approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

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

improved and more feasible numerical methods for riesz space fractional partial differential equations abstract in this article the numerical solutions of fractional order partial differential equations with riesz space fractional derivatives on a finite domain have been considered two kinds of equations have been considered the riesz fractional diffusion equation and the riesz fractional advection dispersion equation the riesz fractional diffusion equation is obtained from the standard diffusion equation by replacing the second order space derivative with the riesz fractional derivative of order where 1 2 the riesz fractional advection dispersion equation is obtained from the standard advection dispersion equation by replacing the first and second order space derivatives with the riesz fractional derivatives of order 1 1 and 1 2 respectively to obtain the solutions firstly the system of odes is obtained using the improved matrix transform method with respect to the space variable after this the 3 1 pade approximation can be used to construct the exponential matrix in the analytic solution of the ode and we obtain two difference schemes it has also been shown that the two difference schemes are unconditionally stable and feasible using the matrix analysis method finally some numerical results are given which demonstrate the effectiveness of the two difference schemes

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

the analytical solution and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation in this paper we consider the analytical and numerical solutions for a two dimensional multi term time fractional diffusion and diffusion wave equation we derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate mittag leffler function an implicit difference approximation is constructed stability and convergence analysis of the numerical scheme are proved by the energy method numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis

a stable three level finite difference scheme for solving the parabolic two step model in a 3d micro sphere heated by ultrashort pulsed lasers ultrashort pulsed lasers with pulse durations of the order of sub picosecond to femtosecond domain possess exclusive capabilities in limiting the undesirable spread of the thermal process zone in the heated sample parabolic two step micro heat transport equations have been widely applied for thermal analysis of thin metal films exposed to picosecond thermal pulses in this study we develop a three level finite difference scheme for solving the heat transport equations in a three dimensional micro sphere heated by ultrashort pulsed lasers it is shown that the scheme is unconditionally stable the method is illustrated by numerical examples

on the stability of boundary conditions for molecular dynamics we study the stability of boundary conditions for molecular dynamics simulations a general stability criterion is established we first consider a one dimensional model with nearest neighbor interaction and multiple neighbor interactions we then generalize the results to more realistic models in 3d with nonlinear atomic interaction

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

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

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

n dimensional fractional fokker planck equation and its solutions for anomalous radial two phase flow in porous media in this paper we present a time fractional fokker planck equation ffpe for radial two phase flow of liquid and gas in porous media the ffpe of order a is solved for both two and three dimensional flow patterns using the laplace transform method the general solutions of the ffpe for both two and three dimensional flows are given as a convolution integral of the input and a kernel in the laplace domain special solutions for a large value and a periodic boundary condition are also given in the time domain when the inverse laplace transform can be found analytically the ffpe for two phase flow in porous media presented in this paper is the first report of its kind

on nonlinear equations for fuzzy mappings in probabilistic normed spaces in this paper we introduce the concept of probabilistic contractor couple which simplifies and weakens that of probabilistic g contractor given by cho et al fuzzy sets and systems 110 2000 115 122 by using this concept we study the existence and iteration convergence of solutions for set valued nonlinear operator equations and nonlinear equations of fuzzy mappings in menger pn spaces the results presented in this paper improve and extend the corresponding results of cho et al in a sense

non linear vibrations of free edge thin spherical shells modal interaction rules and 1 1 2 internal resonance this paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement the analog for thin shallow shells of von karman s theory for large deflection of plates is used the validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution the specific case of a free edge is considered the governing partial differential equations are expanded onto the natural modes of vibration of the shell the problem is replaced by an infinite set of coupled second order differential equations with quadratic and cubic non linear terms analytical expressions of the non linear coefficients are derived and a number of them are found to vanish as a consequence of the symmetry of revolution of the structure then for all the possible internal resonances a number of rules are deduced thus predicting the activation of the energy exchanges between the involved modes finally a specific mode coupling due to a 1 1 2 internal resonance between two companion modes and an axisymmetric mode is studied 2004 elsevier ltd all rights reserved

