g 37 also suffices for l to be of holomorphic l1 type moreover if this condition which was first introduced by j boidol 6 in a different context holds for generic points in the dual g of the lie algebra of g then l is of holomorphic lp type for every p 2 besides the non symmetry of l1 g also the closedness of coadjoint orbits plays a crucial role we also discuss an example of a higher rank an group this example and our results in the rank one case suggest that sub laplacians on exponential lie groups may be of holomorphic l1 type if and only if there exists a closed coadjoint orbit g such that the points of satisfy boidol s condition in the course of the proof of our main results whose principal strategy is similar as in 8 we develop various tools which may be of independent interest and largely apply to more general lie groups some of them are certainly known as folklore results for instance we study subelliptic estimates on representation spaces the relation between spectral multipliers and unitary representations and develop some holomorphic and continuous perturbation theory for images of sub laplacians under smoothly varying families of irreducible unitary representations

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