stability and convergence of the difference methods for the space time fractional advection diffusion equation abstract in this paper we consider a space time fractional advection dispersion equation stfade on a finite domain the stfade is obtained from the standard advection dispersion equation by replacing the first order time derivative by the caputo fractional derivative of order 0 1 and the first order and second order space derivatives by the riemman liouville fractional derivatives of order 0 1 and of order 1 2 respectively for the space fractional derivatives d x u x t and d x u x t we adopted the grunwald formula and the shift grunwald formula respectively we propose an implicit difference method idm and an explicit difference method edm to solve this equation stability and convergence of these methods are discussed using mathematical induction we prove that the idm is unconditionally stable and convergent but the edm is conditionally stable and convergent numerical results are in good agreement with theoretical analysis

an optimal error estimate for the two dimensional nonlinear time fractional advection diffusion equation with smooth and non smooth solutions abstract in this paper a linearized l1 legendre galerkin spectral method for the two dimensional nonlinear time fractional advection diffusion equation is derived the numerical method is stable without the cfl conditions based on the error splitting argument technique and the discrete fractional gronwall type inequality we also consider the case of non smooth solutions by adding some correction terms the numerical experiments are given to verify the theoretical analysis and the effectiveness of the correction method

finite element method for nonlinear riesz space fractional diffusion equations on irregular domains in this paper we consider two dimensional riesz space fractional diffusion equations with nonlinear source term on convex domains applying galerkin finite element method in space and backward difference method in time we present a fully discrete scheme to solve riesz space fractional diffusion equations our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes which can help us to deal with space fractional terms on any convex domain the stability and convergence of the scheme are also discussed numerical examples are given to verify accuracy and stability of our scheme a finite element method is developed to solve 2d space fractional diffusion equations with nonlinear source term we develop a finite element method for fpdes on irregular domain we obtain explicit expressions for fractional derivatives of shape functions we give the details on how to compute fractional stiffness matrix fractional fitzhugh nagumo model is solved on circular domain

optimal error estimates of spectral petrov galerkin and collocation methods for initial value problems of fractional differential equations we present optimal error estimates for spectral petrov galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval we also develop laguerre spectral petrov galerkin methods and collocation methods for fractional equations on the half line numerical results confirm the error estimates

cuspidal ell modular representations of p adic classical groups for a classical group over a non archimedean local field of odd residual characteristic p we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p as representations induced from a cuspidal type we also give a fundamental step towards the classification of cuspidal representations identifying when certain cuspidal types induce to equivalent representations this result is new even in the case of complex representations finally we prove that the representations induced from more general types are quasi projective a crucial tool for extending the results here to arbitrary irreducible representations

multi domain spectral collocation method for variable order nonlinear fractional differential equations abstract spectral and spectral element methods using galerkin type formulations are efficient for solving linear fractional pdes fpdes of constant order but are not efficient in solving nonlinear fpdes and cannot handle fpdes with variable order in this paper we present a multi domain spectral collocation method that addresses these limitations we consider fpdes in the riemann liouville sense and employ jacobi lagrangian interpolants to represent the solution in each element we provide variable order differentiation formulas which can be computed efficiently for the multi domain discretization taking into account the nonlocal interactions we enforce the interface continuity conditions by matching the solution values at the element boundaries via the lagrangian interpolants and in addition we minimize the jump in integer fluxes using a penalty method we analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme we demonstrate the effectiveness of the new method for the fractional helmholtz equation of constant and variable order using h p refinement for different values of the penalty parameter we also solve the fractional burgers equation with constant and variable order and compare with solutions obtained with a single domain spectral collocation method

developing finite element methods for maxwell s equations in a cole cole dispersive medium in this paper we consider the time dependent maxwell s equations when cole cole dispersive medium is involved the cole cole model contains a fractional time derivative term which couples with the standard maxwell s equations in free space and creates some challenges in developing and analyzing time domain finite element methods for solving this model as mentioned in our earlier work j li j sci comput 47 2001 pp 1 26 by adopting some techniques developed for the fractional diffusion equations v j ervin n heuer and j p roop siam j numer anal 45 2007 pp 572 591 y lin and c xu j comput phys 225 2007 pp 1533 1552 f liu p zhuang v anh i turner and k burrage appl math comput 191 2007 pp 12 20 we propose two fully discrete mixed finite element schemes for the cole cole model numerical stability and optimal error estimates are proved for both schemes the proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis