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

solving linear and non linear space time fractional reaction diffusion equations by the adomian decomposition method in this paper we consider linear and non linear space time fractional reaction diffusion equations stfrde on a finite domain the equations are obtained from standard reaction diffusion equations by replacing a second order space derivative by a fractional derivative of order 1 2 and a first order time derivative by a fractional derivative of order 0 1 we use the adomian decomposition method to construct explicit solutions of the linear and non linear stfrde finally some examples are given

convective scheme solution of the boltzmann transport equation for nanoscale semiconductor devices a model for the simulation of the electron energy distribution in nanoscale metal oxide semiconductor field effect transistor mosfet devices using a kinetic simulation technique is implemented the convective scheme cs a method of characteristics is an accurate method of solving the boltzmann transport equation a nonlinear integrodifferential equation for the distribution of electrons in a mosfet device the method is used to find probabilities for use in an iterative scheme which iterates to find collision rates in cells the cs is also a novel approach to 2d semiconductor device simulation the cs has been extended to handle boundary conditions in 2d as well as to calculation of polygon overlap for polygons of more than three sides electron energy distributions in the channel of a mosfet are presented

a mass balance based numerical method for the fractional advection dispersion equation theory and application 1 the inherent heterogeneity of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement and one approach to model solute transport through such media is the continuous time random walk ctrw one special asymptotic case of the ctrw is the fractional advection dispersion equation fade which has proven to be a promising alternative to model anomalous dispersion and has been increasingly used in hydrology to model chemical transport in both surface and subsurface water most practical problems in hydrology have complicated initial and boundary conditions and need to be solved numerically but the numerical solution of the fade is not trivial in this paper we present a finite volume approach to solve the fade where the spatial derivative of the dispersion term is fractional we also give methods to solve different boundary conditions often encountered in practical applications the linear system resulting from the temporal spatial discretization is solved using a semi implicit scheme the numerical method is derived on the basis of mass balance and its accuracy is tested against analytical solutions the method is then applied to simulate tracer movement in a stream and a near saturated hillslope in a naturally structured upland podzol field in northeast scotland

the fractional diffusion equation in one space and in one time dimension a diffusion equation is solved where the first time derivative is replaced by the fractional time derivative 0 1 the solution is given in closed form in terms of fox functions

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

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

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

evidences of global bifurcations of an imperfect circular plate abstract the global bifurcations in modal interactions of an imperfect circular plate with one to one internal resonance are investigated the case of the third order subharmonic resonance in which an excitation frequency is near triple natural frequencies is considered the equations governing nonlinear oscillations of an imperfect circular plate are reduced to a system of non autonomous ordinary differential equations via galerkin s procedure the method of multiple scales is used to obtain a system of autonomous ordinary differential equations and then kovacic and wiggins method is used to investigate the global dynamics of the plate having found a sufficient condition under which silnikov type homoclinic orbit can exist we failed to observe any numerical evidences of global bifurcation

coarse graining molecular dynamics models using an extended galerkin projection method summary we present a new framework for coarse graining molecular dynamics models for crystalline solids the reduction method is based on a galerkin projection to a subspace whose dimension is much smaller than that of the full atomistic model to effectively reduce artificial reflections of phonons at the interface we construct extended subspaces with increasing accuracy by adding more coarse grained variables near the interface between lattice defects and the surrounding region this approach is equivalent to the generalized langevin model but it eliminates the need to precompute the memory function a well known practical difficulty further the variational formulation preserves the stability of the molecular models copyright 2014 john wiley sons ltd

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

wright functions as scale invariant solutions of the diffusion wave equation the time fractional diffusion wave equation is obtained from the classical diffusion or wave equation by replacing the first or second order time derivative by a fractional derivative of order 0 2 using the similarity method and the method of the laplace transform it is shown that the scale invariant solutions of the mixed problem of signalling type for the time fractional diffusion wave equation are given in terms of the wright function in the case 0 1 and in terms of the generalized wright function in the case 1 2 the reduced equation for the scale invariant solutions is given in terms of the caputo type modification of the

research on some problems for nonlinear operators abstract in this paper we discuss some problems for nonlinear operators in the menger pn space meanwhile we prove existence theorems for the solution of an operator equation and obtain some new results by means of the theory of topological degree for the z p s space

the analytical solution and numerical solution of the fractional diffusion wave equation with damping fractional partial differential equations have been applied to many problems in physics finance and engineering numerical methods and error estimates of these equations are currently a very active area of research in this paper we consider a fractional diffusionwave equation with damping we derive the analytical solution for the equation using the method of separation of variables an implicit difference approximation is constructed stability and convergence are proved by the energy method finally two numerical examples are presented to show the effectiveness of this approximation

