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

deterministic extractors for bit fixing sources and exposure resilient cryptography we give an efficient deterministic algorithm that extracts omega n 2 gamma almost random bits from sources where n frac 1 2 gamma of the n bits are uniformly random and the rest are fixed in advance this improves upon previous constructions which required that at least n 2 of the bits be random in order to extract many bits our construction also has applications in exposure resilient cryptography giving explicit adaptive exposure resilient functions and in turn adaptive all or nothing transforms for sources where instead of bits the values are chosen from d for d 2 we give an algorithm that extracts a constant fraction of the randomness we also give bounds on extracting randomness for sources where the fixed bits can depend on the random bits

numerical solutions of variable order time fractional 1 1 and 1 2 dimensional advection dispersion and diffusion models abstract a numerical scheme based on haar wavelets coupled with finite differences is suggested to study variable order time fractional partial differential equations tfpdes the technique is tested on 1 1 dimensional advection dispersion and 1 2 dimensional advection diffusion equations in the proposed scheme time fractional derivative is firstly approximated by quadrature formula and then finite differences are combined with one and two dimensional haar wavelets with the help of suggested method the tfpdes convert to a system of algebraic equations which is easily solvable also convergence of the proposed scheme has been discussed which is an important part of the present work for validation the obtained results are matched with earlier work and exact solutions computations illustrate that the proposed scheme has better outcomes

gate elimination circuit size lower bounds and sat upper bounds abstract most of the known lower bounds for binary boolean circuits with unrestricted depth are proved by the gate elimination method the most efficient known algorithms for the sat problem on binary boolean circuits use similar case analyses to the ones in gate elimination chen and kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds that is lower bounds on the size of approximations of an explicit function in this paper we provide a general framework for proving worst average case lower bounds for circuits and upper bounds for sat that is built on ideas of chen and kabanets a proof in such a framework goes as follows one starts by fixing three parameters a class of circuits a circuit complexity measure and a set of allowed substitutions the main ingredient of a proof goes as follows by going through a number of cases one shows that for any circuit from the given class one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount this case analysis immediately implies an upper bound for sat to obtain worst average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser extractor for the class of sources defined by the set of substitutions under consideration we show that many known proofs of circuit size lower bounds and upper bounds for sat fall into this framework using this framework we prove the following new bounds average case lower bounds of 3 24 n and 2 59 n for circuits over u 2 and b 2 respectively though the lower bound for the basis b 2 is given for a quadratic disperser whose explicit construction is not currently known and faster than 2 n sat algorithms for circuits over u 2 and b 2 of size at most 3 24 n and 2 99 n respectively here by b 2 we mean the set of all bivariate boolean functions and by u 2 the set of all bivariate boolean functions except for parity and its complement

notes on implicit finite difference approximation for time fractional diffusion equations comput math appl 56 2008 1138 1145 in a recent paper diego a murio implicit finite difference approximation for time fractional diffusion equations comput math appl 56 2008 1138 1145 the author has proposed an unconditional stable implicit difference scheme for solving time fractional diffusion equations in this paper we show that the method of stability analysis by the author is not accurate so we analyze the stability by using a correct method

a space time fractional phase field model with tunable sharpness and decay behavior and its efficient numerical simulation abstract we present a space time fractional allen cahn phase field model that describes the transport of the fluid mixture of two immiscible fluid phases the space and time fractional order parameters control the sharpness and the decay behavior of the interface via a seamless transition of the parameters although they are shown to provide more accurate description of anomalous diffusion processes and sharper interfaces than traditional integer order phase field models do fractional models yield numerical methods with dense stiffness matrices consequently the resulting numerical schemes have significantly increased computational work and memory requirement we develop a lossless fast numerical method for the accurate and efficient numerical simulation of the space time fractional phase field model numerical experiments shows the utility of the fractional phase field model and the corresponding fast numerical 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

a new operator splitting method for american options under fractional black scholes models abstract a new operator splitting method is proposed for american options under time fractional black scholes models the fractional linear complementarity problem is split into two easy sub problems with the leading coefficients separated from the convolution sum and matched through a general correction step the method is implementation friendly in the sense that one can easily modify a fractional european solver to obtain the proposed method since the correction step is decoupled and is trivial to solve the method is validated through numerical experiments and demonstrated to be superior to the traditional approach the paper also provides numerical studies including the effect of fractional orders and the comparison of fractional models