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

numerical solution of the time fractional black scholes model governing european options when considering the price change of the underlying fractal transmission system a fractional black scholes b s model with an order time fractional derivative is derived in this paper we discuss the numerical simulation of this time fractional black scholes model tfbsm governing european options a discrete implicit numerical scheme with a spatially second order accuracy and a temporally 2 order accuracy is constructed then the stability and convergence of the proposed numerical scheme are analyzed using fourier analysis some numerical examples are chosen in order to demonstrate the accuracy and effectiveness of the proposed method finally as an application we use the tfbsm and the above numerical technique to price several different european options

a fractional order darcy s law abstract by using spatial averaging methods in this work we derive a darcy s type law from a fractional newton s law of viscosity which is intended to describe shear stress phenomena in non homogeneous porous media as a prerequisite towards this end we derive an extension of the spatial averaging theorem for fractional order gradients the usage of this tool for averaging continuity and momentum equations yields a darcy s law with three contributions i similar to the classical darcy s law a term depending on macroscopic pressure gradients and gravitational forces ii a fractional convective term induced by spatial porosity gradients and iii a fractional brinkman type correction in the three cases the corresponding permeability tensors should be computed from a fractional boundary value problem within a representative cell consistency of the resulting darcy s type law is demonstrated by showing that it is reduced to the classical one in the case of integer order velocity gradients and homogeneous porous media

spectral analysis of fractional kinetic equations with random data we present a spectral representation of the mean square solution of the fractional kinetic equation also known as fractional diffusion equation with random initial condition gaussian and non gaussian limiting distributions of the renormalized solution of the fractional in time and in space kinetic equation are described in terms of multiple stochastic integral representations

a preconditioned fast quadratic spline collocation method for two sided space fractional partial differential equations abstract quadratic spline collocation methods for the one and two dimensional fractional diffusion equations are proposed by carefully exploring the mathematical structure of the coefficient matrix we propose a matrix free fast krylov subspace iterative solver for the corresponding quadratic spline collocation scheme and then preconditioning technique is applied to further accelerate the convergence of the fast krylov subspace iterative method it shows that the fast quadratic spline collocation scheme has greatly reduced the computational cost from o n 2 to o n log n per iteration and memory requirement from o n 2 to o n while still well approximates the fractional diffusion equations without any accuracy lost numerical experiments are given to demonstrate the efficiency and effectiveness of the fast method

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

a finite difference scheme for semilinear space fractional diffusion equations with time delay a linearized quasi compact finite difference scheme is proposed for semilinear space fractional diffusion equations with a fixed time delay the nonlinear source term is discretized and linearized by taylor s expansion to obtain a second order discretization in time the space fractional derivatives are approximated by a weighted shifted grunwald letnikov formula which is of fourth order approximation under some smoothness assumptions of the exact solution under the local lipschitz conditions the solvability and convergence of the scheme are proved in the discrete maximum norm by the energy method numerical examples verify the theoretical predictions and illustrate the validity of the proposed scheme

high order time stepping methods for two dimensional riesz fractional nonlinear reaction diffusion equations abstract anomalous diffusion and long range spatial interactions in anisotropic media could be captured by considering the riesz fractional derivatives rather than the classical laplacian while analytical solutions of the resulting fractional reaction diffusion models may not be available the numerical methods for approximating them are challenging in this paper fourth order l stable etdrk04 and a stable etdrk22 linearly implicit predictor corrector type time stepping methods are presented these methods are implemented on two dimensional riesz fractional nonlinear reaction diffusion equations with smooth and non smooth initial data three types of nonlinear reaction diffusion models are considered allen cahn equation with cubic nonlinearity fisher s equation with quadratic nonlinearity and enzyme kinetics equation with rational nonlinearity fourth order temporal convergence rate of the methods is proved analytically and computed numerically profiles of the numerical solutions corresponding to different orders and rates of diffusion are included computational efficiency of the etdrk04 and etdrk22 methods over the well known cox matthews etdrk4 is presented the superiority of the provided methods in terms of computational accuracy efficiency and reliability is demonstrated through the numerical experiments