numerical solution of the time fractional reaction diffusion equation with a moving boundary abstract a fractional reaction diffusion model with a moving boundary is presented in this paper an efficient numerical method is constructed to solve this moving boundary problem our method makes use of a finite difference approximation for the temporal discretization and spectral approximation for the spatial discretization the stability and convergence of the method is studied and the errors of both the semi discrete and fully discrete schemes are derived numerical examples motivated by problems from developmental biology show a good agreement with the theoretical analysis and illustrate the efficiency of our method

preconditioned iterative methods for space time fractional advection diffusion equations in this paper we propose practical numerical methods for solving a class of initial boundary value problems of space time fractional advection diffusion equations first we propose an implicit method based on two sided grunwald formulae and discuss its stability and consistency then we develop the preconditioned generalized minimal residual preconditioned gmres method and preconditioned conjugate gradient normal residual preconditioned cgnr method with easily constructed preconditioners importantly because resulting systems are toeplitz like fast fourier transform can be applied to significantly reduce the computational cost we perform numerical experiments to demonstrate the efficiency of our preconditioners even in cases with variable coefficients

numerical simulation of the fractional bloch equations in physics and chemistry specifically in nmr nuclear magnetic resonance or mri magnetic resonance imaging or esr electron spin resonance the bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization m m x m y m z as a function of time when relaxation times t 1 and t 2 are present recently some fractional models have been proposed for the bloch equations however effective numerical methods and supporting error analyses for the fractional bloch equation fbe are still limited in this paper the time fractional bloch equations tfbe and the anomalous fractional bloch equations afbe are considered firstly we derive an analytical solution for the tfbe with an initial condition secondly we propose an effective predictor corrector method pcm for the tfbe and the error analysis for pcm is investigated furthermore we derive an effective implicit numerical method inm for the anomalous fractional bloch equations afbe and the stability and convergence of the inm are investigated we prove that the implicit numerical method for the afbe is unconditionally stable and convergent finally we present some numerical results that support our theoretical analysis

a fourth order compact solution of the two dimensional modified anomalous fractional sub diffusion equation with a nonlinear source term this work is concerned to the study of high order difference scheme for the solution of a two dimensional modified anomalous sub diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses the space fractional derivatives are described in the riemann liouville sense in the proposed scheme we discretize the space derivatives with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivatives to obtain a fully discrete implicit scheme we prove the stability and convergence of proposed scheme using the fourier analysis the convergence order of the proposed method is o t h x 4 h y 4 comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme

a fast semi implicit difference method for a nonlinear two sided space fractional diffusion equation with variable diffusivity coefficients in this paper we derive a new nonlinear two sided space fractional diffusion equation with variable coefficients from the fractional fick s law a semi implicit difference method sidm for this equation is proposed the stability and convergence of the sidm are discussed for the implementation we develop a fast accurate iterative method for the sidm by decomposing the dense coefficient matrix into a combination of toeplitz like matrices this fast iterative method significantly reduces the storage requirement of o n 2 and computational cost of o n 3 down to n and o n log n where n is the number of grid points the method retains the same accuracy as the underlying sidm solved with gaussian elimination finally some numerical results are shown to verify the accuracy and efficiency of the new method

analytical and numerical solutions of the unsteady 2d flow of mhd fractional maxwell fluid induced by variable pressure gradient abstract the nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations this paper studies the analytical and numerical methods of unsteady 2d flow of magnetohydrodynamic mhd fractional maxwell fluid in a rectangular pipe driven by variable pressure gradient the governing equation is formulated with caputo time dependent fractional derivatives whose orders are distributed in interval 0 2 a challenge is to firstly obtain the exact solution by combining modified separation of variables method with mikusinski type operational calculus meanwhile the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed different to the most classical works both the stability and convergence analysis of two dimensional multi term time fractional momentum equation are derived based on numerical analysis the results show that the velocity increases with the rise of the fractional parameter and relaxation time while an increase in the values of hartmann number leads to a slower velocity in the rectangular pipe

non linear interactions in asymmetric vibrations of a circular plate abstract in order to investigate the non linear asymmetric vibrations of a clamped circular plate on an elastic foundation the primary resonances of a plate with an internal resonance in which the natural frequencies of two asymmetric modes are commensurable are considered the response is expressed as an expansion in terms of the linear free oscillation modes and its amplitude is considered to be small but finite the method of multiple scales is used to reduce the non linear governing equations to a system of autonomous ordinary differential equations for amplitude and phase variables for a numerical example the case of internal resonance a commensurable relationship between natural frequencies 32 3 11 where the first subscript refers to the number of nodal diameters and the second subscript the number of nodal circles including boundary is considered when the frequency of excitation is near 11 there exist at most five stable steady state responses four of them are superpositions of traveling wave components and one is a superposition of standing wave components the result shows the interaction between modes corresponding to 11 and 32 by showing non vanishing amplitudes of the mode not directly excited when the frequency of excitation is near 32 similarly the interaction between modes is shown to exist all of the responses with non vanishing amplitudes of modes excited indirectly however turn out to be unstable which is a peculiar phenomenon