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

deterministic extractors for bit fixing sources by obtaining an independent seed an n k bit fixing source is a distribution x over 0 1 n such that there is a subset of k variables in x 1 ldots x n which are uniformly distributed and independent of each other and the remaining n k variables are fixed a deterministic bit fixing source extractor is a function e 0 1 n rightarrow 0 1 m which on an arbitrary n k bit fixing source outputs m bits that are statistically close to uniform recently kamp and zuckerman proceedings of the 44th annual ieee symposium on foundations of computer science 2003 pp 92 101 gave a construction of a deterministic bit fixing source extractor that extracts omega k 2 n bits and requires k sqrt n in this paper we give constructions of deterministic bit fixing source extractors that extract 1 o 1 k bits whenever k log n c for some universal constant c 0 thus our constructions extract almost all the randomness from bit fixing sources and work even when k is small for k gg sqrt n the extracted bits have statistical distance 2 n omega 1 from uniform and for k le sqrt n the extracted bits have statistical distance k omega 1 from uniform our technique gives a general method to transform deterministic bit fixing source extractors that extract few bits into extractors which extract almost all the bits

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

preconditioned iterative methods for fractional diffusion equation in this paper we are concerned with numerical methods for the solution of initial boundary value problems of anomalous diffusion equations of order 1 2 the classical crank nicholson method is used to discretize the fractional diffusion equation and then the spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates two preconditioned iterative methods namely the preconditioned generalized minimal residual preconditioned gmres method and the preconditioned conjugate gradient for normal residual preconditioned cgnr method are proposed to solve relevant linear systems numerical experiments are given to illustrate the efficiency of the methods

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 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

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

adaptive dead zone compensation for output feedback canonical systems an adaptive output feedback dead zone compensation scheme is designed for systems with an unknown dead zone at the input of an n th order smooth nonlinear dynamics in the output feedback canonical form the proposed adaptive controller employs an adaptive dead zone inverse to cancel the dead zone and uses a backstepping design for adaptive output feedback control it has a new state observer parametrization that is needed to handle the dead zone uncertainty and a new robust control law that is suitable for the parameter projection needed to implement an adaptive dead zone inverse the adaptive dead zone compensation design ensures closed loop signal boundedness and improves system tracking performance

boundary particle method for laplace transformed time fractional diffusion equations highlights error analysis and numerical experiments demonstrate the efficiency of our method for time fractional diffusion equations our method avoids costly convolution integral calculation and dilemmatic selection of time step size a truly boundary only meshless method is applied to laplace transformed inhomogeneous problem our method effectively simulates 3d long time history fractional diffusion systems this paper develops a novel boundary meshless approach laplace transformed boundary particle method ltbpm for numerical modeling of time fractional diffusion equations it implements laplace transform technique to obtain the corresponding time independent inhomogeneous equation in laplace space and then employs a truly boundary only meshless boundary particle method bpm to solve this laplace transformed problem unlike the other boundary discretization methods the bpm does not require any inner nodes since the recursive composite multiple reciprocity technique rc mrm is used to convert the inhomogeneous problem into the higher order homogeneous problem finally the stehfest numerical inverse laplace transform nilt is implemented to retrieve the numerical solutions of time fractional diffusion equations from the corresponding bpm solutions in comparison with finite difference discretization the ltbpm introduces laplace transform and stehfest nilt algorithm to deal with time fractional derivative term which evades costly convolution integral calculation in time fractional derivation approximation and avoids the effect of time step on numerical accuracy and stability consequently it can effectively simulate long time history fractional diffusion systems error analysis and numerical experiments demonstrate that the present ltbpm is highly accurate and computationally efficient for 2d and 3d time fractional diffusion equations

the shrinkage exponent of de morgan formulas is 2 we prove that if we hit a de morgan formula of size l with a random restriction from rp then the expected remaining size is at most o p 2 log frac 1 p 3 2 l p sqrt l as a corollary we obtain an omega n 3 o 1 formula size lower bound for an explicit function in p this is the strongest known lower bound for any explicit function in np