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

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

stability and convergence of a new explicit finite difference approximation for the variable order nonlinear fractional diffusion equation in this paper we consider the variable order nonlinear fractional diffusion equation u x t t b x t xr a x t u x t f u x t where xr a x t is a generalized riesz fractional derivative of variable order a x t 1 a x t 2 and the nonlinear reaction term f u x t satisfies the lipschitz condition f u 1 x t f u 2 x t

a finite volume method for two dimensional riemann liouville space fractional diffusion equation and its efficient implementation abstract we develop a finite volume method based on crank nicolson time discretization for the two dimensional nonsymmetric riemann liouville space fractional diffusion equation stability and convergence are then carefully discussed we prove that the finite volume scheme is unconditionally stable and convergent with second order accuracy in time and min 1 1 order in space with respect to a weighted discrete norm here 0 1 are the space fractional order indexes in x and y directions respectively furthermore we rewrite the finite volume scheme into a matrix form and develop a matrix free preconditioned fast krylov subspace iterative method which only requires storage of o n and computational cost of o n log n per iteration without losing any accuracy compared to the direct gaussian elimination method here n is the total number of spatial unknowns consequently the fast finite volume method is particularly suitable for large scale modeling and simulation numerical experiments verify the theoretical results and show strong potential of the fast method

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

crank nicolson adi galerkin finite element method for two dimensional fractional fitzhugh nagumo monodomain model in this paper a two dimensional fractional fitzhugh nagumo monodomain model 2d ffhnmm with zero dirichlet boundary condition is considered the model consists of a coupled two dimensional space fractional nonlinear reaction diffusion model 2d sfnrdm and an ordinary differential equation the 2d sfnrdm and ordinary differential equation are decoupled at each time step a new crank nicolson alternating direction implicit adi galerkin finite element method for the 2d sfnrdm is developed the stability and convergence of the numerical method are discussed finally some numerical examples on 2d sfnrdm and 2d ffhnmm are given for verification of our theoretical analysis

fractional vector calculus for fractional advection dispersion abstract we develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient divergence and curl and a fractional divergence theorem and stokes theorem these basic tools are then applied to provide a physical explanation for the fractional advection dispersion equation for flow in heterogeneous porous media

local discontinuous galerkin method for a nonlinear time fractional fourth order partial differential equation abstract in this article a fully discrete local discontinuous galerkin ldg method with high order temporal convergence rate is presented and developed to look for the numerical solution of nonlinear time fractional fourth order partial differential equation pde in the temporal direction for approximating the fractional derivative with order 0 1 the weighted and shifted grunwald difference wsgd scheme with second order convergence rate is introduced and for approximating the integer time derivative two step backward euler method with second order convergence rate is used for the spatial direction the ldg method is used for the numerical theories the stability is derived and a priori error results are proved further some error results and convergence rates are calculated by numerical procedure to illustrate the effectiveness of proposed method

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

efficient two dimensional simulations of the fractional szabo equation with different time stepping schemes the modified szabo wave equation is one of the various models that have been developed to model the power law frequency dependent attenuation phenomena in lossy media the purpose of this study is to develop two different efficient numerical methods for the two dimensional szabo equation and to compare the relative merits of each method in both methods we employ the adi scheme to split directions however we use different time discretization specifically in the first adi method adi i we include a third order correction term to achieve second order convergence for smooth solutions hence extending the work of sun and wu 2006 in the second adi method adi ii we employ the scheme in zeng et al submitted for publication to two dimensional fractional wave equation using multiple correction terms to enhance accuracy for non smooth solutions our simulation results show that both methods are computationally efficient for the fractional wave equation but have different advantages in terms of accuracy specifically adi ii seems to produce more accurate results than adi i for non smooth solutions however for smooth solutions and fractional order close to two adi i seems to outperform adi ii