stable evaluations of fractional derivative of the muntz legendre polynomials and application to fractional differential equations abstract the aim of this paper is to present efficient and stable methods to compute caputo fractional derivative cfd of the muntz legendre polynomials based on three term recurrence relations and gauss jacobi quadrature rules this approach with collocation method at chebyshev gauss lobatto points has been applied for solving linear and nonlinear fractional multi order differential equations fdes described in caputo sense the main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations in this work for the first time we present the new rates of convergence for projection error which are more accurate than the rate presented by shen and wang in shen and wang 2016 moreover we present convergence rate for spectral collocation method for linear fdes with initial value on a finite interval and endpoint singularities also we propose an error analysis for jacobi gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper finally the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples

a new approach of superconvergence analysis for two dimensional time fractional diffusion equation abstract in this paper a new approach of superconvergent estimate of bilinear finite element is established for two dimensional time fractional diffusion equation under fully discrete scheme the novelty of this approach is the combination technique of the interpolation and ritz projection as well as the superclose estimate in h 1 norm between them which avoids the difficulty of constructing a postprocessing operator for ritz projection operator and reduces the regularity requirement of the exact solution at the same time three numerical examples are carried out to verify the theoretical analysis

the effect of the number of nodal diameters on non linear interactions in two asymmetric vibration modes of a circular plate in order to investigate the effect of the number of nodal diameters on non linear interactions in asymmetric vibrations of a circular plate a primary resonance of the plate is considered the plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one the response of the plate is expressed as an expansion in terms of the linear free oscillation modes and its amplitude is considered to be small but finite and the method of multiple scales is used in view of the corrected solvability conditions for the responses it has been found that in order for the modes to interact the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one in this study the one to one case in which the modes have the same number of nodal diameters is examined the non linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions the steady state responses and their stability are determined by using this system the result shows very complicated interactions between two modes by telling existence of non vanishing amplitudes of the mode not directly excited

convergence and superconvergence of a fully discrete scheme for multi term time fractional diffusion equations using finite element method in spatial direction and classical l1 approximation in temporal direction a fully discrete scheme is established for a class of two dimensional multi term time fractional diffusion equations with caputo fractional derivatives the stability analysis of the approximate scheme is proposed the spatial global superconvergence and temporal convergence of order o h2 2 for the original variable in h1 norm is presented by means of properties of bilinear element and interpolation postprocessing technique where h and are the step sizes in space and time respectively finally several numerical examples are implemented to evaluate the efficiency of the theoretical results

block pulse operational matrix method for solving fractional partial differential equation in this paper we first introduce two dimensional block pulse functions and the block pulse operational matrices of the fractional order integration also the block pulse operational matrices of the fractional order differentiation are obtained then we present a computational method based on the above results for solving a class of fractional partial differential equations transforming the initial equation into a sylvester equation the error analysis of the method is given the method is computationally attractive and applications are demonstrated by some numerical examples moreover comparing the methodology with the known technique shows that our approach is more efficient and more convenient

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

calculations of random fixed point index in this paper we prove an important inequality and investigate some new calculating problems of random fixed point index and generalize famous theorem by means of the theory of random fixed point index

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 methods for fractional dispersion equations abstract in this paper the fractional weighted average finite difference method for space fractional advection dispersion equation is proposed which is based on shifted grunwald formula this method is unconditionally stable consistent and convergent a numerical example is given and the numerical results verify the theoretical conclusions

modal interaction in circular plates abstract the dynamic analogue of the von karman equations in polar form is used to study the symmetric response of a circular plate to a harmonic external excitation the method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the phases and amplitudes of oscillation for the case of an internal or autoparametric resonance the results show that the internal resonance is responsible for coupling of the modes involved and that the excited mode is not necessarily the dominant one furthermore the results show that for a choice of parameters the multi mode response loses its stability through a hopf bifurcation resulting in periodically or chaotically modulated motions of the plate

