a numerical approach to the generalized nonlinear fractional fokker planck equation in this paper we propose a fully discrete galerkin finite element method to solve the generalized nonlinear fractional fokker planck equation which has a multi fractional spatial operator characteristic that describes the levy flight in the time direction we use the finite difference method and in the spatial direction we use the fractional finite element method in the framework of the fractional sobolev spaces we derive a fully discrete scheme for the considered equation we prove the existence and uniqueness of the discrete solution and give the error estimates the numerical examples are also included which support the theoretical analysis

truncation error bounds for the composition of limit periodic linear fractional transformations abstract baltus c truncation error bounds for the composition of limit periodic linear fractional transformations journal of computational and applied mathematics 46 1993 395 404 suppose i tn w is a sequence of linear fractional transformations tn w t w aw b w d ad b or a b 0 d where fixed points un u vn v with d v d u

approximations of fractional integrals and caputo fractional derivatives abstract in this paper we propose two algorithms for numerical fractional integration and caputo fractional differentiation we present a modification of trapezoidal rule that is used to approximate finite integrals the new modification extends the application of the rule to approximate integrals of arbitrary order 0 we then using the new modification derive an algorithm to approximate fractional derivatives of arbitrary order 0 where the fractional derivative based on caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points the study is conducted through illustrative examples and error analysis

fractional derivatives in complex planes abstract many models are formulated in terms of fractional derivatives such as in viscoelasticity electrochemistry electrode electrolyte polarization signal processing diffusion processes control processing etc in this paper we first study important properties of the caputo derivative in real line then we study the recently developed fractional derivative in complex plane by ortigueira which is very useful in signal processing we also generalize the caputo derivative in real line to that in complex plane then study its properties these discussions are helpful in understanding fractional calculus and establishing fractional models in science and engineering

analytical solution for the time fractional telegraph equation by the method of separating variables in this paper a method of separating variables is effectively implemented for solving a time fractional telegraph equation tfte we discuss and derive the analytical solution of the tfte with three kinds of nonhomogeneous boundary conditions namely dirichlet neumann and robin boundary conditions

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

a rbf based differential quadrature method for solving two dimensional variable order time fractional advection diffusion equation abstract numerical simulation technique of two dimensional variable order time fractional advection diffusion equation is developed in this paper using radial basis function based differential quadrature method rbf dq to the best of the authors knowledge this is the first application of this method to variable order time fractional advection diffusion equations for the general case of irregular geometries the meshless local form of rbf dq is used and the multiquadric type of radial basis functions is selected for the computations this approach allows one to define a reconstruction of the local radial basis functions to treat accurately both the dirichlet and neumann boundary conditions on the irregular boundaries the method is validated by the well documented test examples involving variable order fractional modeling of air pollution the numerical results demonstrate that the proposed method provides accurate solutions for two dimensional variable order time fractional advection diffusion equations

finite difference finite element method for two dimensional space and time fractional bloch torrey equations in this paper a class of two dimensional space and time fractional bloch torrey equations 2d stfbtes are considered some definitions and properties of fractional derivative spaces are presented by finite difference method and galerkin finite element method a semi discrete variational formulation for 2d stfbtes is obtained the stability and convergence of the semi discrete form are discussed then a fully discrete scheme of 2d stfbtes is derived and the convergence is investigated finally some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis

complex dynamics of the limit periodic system f n z f n 1 f n z f n f abstract the sequence f n z is one kind of generalization of limit periodic continued fractions the convergence divergence of f n z relative to that of the iterative sequence f n z is determined when n uniformly on a domain s when maps the closure of s into s f n z for all z s if n rapidly on s the sequence f n uniformly shadows n f n z n z and the two sequences show similar complex dynamics a priori uniform tail end bounds are provided n 1 o o n m z g n 1 o o g n m z n m where for example the first expansion may represent a modified continued fraction and the second another limit periodic structure with g n

solving fractional optimal control problems within a chebyshev legendre operational technique abstractin this manuscript we report a new operational technique for approximating the numerical solution of fractional optimal control foc problems the operational matrix of the caputo fractional derivative of the orthonormal chebyshev polynomial and the legendre gauss quadrature formula are used and then the lagrange multiplier scheme is employed for reducing such problems into those consisting of systems of easily solvable algebraic equations we compare the approximate solutions achieved using our approach with the exact solutions and with those presented in other techniques and we show the accuracy and applicability of the new numerical approach through two numerical examples

constructive approximations of markov operators we construct piecewise linear markov finite approximations of markov operators defined on l1 0 1 n and we study various properties such as consistency stability and convergence for the purpose of numerical analysis of markov operators

differentiation to fractional orders and the fractional telegraph equation using methods of differential and integral calculus we present and discuss the calculation of a fractional green function associated with the one dimensional case of the so called general fractional telegraph equation with one space variable this is a fractional partial differential equation with constant coefficients the equation is solved by means of juxtaposition of transforms i e we introduce the laplace transform in the time variable and the fourier transform in the space variable several particular cases are discussed in terms of the parameters some known results are recovered as a by product of our main result we obtain two new relations involving the two parameter mittag leffler function