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

improved average case lower bounds for de morgan formula size matching worst case lower bound we give an explicit function h 0 1 n to 0 1 such that every de morgan formula of size n 3 o 1 r 2 agrees with h on at most a fraction of frac 1 2 2 omega r of the inputs our technical contributions include a theorem that shows that the expected shrinkage result of h aastad siam j comput 27 1998 pp 48 64 actually holds with very high probability where the restrictions are chosen from a certain distribution that takes into account the structure of the formula using ideas of impagliazzo meka and zuckerman proceedings of focs 2012 pp 111 119

adaptive fuzzy robust output feedback control of nonlinear systems with unknown dead zones based on a small gain approach in this paper an adaptive fuzzy robust output feedback control problem is considered for a class of single input and single output nonlinear systems in a strict feedback form the considered systems possess the unstructured uncertainties unknown dead zone and the dynamics uncertainties and they do not assume the states being available for the controller design in the controller design fuzzy logic systems are first used to approximate the unstructured uncertainties and by utilizing the information of the bounds of the dead zone slopes and treating the time varying inputs coefficients as a system uncertainty a fuzzy state observer is designed to estimate the unmeasured states by combining a backstepping technique with a nonlinear small gain approach a new adaptive fuzzy robust output feedback control has been developed it is proved that the proposed fuzzy adaptive control approach can guarantee the semiglobal uniform ultimate boundedness for all the solutions of the closed loop systems simulation studies and comparisons with previous methods are included to illustrate the effectiveness of the proposed approach

adaptive fuzzy output feedback control of mimo nonlinear systems with unknown dead zone inputs this paper is concerned with the problem of adaptive fuzzy tracking control for a class of multi input and multi output mimo strict feedback nonlinear systems with both unknown nonsymmetric dead zone inputs and immeasurable states in this research fuzzy logic systems are utilized to evaluate the unknown nonlinear functions and a fuzzy adaptive state observer is established to estimate the unmeasured states based on the information of the bounds of the dead zone slopes as well as treating the time varying inputs coefficients as a system uncertainty a new adaptive fuzzy output feedback control approach is developed via the backstepping recursive design technique it is shown that the proposed control approach can assure that all the signals of the resulting closed loop system are semiglobally uniformly ultimately bounded it is also shown that the observer and tracking errors converge to a small neighborhood of the origin by selecting appropriate design parameters simulation examples are also provided to illustrate the effectiveness of the proposed approach

selection of shape parameter in radial basis functions for solution of time fractional black scholes models abstract the current work aims to exploit two techniques namely residual power series method rpsm and collocation based meshfree method for the solution of time fractional black scholes models with constant and variable coefficients firstly using rpsm we obtain exact solutions of the considered models and then numerical solution by meshfree method computer simulations are performed for three test problems of european options pricing the simulations features excellent agreement with exact solutions accuracy and efficiency of the proposed numerical method is assessed via e2 e and erms error norms convergence of the proposed methods is also analyzed

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the 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

adaptive fuzzy output feedback tracking backstepping control of strict feedback nonlinear systems with unknown dead zones in this paper an adaptive fuzzy backstepping control approach is considered for a class of nonlinear strict feedback systems with unknown functions unknown dead zones and immeasurable states fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a fuzzy filters state observer is designed to estimate the immeasurable states by using the adaptive backstepping recursive design technique and constructing the dead zone inverse a new adaptive fuzzy backstepping output feedback control approach is developed it is mathematically proved that all the signals of the resulting closed loop adaptive control system are semiglobally uniformly ultimately bounded and the tracking error converges to a small neighborhood of the origin by appropriate choice of design parameters the proposed approach cannot only solve the problem of the dead zones but also cancel the restrictive assumption in the previous literature that the states are all available for measurement two simulation examples are provided to show the effectiveness of the proposed approach

quantum lower bounds by polynomials we examine the number of queries to input variables that a quantum algorithm requires to compute boolean functions on 0 1 n in the black box model we show that the exponential quantum speed up obtained for partial functions i e problems involving a promise on the input by deutsch and jozsa simon and shor cannot be obtained for any total function if a quantum algorithm computes some total boolean function f with small error probability using t black box queries then there is a classical deterministic algorithm that computes f exactly with o ts6 queries we also give asymptotically tight characterizations of t for all symmetric f in the exact zero error and bounded error settings finally we give new precise bounds for and or and parity our results are a quantum extension of the so called polynomial method which has been successfully applied in classical complexity theory and also a quantum extension of results by nisan about a polynomial relationship between randomized and deterministic decision tree complexity