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

fast algorithm based on tt m fe system for space fractional allen cahn equations with smooth and non smooth solutions abstract in this article a fast algorithm based on time two mesh tt m finite element fe scheme which aims at solving nonlinear problems quickly is considered to numerically solve the nonlinear space fractional allen cahn equations with smooth and non smooth solutions the implicit second order scheme containing both implicit crank nicolson scheme and second order backward difference method is applied to time direction a fast tt m method is used to increase the speed of calculation and the fe method is developed to approximate the spacial direction the tt m fe algorithm includes the following main computing steps firstly a nonlinear implicit second order fe scheme on the time coarse mesh c is solved by a nonlinear iterative method secondly based on the chosen initial iterative value a linearized fe system on time fine mesh c is solved where some useful coarse numerical solutions are found by the lagrange s interpolation formula the analysis for both stability and a priori error estimates are made in detail finally three numerical examples with smooth and non smooth solutions are provided to illustrate the computational efficiency in solving nonlinear partial differential equations from which it is easy to find that the computing time can be saved

high order local discontinuous galerkin method combined with wsgd approximation for a fractional subdiffusion equation abstract in this paper a high order local discontinuous galerkin ldg method combined with weighted and shifted grunwald difference wsgd approximation is developed and discussed for a caputo time fractional subdiffusion equation the time fractional derivative of order 0 1 is approximated by a third order method based on the idea of wsgd operator while the spatial operator is approximated by the ldg method some useful lemmas are first introduced and proved then the analysis of stability and optimal error estimate o t 3 h k 1 are obtained for the ldg method extensive numerical results using p k k 0 1 2 3 elements are presented to validate our theoretical analysis

certain new models of the multi space fractional gardner equation abstract in this paper we study an accurate numerical method for solving the multi space fractional gardner equation msfge with the caputo fabrizio cf and atangana baleanu ab fractional derivatives where the space fractional terms are under the sense of caputo to the best knowledge of the reader we are first to use the spectral collocation method based on the third chebyshev approximations to reduced the multi space fractional gardner equation to a system of ordinary differential equations by using the properties of chebyshev polynomials and then solved them via the finite difference method fdm by computing the absolute errors we present the effectiveness and accuracy of the proposed methods the present paper investigates the dynamics of gardner equation by considering two fractional operators that is the caputo fabrizio and atangana baleanu this is entirely new idea by using the two operators on a gardner equation our results prove that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non local singular kernels

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

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

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

a novel finite element method for the distributed order time fractional cable equation in two dimensions abstract in this paper the unstructured mesh galerkin finite element method with a weighted and shifted grunwald difference approximation and composite trapezoid formula is presented to solve the nonhomogeneous two dimensional distributed order time fractional cable equation on irregular convex domains the crank nicolson type discretization of the finite element scheme is implemented to obtain the numerical solution the stability and convergence of the numerical scheme are discussed and derived finally some numerical examples on irregular convex domains are given to confirm our theoretical results

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

fast alternating direction finite difference methods for three dimensional space fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations because of the nonlocal property of fractional differential operators numerical methods for space fractional diffusion equations generate dense or even full coefficient matrices with complicated structures traditionally these methods were solved via 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 we present an alternating direction implicit adi finite difference formulation for space fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x y z directions commute we base on the adi formulation to develop a fast iterative adi finite difference method which has a computational work count of o n log n per iteration at each time step and a memory requirement of o n we also develop a fast multistep adi finite difference method which has a computational work count of o n log 2 n per time step and a memory requirement of o n log n numerical experiments of a three dimensional space fractional diffusion equation show that these both fast methods retain the same accuracy as the regular three dimensional implicit finite difference method but have significantly improved computational cost and memory requirement these numerical experiments show the utility of the fast method

a novel compact adi scheme for two dimensional riesz space fractional nonlinear reaction diffusion equations abstract this paper is concerned with the construction and analysis of a novel linearized compact adi scheme for the two dimensional riesz space fractional nonlinear reaction diffusion equations convergence of the proposed scheme is proved the highlight is that the time discretization is achieved by applying a second order one step and linearized method the time discretization requires only one starting value which is sharp contrast to the extrapolated crank nicolson method or the usual second order linearized schemes numerical examples on several fractional models are presented to confirm our theoretical results