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

an unstructured mesh control volume method for two dimensional space fractional diffusion equations with variable coefficients on convex domains in this paper we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains which to the best of our knowledge is a new contribution to the literature firstly we present the finite volume scheme for the two dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements secondly we explore the property of the stiffness matrix generated by the integral of the space fractional derivative we find that the stiffness matrix is sparse and not regular therefore we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system which is more efficient than using the gaussian elimination method finally we present several examples to verify our method in which we make a comparison of our method with the finite element method for solving a riesz space fractional diffusion equation on a circular domain the numerical results demonstrate that our method can reduce cpu time significantly while retaining the same accuracy and approximation property as the finite element method the numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains

boundary layer flows of viscoelastic fluids over a non uniform permeable surface abstract this study investigates viscoelastic fluid which possesses both viscous and elastics properties by employing a fractional derivative model to reveal the stress relaxation phenomenon with distance the spatial fractional derivative used in the momentum conservation equation is the riemann liouville type derivative further we use non uniform boundary conditions subject to the boundary layer equations of the fluid flowing through a semi infinite permeable flat surface owing to the fractional derivative model and non uniform boundary conditions this problem is complex thus a finite difference scheme is applied after the coupled continuity equation and momentum equation are decoupled and linearized the accuracy convergence and stability of the numerical method are presented it is shown that non uniform mass transfer through a permeable surface considerably affects the velocity boundary layer by illustrating the physical interactions between the velocity fields in the boundary layer and the permeation mode in the surface this paper predicts the velocity distributions with varying permeable surface and also provides the possibility of changing the velocity fields by altering the permeable sheet the results and numerical technique used in this study will help in the understanding of fractional calculus investigation in engineering

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

boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

space fractional advection dispersion equations by the kansa method the paper makes the first attempt at applying the kansa method a radial basis function meshless collocation method to the space fractional advection dispersion equations which have recently been observed to accurately describe solute transport in a variety of field and lab experiments characterized by occasional large jumps with fewer parameters than the classical models of integer order derivative however because of non local property of integro differential operator of space fractional derivative numerical solution of these novel models is very challenging and little has been reported in literature it is stressed that local approximation techniques such as the finite element and finite difference methods lose their sparse discretization matrix due to this non local property thus the global methods appear to have certain advantages in numerical simulation of these non local models because of their high accuracy and smaller size resultant matrix equation compared with the finite difference method popular in the solution of fractional equations the kansa method is a recent meshless global technique and is promising for high dimensional irregular domain problems in this study the resultant matrix of the kansa method is accurately calculated by the gauss jacobi quadrature rule numerical results show that the kansa method is highly accurate and computationally efficient for space fractional advection dispersion problems

basin boundaries in asymmetric vibrations of a circular plate abstract in order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation we reexamine a primary resonance studied by yeo and lee corrected solvability conditions for non linear asymmetric vibrations of a circular plate journal of sound and vibration 257 2002 653 665 in which at most three stable steady state responses one standing wave and two traveling waves are observed to exist further examination however tells that there exist at most five stable steady state responses one standing wave and four traveling waves two of the traveling waves lose their stability by hopf bifurcation and have a sequence of period doubling bifurcations leading to chaos when the system has five attractors three equilibrium solutions one standing wave and two traveling waves and two chaotic attractors two modulated traveling waves the basin boundaries of the attractors on the principal plane are obtained also examined is how basin boundaries of the modulated motions quasi periodic and chaotic motions evolve as a system parameter varies the basin boundaries of the modulated motions turn out to have the fractal nature

a meshless method for solving the time fractional advection diffusion equation with variable coefficients abstract in this paper an efficient and accurate meshless method is proposed for solving the time fractional advection diffusion equation with variable coefficients which is based on the moving least square mls approximation in the proposed method firstly the time fractional derivative is approximated by a finite difference scheme of order o t 2 0 1 and then the mls approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the caputo sense also the validity of the proposed method is investigated in error analysis discussion the main aim is to show that the meshless method based on the mls shape functions is highly appropriate for solving fractional partial differential equations fpdes with variable coefficients the efficiency and accuracy of the proposed method are verified by solving several examples