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

determination of the initial data in a time fractional diffusion wave problem by a final time data abstract this paper is to determine an initial value for a time fractional diffusion wave equation from the final time data the regularity of the weak solution for the direct problem is improved the existence and uniqueness of a weak solution for the adjoint problem are proved by using the fourier method the tikhonov regularization method is applied for finding a stable approximate solution in order to get the minimizer of the tikhonov regularization functional we propose a conjugate gradient algorithm based on the strict deductions of the sensitive problem and the adjoint problem four numerical examples in one dimensional and two dimensional cases are provided to show the effectiveness and stability of the proposed algorithm

simultaneous identification of three parameters in a time fractional diffusion wave equation by a part of boundary cauchy data abstract this paper is devoted to determine the fractional order the initial flux speed and the boundary neumann data simultaneously in a one dimensional time fractional diffusion wave equation from part boundary cauchy observation data we prove the uniqueness result for this inverse problem by using a new result for the mittag leffler function and laplace transform combining with analytic continuation then we use the iterative regularizing ensemble kalman method in bayesian framework to solve the inverse problem numerically and four numerical examples are provided to show the effectiveness and stability of the proposed algorithm

on the solitary wave solution of fractional kudryashov sinelshchikov equation describing nonlinear wave processes in a liquid containing gas bubbles in the present paper the nonlinear time fractional kudryashov sinelshchikov equation has been solved numerically by using radial basis function rbf method the rbf method is successfully implemented to the time fractional kudryashov sinelshchikov equation in order to achieve its approximate solution as the exact solution of fractional kudryashov sinelshchikov equation is not recorded earlier in literature the analytical exact solutions have been constructed by using sech tanh method via fractional complex transform the numerical outcomes obtained by rbf are observed to be in good agreement with the exact solutions as well as with the acquired sech tanh solutions

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

solving the time fractional diffusion equation via sinc haar collocation method expanding the approximation with the elements of sinc and haar functions an efficient algorithm with exponential convergence rate computational speed is high due to using the haar operational matrices to improve the accuracy by increasing the number of collocation points the present study investigates the sinc haar collocation method for the solution of the time fractional diffusion equation the advantages of this technique are that not only the convergence rate of sinc approximation is exponential but the computational speed also is high due to the use of the haar operational matrices this technique is used to convert the problem to the solution of linear algebraic equations via expanding the required approximation based on the elements of sinc functions in space and haar functions in time with unknown coefficients the effectiveness of the proposed method is examined by comparing the numerical results with the exact solutions

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

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

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations 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 a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

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

identifying an unknown time dependent boundary source in time fractional diffusion equation with a non local boundary condition abstract this paper is devoted to determining a time dependent factor of an unknown source on some subboundary conditions for a time fractional diffusion equation from the non local measurement data we prove existence and uniqueness of the solution of the inverse boundary source by using the lax milgram lemma in some suitable sobolev spaces consequently we use the profitable meshless method based on radial basis functions to solve the inverse problem the numerical experimental results and numerical simulations show that the proposed scheme is effective and stable when one considers measurements input data contaminated with noise

adaptive fuzzy decentralized output feedback control for nonlinear large scale systems with unknown dead zone inputs in this paper the problem of adaptive fuzzy decentralized backstepping control is considered for a class of nonlinear large scale strict feedback systems with unknown dead zones and immeasurable states fuzzy logic systems are used to approximate the unknown nonlinear functions and a state filter is designed to estimate the immeasurable states applying an adaptive backstepping design technique and combining it with the dead zone inverse method an adaptive fuzzy decentralized output feedback backstepping control is developed it is proved that all the signals of the resulting closed loop adaptive control system are semiglobally uniformly ultimately bounded and tracking errors converge to a small neighborhood of the origin by appropriate choice of design parameters simulation results are given to demonstrate that the proposed adaptive decentralized control approach has a satisfactory control performance