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

semisimple types for p adic classical groups we construct for any symplectic unitary or special orthogonal group over a locally compact nonarchimedean local field of odd residual characteristic a type for each bernstein component of the category of smooth representations using bushnell kutzko s theory of covers moreover for a component corresponding to a cuspidal representation of a maximal levi subgroup we prove that the hecke algebra is either abelian or a generic hecke algebra on an infinite dihedral group with parameters which are at least in principle computable via results of lusztig in an appendix we make a correction to the proof of a result of the second author that every irreducible cuspidal representation of a classical group as considered here is irreducibly compactly induced from a type

dissipation preserving galerkin legendre spectral methods for two dimensional fractional nonlinear wave equations abstract in this paper we consider implicit and linearized dissipation preserving galerkin legendre spectral methods to solve space fractional nonlinear damped wave equation in two dimensions the full discrete schemes preserve dissipation of energy in damped case and conservation of energy in undamped case moreover the stability and convergence analysis of the full discrete schemes are rigorously given furthermore we get that two methods are convergent with second order accuracy in time and optimal error estimates in space in order to reduce the computational cost we adopt matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation numerical results are presented to validate the theoretical analysis

numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term in this paper we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term a new implicit difference method is constructed the stability and convergence are discussed using a new energy method finally some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis

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

a fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

the supercuspidal representations of p adic classical groups let g be a unitary symplectic or special orthogonal group over a locally compact non archimedean local field of odd residual characteristic we construct many new supercuspidal representations of g and bushnell kutzko types for these representations moreover we prove that every irreducible supercuspidal representation of g arises from our constructions

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

wellposedness of variable coefficient conservative fractional elliptic differential equations fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second order diffusion equations fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second order differential equations there are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second order differential equations these are i fractional differential operators are nonlocal operators and ii the adjoint of a fractional differential operator is not the negative of itself the wellposedness of a galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously many subsequent works were carried out to extend the analysis to other numerical methods a constant diffusivity coefficient has been assume

a time space spectral method for the time space fractional fokker planck equation and its inverse problem abstract in the paper we consider a time space spectral method to get the numerical solution of time space fractional fokker planck initial boundary value problem the temporal discretization is constructed by jacobi polynomials and the spatial discretization is composed by legendre polynomials moreover we present the stability and convergence analysis strictly the main advantages of the provided method are spectrally accurate in time and space and high computational efficiency in addition we introduce the inverse problem based on the spectral form with high order accuracy of the direct problem for the first time the levenberg marquardt l m method is proposed to estimate the two fractional derivatives and 2 some numerical results presented are consistent with the theoretical analysis

semisimple characters for p adic classical groups let g be a unitary symplectic or orthogonal group over a non archimedean local field of residual characteristic different from 2 considered as the fixed point subgroup in a general linear group of an involution following previous work of bushnell and kutzko and of the author we generalize the notion of a semisimple character for and for g in particular following the formalism of bushnell and henniart we show that these semisimple characters have certain functorial properties finally we show that any positive level supercuspidal representation of g contains a semisimple character

a pseudo spectral method for a non local kdv burgers equation posed on r abstract in this paper we present a new pseudo spectral method to solve the initial value problem associated to a non local kdv burgers equation involving a caputo type fractional derivative the basic idea is using an algebraic map to transform the whole real line into a bounded interval where we can apply a fourier expansion special attention is given to the correct computation of the fractional derivative in this setting

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

fractional pearson diffusions pearson diffusions are governed by diffusion equations with polynomial coefficients fractional pearson diffusions are governed by the corresponding time fractional diffusion equation they are useful for modeling sub diffusive phenomena caused by particle sticking and trapping this paper provides explicit strong solutions for fractional pearson diffusions using spectral methods it also presents stochastic solutions using a non markovian inverse stable time change

a crank nicolson adi spectral method for a two dimensional riesz space fractional nonlinear reaction diffusion equation in this paper a new alternating direction implicit galerkin legendre spectral method for the two dimensional riesz space fractional nonlinear reaction diffusion equation is developed the temporal component is discretized by the crank nicolson method the detailed implementation of the method is presented the stability and convergence analysis is strictly proven which shows that the derived method is stable and convergent of order 2 in time an optimal error estimate in space is also obtained by introducing a new orthogonal projector the present method is extended to solve the fractional fitzhugh nagumo model numerical results are provided to verify the theoretical analysis