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

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

path integral solutions of the governing equation of sdes excited by levy white noise abstract in this paper the probability density functions pdfs of scalar stochastic differential equations sdes subject to stable levy white noise are investigated the path integral pi method is extended to solve one dimensional space fractional fokker planck kolmogorov fpk equations which are the governing equations corresponded to scalar sdes excited by stable levy white noise first we derive a short time solution of the one dimensional space fractional fpk equation which is used in the chapman kolmogorov smoluchowski cks equation to obtain the pi solution then the accuracy of the pi solution is analyzed theoretically in terms of its characteristic function our results demonstrate that the pi method has a higher accuracy than the first order finite difference method for one step iteration in time finally several illustrative examples are carried out in detail to verify the feasibility and effectiveness of the pi method for solving one dimensional space fractional fpk equations we find that the pi solution agrees well with the exact solution or the monte carlo one

relative controllability of fractional dynamical systems with distributed delays in control this paper is concerned with the global relative controllability of fractional dynamical systems with multiple delays in control for finite dimensional spaces sufficient conditions for controllability results are obtained using schauder s fixed point theorem and the controllability grammian matrix which is defined by the mittag leffler matrix function an example is provided to illustrate the theory

convergence of the variational iteration method for solving multi order fractional differential equations in this paper the variational iteration method vim is applied to obtain approximate solutions of multi order fractional differential equations m fdes we can easily obtain the satisfying solution just by using a few simple transformations and applying the vim a theorem for convergence and error estimates of the vim for solving m fdes is given moreover numerical results show that our theoretical analysis are accurate and the vim is a powerful method for solving m fdes

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

several nonlinear operator problems in the menger pn space abstract in this paper we investigate some nonlinear operator problems and the theory of topological degree meanwhile we obtain some new fixed point theorems in the m pn space

corrected solvability conditions for non linear asymmetric vibrations of a circular plate an investigation into non linear asymmetric vibrations of a clamped circular plate under a harmonic excitation is made we re examined a primary resonance studied by sridhar mook and nayfeh in which the frequency of excitation is near the natural frequency of an asymmetric mode of the plate we corrected their solvability conditions and found that in the absence of internal resonance the steady state response can have not only the form of standing wave but also the form of travelling wave which is a remarkable contrast to their conclusion

maximum norm error analysis of difference schemes for fractional diffusion equations in this study we present several numerical approximation methods for solving the fractional diffusion equations with the riesz space fractional derivative based on the fractional centered difference approximation to the riesz fractional derivative an explicit and an implicit finite difference schemes for solving the space fractional diffusion equations are obtained the stability and global convergence of these two schemes in the maximum norm are analyzed rigorously the convergence order is o h 2 where is the temporal grid size and h is spatial grid size respectively furthermore a numerical scheme for the one dimensional multi term time space fractional diffusion equations is obtained the unconditionally stability and convergence of the scheme in maximum norm are established the convergence orders are 2 in the temporal direction and two in the spatial direction where is the maximum time fractional derivative orders in the equation in addition the extension to the two dimensional case is also discussed finally several numerical examples are provided to show the effectiveness and accuracy of our methods

fractional parabolic two step model and its accurate numerical scheme for nanoscale heat conduction abstract simulation of nanoscale heat transfer phenomena has attracted great attention particularly the nanoscale heat conduction induced by ultrashort pulsed laser heating in this article we present a nanoscale fractional parabolic two step model and obtain an accurate numerical scheme for nanoscale heat conduction in metals the model is obtained by introducing the knudsen number k n and the fractional order derivative in time to the original parabolic two step energy transport and then coupling them with the k n dependent and temperature jump boundary condition the numerical scheme is developed based on the fourth order compact finite difference method and the l 1 approximation for fractional derivatives the well posedness of the model and the stability and convergence of the scheme are analyzed theoretically we finally test the accuracy and applicability of the new model and the obtained numerical method by two examples by changing values of the knudsen number and fractional order derivative as well as the parameter in the boundary condition the simulation could be a tool for analyzing the nanoscale heat conduction in porous media such as porous thin metal film exposed to ultrashort pulsed lasers

a compact fourth order in space energy preserving method for riesz space fractional nonlinear wave equations in this work we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the riesz type the model under consideration generalizes various nonlinear wave equations including the sine gordon and the nonlinear klein gordon models the system considered in this work is conservative when homogeneous dirichlet boundary conditions are imposed motivated by this fact we propose a finite difference method based on fractional centered differences that is capable of preserving the discrete energy of the system the method under consideration is a nonlinear implicit scheme which has various numerical properties among the most interesting numerical features we show that the methodology is consistent of second order in time and fourth order in space moreover we show that the technique is stable and convergent some numerical simulations show that the method is capable of preserving the energy of the discrete system this characteristic of the technique is in obvious agreement with the properties of its continuous counterpart