a novel high order approximate scheme for two dimensional time fractional diffusion equations with variable coefficient abstract based on the spatial quasi wilson nonconforming finite element method and temporal l 2 1 formula a fully discrete approximate scheme is proposed for a two dimensional time fractional diffusion equations with variable coefficient on anisotropic meshes in order to demonstrate the stable analysis and error estimates several lemmas are provided which focus on high accuracy about projection and superclose estimate between the interpolation and projection unconditionally stable analysis are derived in l 2 norm and broken h 1 norm moreover convergence result of accuracy o h 2 2 and superclose property of accuracy o h 2 2 are deduced by combining interpolation with projection where h and are the step sizes in space and time respectively and then the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

a kronecker product splitting preconditioner for two dimensional space fractional diffusion equations abstract we study preconditioned iterative methods for the linear system arising in the numerical discretization of a two dimensional space fractional diffusion equation our approach is based on a formulation of the discrete problem that is shown to be the sum of two kronecker products by making use of an alternating kronecker product splitting iteration technique we establish a class of fixed point iteration methods theoretical analysis shows that the new method converges to the unique solution of the linear system moreover the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real the basic iteration is accelerated by a krylov subspace method like gmres the corresponding preconditioner is in a form of a kronecker product structure and requires at each iteration the solution of a set of discrete one dimensional fractional diffusion equations we use structure preserving approximations to the discrete one dimensional fractional diffusion operators in the action of the preconditioning matrix numerical examples are presented to illustrate the effectiveness of this approach

time fractional diffusion equation for signal smoothing the time fractional diffusion equation is used for signal smoothing compared to the classical diffusion equation the time fractional diffusion equation has another adjustable time fractional derivative order to control the diffusion process therefore some simulated signals are used to compare the smoothing performance between the time fractional diffusion equation and the classical diffusion equation as well as between classical smoothing methods regularization method savitzky golay method and wavelet method in the end the time fractional diffusion filtering is applied in an nmr spectrum smoothing results indicate that the time fractional diffusion filtering is advantage over the classical diffusion filtering and their smoothing performance is better than that of classical smoothing methods

efficient numerical solution of the time fractional diffusion equation by mapping from its brownian counterpart the solution of a caputo time fractional diffusion equation of order 0 1 is expressed in terms of the solution of a corresponding integer order diffusion equation we demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem in the context of an n point finite difference time discretisation the mapping allows for an improvement in time computational complexity from o n 2 to o n given a precomputation of o n 1 ln n the mapping is applied successfully to the least squares fitting of a fractional advection diffusion model for the current in a time of flight experiment resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes a mapping between fractional diffusion and normal diffusion is explored numerically an efficient solution method for fractional diffusion equations is proposed a benchmark fractional order model fit for scher montroll transport is presented

a cubic trigonometric b spline collocation approach for the fractional sub diffusion equations an algorithm for fractional sub diffusion model using cubic trigonometric b spline approach is presented the proposed scheme is unconditionally stable using von neumann approach error norms confirm the accuracy and efficiency of presented scheme the proposed method provides more accurate results as compare to cubic b spline method a cubic trigonometric b spline collocation approach for the numerical solution of fractional sub diffusion equation is presented in this paper the approach is based on the usual finite difference scheme to discretize the time derivative while the approximation of the second order derivative with respect to space is obtained by the cubic trigonometric b spline functions with the help of grunwald letnikov discretization of the riemann liouville derivative the scheme is shown to be stable using the fourier method and the accuracy of the scheme is tested by application to a test problem the results of the numerical test verify the accuracy and efficiency of the proposed algorithm

high accuracy analysis of an h1 galerkin mixed finite element method for two dimensional time fractional diffusion equations abstract in this paper an h 1 galerkin mixed finite element approximate scheme is established for a class of two dimensional time fractional diffusion equations by using the bilinear element raviart thomas element and l 1 time stepping method which is unconditionally stable and free of lbb condition and then without the classical ritz projection superclose results for the original variable u in h 1 norm and the flux p u in h d i v norm are derived by means of properties of the elements and l 1 approximation furthermore with the help of the interpolation postprocessing operator the global superconvergence results for the original variable u in h 1 norm are obtained finally numerical simulations verify that the theoretical results are true on both regular meshes and anisotropic meshes

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

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

the method of approximate particular solutions for the time fractional diffusion equation with a non local boundary condition in this paper we consider the numerical solution of the time fractional diffusion equation with a non local boundary condition the method of approximate particular solutions maps using multiquadric radial basis function mq rbf is employed for this equation due to the accuracy of the mq based meshless methods is severely influenced by the shape parameter we adopt a leave one out cross validation loocv algorithm proposed by rippa 34 to enhance the performance of the maps the numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time fractional diffusion equation with a non local boundary condition