hecke algebras and rm sl _n types let f be a non archimedean local field in an earlier paper we constructed certain types in the sense of bushnell and kutzko in sl n f in fact enough to describe the non supercuspidal components of the bernstein decomposition of this group we now give an almost fully explicit presentation of the hecke algebras of these types

on depth zero l packets for classical groups by computing reducibility points of parabolically induced representations we construct to within at most two unramified quadratic characters the langlands parameter of an arbitrary depth zero irreducible cuspidal representation formula presented of a classical group which may be not quasi split over a non archimedean local field of odd residual characteristic from this we can explicitly describe all the irreducible cuspidal representations in the union of one two or four formula presented packets containing formula presented these results generalize the work of debacker reeder in the case of classical groups from regular to arbitrary tame langlands parameters

numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term in this paper we consider a variable order fractional advection diffusion equation with a nonlinear source term on a finite domain explicit and implicit euler approximations for the equation are proposed stability and convergence of the methods are discussed moveover we also present a fractional method of lines a matrix transfer technique and an extrapolation method for the equation some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis

optimal error estimates of the chebyshev legendre spectral method for solving the generalized burgers equation in this paper the chebyshev legendre collocation method is applied to the generalized burgers equation optimal error estimate of the method is proved for the problem with the dirichlet boundary conditions also a legendre galerkin chebyshev collocation method is given for the generalized burgers equation the scheme is basically formulated in the legendre spectral form but with the nonlinear term being treated by the chebyshev collocation method so that the scheme can be implemented at chebyshev gauss lobatto points efficiently optimal order convergence is also obtained through coupling estimates in the l2 norm and the h1 norm

a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of variable coefficient conservative fractional diffusion equations fractional diffusion equations were shown to provide an adequate and accurate description of transport processes exhibiting anomalous diffusion behavior recently spectral galerkin methods were developed for space fractional diffusion equations aiming at achieving exponential convergence an optimal order error estimate in the fractional energy norm was proved under the assumption that the true solution to the fractional diffusion equation has the desired regularity an optimal order error estimate in the l 2 norm was proved via the well known nitsche lifting technique under the assumption that the true solution to the corresponding boundary value problem of the fractional diffusion equation has the required regularity for each right hand side in this paper we show that the true solution to the dirichlet boundary value problem of a conservative fractional diffusion equation of order 2 with 0 1 as well as a constant diffusivity coefficient and a constant source term is not in the fractional sobolev space h 3 2 in general but is still in the besov space b 3 2 l 2 hence the provable convergence rate of a spectral galerkin method in the l 2 norm is at most of the order o n 3 2 where n is the degree of the polynomial space in the numerical method numerical experiments show that the spectral galerkin method exhibits a subquadratic convergence in the l 2 norm for any 0 1 we develop a high accuracy preserving spectral galerkin method for the dirichlet boundary value problem of one sided variable coefficient conservative fractional diffusion equations the method has a proved high order convergence rate of arbitrary order i without requiring the smoothness of the true solution u to the given boundary value problem but only assuming that the diffusivity coefficient and the right hand source term have the desired regularity ii for a variable diffusivity coefficient and iii for an inhomogeneous dirichlet boundary condition numerical experiments substantiate the theoretical analysis and show that the method exhibits exponential convergence provided the diffusivity coefficient and the right hand source term have the desired regularity

a novel high order space time spectral method for the time fractional fokker planck equation the fractional fokker planck equation is an important physical model for simulating anomalous diffusions with external forces because of the nonlocal property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations in this paper a space time spectral method is presented for the numerical solution of the time fractional fokker planck initial boundary value problem the proposed method employs the jacobi polynomials for the temporal discretization and fourier like basis functions for the spatial discretization due to the diagonalizable trait of the fourier like basis functions this leads to a reduced representation of the inner product in the galerkin analysis we prove that the time fractional fokker planck equation attains the same approximation order as the time fractional diffusion equation developed in x li and c xu siam j numer anal 47 2009 pp 2108 2131 by using the present method that indicates an e

superconvergence of a discontinuous galerkin method for fractional diffusion and wave equations we consider an initial boundary value problem for partial tu partial t alpha nabla 2u f t that is for a fractional diffusion 1 alpha 0 or wave 0 alpha 1 equation a numerical solution is found by applying a piecewise linear discontinuous galerkin dg method in time combined with a piecewise linear conforming finite element method in space the time mesh is graded appropriately near t 0 but the spatial mesh is quasi uniform previously we proved that the error measured in the spatial l 2 norm is of order k 2 alpha h 2 ell k uniformly in t where k is the maximum time step h is the maximum diameter of the spatial finite elements alpha min alpha 0 le0 and ell k max 1 log k here we prove convergence of order k 3 2 alpha ell k h 2 at each time level t n for 1 alpha 1 thus if 1 2 alpha 1 then the dg solution is superconvergent which generalizes a known result for the classical heat equation i e the case alpha 0 a

twisted endoscopy and reducibility of induced representations for p adic groups 1 introduction this is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the levi factors ofmaximal parabolic subgroups of p adic groups on the unitary axis the problem is equivalent to determining certain local langlands l functions which are of arithmetic significance by themselves one of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable when the representation is supercuspidal we also study the reducibility of the induced representations off the unitary axis let f be a p adic field of characteristic zero fix a positive integer n 1 and let g be either of the groups sp2n s02n or s02n l in all three cases there is a conjugacy class of maximal parabolic subgroups which has gl as its levi factor let p mn be the standard parabolic subgroup ofg in this conjugacy class then m gl let tr be a discrete series representation of m gln f and given s c let i s tr be the representation of g g f induced from tr r idet ls let i tr i 0 tr

finite difference finite element method for two dimensional time space fractional bloch torrey equations with variable coefficients on irregular convex domains abstract in magnetic resonance imaging of the human brain the diffusion process of tissue water is considered in the complex tissue environment of cells membranes and connective tissue models based on fractional order bloch torrey equations are known to provide insights into tissue structures and the microenvironment in this paper we consider new two dimensional multi term time and space fractional bloch torrey equations with variable coefficients on irregular convex domains which involve the caputo time fractional derivative and the riemann liouville space fractional derivative an unstructured mesh galerkin finite element method is used to discretize the spatial fractional derivative while for each time fractional derivative we use the l 1 scheme on a temporal graded mesh the stability and convergence of the fully discrete scheme are proved numerical examples are given to verify the efficiency of our method

modelling anomalous diffusion using fractional bloch torrey equations on approximate irregular domains abstract diffusion weighted imaging is an in vivo non invasive medical diagnosis technique that uses the brownian motion of water molecules to generate contrast in the image and therefore reveals exquisite details about the complex structures and adjunctive information of its surrounding biological environment recent work highlights that the diffusion induced magnetic resonance imaging signal loss deviates from the classic monoexponential decay to investigate the underlying mechanism of this deviated signal decay diffusion is re examined through the bloch torrey equation by using fractional calculus with respect to both time and space in this study we explore the influence of the complex geometrical structure on the diffusion process an effective implicit alternating direction method implemented on approximate irregular domains is proposed to solve the two dimensional time space riesz fractional partial differential equation with dirichlet boundary conditions this scheme is proved to be unconditionally stable and convergent numerical examples are given to support our analysis we then applied the proposed numerical scheme with some decoupling techniques to investigate the magnetisation evolution governed by the time space fractional bloch torrey equations on irregular domains

alternating direction implicit spectral element method adi sem for solving multi dimensional generalized modified anomalous sub diffusion equation abstract the main aim of the current paper is to solve the multi dimensional generalized modified anomalous sub diffusion equation by using a new spectral element method at first the time variable has been discretized by a finite difference scheme with second order accuracy the stability and convergence of the time discrete scheme have been investigated we show that the time discrete scheme is unconditionally stable and the convergence order is o 2 in the temporal direction secondly the galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable the unconditional stability and convergence of the full discrete scheme have been proved by developing the proposed scheme we need to calculate one dimensional integrals for two dimensional problems and two dimensional integrals for three dimensional problems thus the used cpu time for the presented numerical procedure is lower than the two and three dimensional galerkin spectral element methods also the proposed method is suitable for computational domains obtained from the tensor product finally two examples are analyzed to check the theoretical results