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

efficient spectral galerkin methods for fractional partial differential equations with variable coefficients efficient spectral galerkin algorithms are developed to solve multi dimensional fractional elliptic equations with variable coefficients in conserved form as well as non conserved form these algorithms are extensions of the spectral galerkin algorithms for usual elliptic pdes developed in 24 more precisely for separable fpdes we construct a direct method by using a matrix diagonalization approach while for non separable fpdes we employ a preconditioned bicgstab method with a suitable separable fpde with constant coefficients as preconditioner the cost of these algorithms is of o n d 1 flops where d is the space dimension we derive rigorous weighted error estimates which provide more precise convergence rate for problems with singularities at boundaries we also present ample numerical results to validate the algorithms and error estimates

a modified memory based mathematical model describing fluid flow in porous media abstract this study presents a modified memory based mathematical model describing fluid flow in porous media for the first time such model is derived employing the grunwald letnikov g l definition of the riemann liouville r l time fractional operator via a generalized darcy s equation the proposed mathematical model is suitable to describe the anomalous diffusion behavior observed in a medium of fractal geometry as well as in disordered and highly heterogeneous porous media a numerical scheme based on existing discretization method is employed to handle the modified memory based mathematical model the accuracy of the numerical model is validated through the analytical solution for a simplified problem in addition numerical experiments are presented to demonstrate the effect of the anomalous diffusion exponent on the predicted reservoir pressure wellbore pressure and volumetric flux considering the flow of an under saturated oil in a hydrocarbon reservoir as an example results show that decrease in magnitude of the anomalous diffusion exponent results in larger pore and wellbore pressure this study would open a door to apply the g l interpretation of the time fractional operator in numerical modeling of fluid flow through porous media

a circulant preconditioner for fractional diffusion equations the implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable is employed to discretize fractional diffusion equations the resulting systems are toeplitz like and then the fast fourier transform can be used to reduce the computational cost of the matrix vector multiplication the preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant hence the convergence rate of the proposed iterative algorithm is superlinear numerical experiments are carried out to demonstrate that our circulant preconditioner works very well even though for cases of variable diffusion coefficients

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

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

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

optimal error bound and simplified tikhonov regularization method for a backward problem for the time fractional diffusion equation in this paper we consider a backward problem for a time fractional diffusion equation such a problem is ill posed the optimal error bound for the problem under a source condition is analyzed a simplified tikhonov regularization method is utilized to solve the problem and its convergence rates are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule respectively numerical examples show that the proposed regularization method is effective and stable and both parameter choice rules work well

identification of time dependent convection coefficient in a time fractional diffusion equation abstract in the present paper we devote our effort to solve a nonlinear inverse problem for identifying a time dependent convection coefficient in a time fractional diffusion equation from the measured data at an interior point for one dimensional case we prove the existence uniqueness and regularity of solution for the direct problem by using the fixed point theorem the stability of inverse convection coefficient problem is obtained based on the regularity of solution for the direct problem and some generalized gronwall s inequalities we use a modified optimal perturbation regularization algorithm to solve the inverse convection coefficient problem two numerical examples are provided to show the effectiveness of the proposed method

on tikhonov s method and optimal error bound for inverse source problem for a time fractional diffusion equation abstract we investigate the linear but ill posed inverse problem of determining a multi dimensional space dependent heat source in a time fractional diffusion equation we show that the problem is ill posed in the hilbert scale h r r n and establish global order optimal lower bound for the worst case error next we use the tikhonov regularization method to deal with this problem in the hilbert scale h r r n locally optimal choices of parameters for the family of regularization operator in the hilbert scales h r r n are analyzed by a priori and a posteriori methods numerical implementations are presented to illustrate our theoretical findings

fractional diffusion equations by the kansa method this study makes the first attempt to apply the kansa method in the solution of the time fractional diffusion equations in which the multiquadrics and thin plate spline serve as the radial basis function in the discretization formulation the finite difference scheme and the kansa method are respectively used to discretize time fractional derivative and spatial derivative terms the numerical solutions of one and two dimensional cases are presented and discussed which agree well with the corresponding analytical solution

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