a convergence question inspired by stieltjes and by value sets in continued fraction theory abstract let v be a subset of the complex plane c let fn n 1 be a sequence of self mappings of v i e fn v v the question is then under what conditions will the sequence fn w f1 f2 fn w n 1 2 3 of composite maps converge to a constant function in v in this paper we give a survey of some of the answers and open problems connected with this question such answers have applications in dynamical systems schur analysis continued fractions and other similar structures like infinite exponentials infinite radicals

fractional telegraph equations we investigate several aspects of the fractional telegraph equations in an effort to better understand the anomalous diffusion processes observed in blood flow experiments in the earlier work eckstein et al electron j differential equations conf 03 1999 39 50 the telegraph equation d 2 u 2adu au 0 was used where d d dt a nd it was shown that as t tends to infinity u is approximated by v w here 2adv av 0 here a d 2 dx 2 on l 2 r o ra can be a more general nonnegative selfadjoint operator in this paper the concern is with the fractional telegraph equation e 2 u 2aeu au 0 where e d and 0 1 after solving this equation it is shown that u is approximated

fourth order finite difference schemes for time space fractional sub diffusion equations in this paper we devote to the study of high order finite difference schemes for one and two dimensional time space fractional sub diffusion equations a fourth order finite difference scheme is invoked for the spatial fractional derivatives and the l 1 approximation is applied to the temporal fractional parts for the two dimensional case an alternating direction implicit scheme based on l 1 approximation is proposed the stability and convergence of the proposed methods are studied numerical experiments are performed to verify the effectiveness and accuracy of the proposed difference schemes

fractional difference finite element approximations for the time space fractional telegraph equation abstract in this paper we study the numerical solution of the time space fractional order fractional for simplicity telegraph equation which can be used in signal analysis for transmission and propagation of electrical signals also the modeling of the reaction diffusion and the random walk of suspension flows and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the spatial riemann liouville fractional derivative with order 1 1 2 and the finite difference schemes for the temporal caputo derivatives with orders 1 2 1 and 2 are analyzed respectively results on the existence and uniqueness of the solution the numerical stability and the error estimates are displayed in details numerical examples are included to confirm the theoretical analysis

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

galerkin finite element method for two dimensional riesz space fractional diffusion equations abstract in this article a class of two dimensional riesz space fractional diffusion equations is considered some fractional spaces are established and some equivalences between fractional derivative spaces and fractional sobolev space are presented by the galerkin finite element method and backward difference method a fully discrete scheme is obtained according to lax milgram theorem the existence and uniqueness of the solution to the fully discrete scheme are investigated the stability and convergence of the scheme are also derived finally some numerical examples are given for verification of our theoretical analysis

compositions of analytic functions of the form f n z f n 1 f n z f n z f z abstract frequently in applications a function is iterated in order to determine its fixed point which represents the solution of some problem in the variation of iteration presented in this paper fixed points serve a different purpose the sequence f n z is studied where f 1 z f 1 z and f n z f n 1 f n z with f n f many infinite arithmetic expansions exhibit this form and the fixed point of f may be used as a modifying factor z to influence the convergence behaviour of these expansions thus one employs rather than seeks the fixed point of the function f

a modified piecewise linear markov approximation of markov operators in this paper we propose a modified piecewise linear markov method for approximating markov operators defined on l 1 0 1 we present numerical results with the new method as compared to those using the original piecewise linear markov method

trapezoidal scheme for time space fractional diffusion equation with riesz derivative abstract in this paper a finite difference scheme is proposed to solve time space fractional diffusion equation which has second order accuracy in both time and space direction the time and space fractional derivatives are considered in the senses of caputo and riesz respectively first the centered difference approach is used to approximate the riesz fractional derivative in space then the obtained fractional ordinary differential equations are transformed into equivalent volterra integral equations and then the trapezoidal rule is utilized to approximate the volterra integral equations the stability and convergence of our scheme are proved via mathematical induction method finally numerical experiments are performed to confirm the high accuracy and efficiency of our scheme

compositions of contractions we prove that if d is a simply connected open domain in the complex plane c e is a closed subset of d and fn n 1 are functions analytic in d such that fn d e for all n then the compositions fn z f1of2o ofn z for n 1 2 3 converge uniformly in d to a constant function some generalizations are also presented

random iterations fixed points and invariant crf horospheres in complex banach spaces abstract for holomorphic noncontractive maps on not necessarily bounded domains in complex banach spaces we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations in particular we show that the problem concerning the existence of the horospheres determined by caratheodory reiffen finsler pseudometrics defined on unbounded domains has the solution and we prove new results of type of julia s lemma and wolff s theorem

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

analytic and approximate solutions of the space and time fractional telegraph equations the adomian decomposition method is used to obtain analytic and approximate solutions of the space and time fractional telegraph equations the space and time fractional derivatives are considered in the caputo sense the analytic solutions are calculated in the form of series with easily computable terms some examples are given the results reveal that the adomian method is very effective and convenient

finite approximations of markov operators in this paper we survey some recent developments in the numerical analysis of markov operators and in particular frobenius perron operators associated with chaotic discrete dynamical systems

a finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation abstract an efficient numerical technique is proposed to solve one and two dimensional space fractional tempered fractional diffusion wave equations the space fractional is based on the riemann liouville fractional derivative at first the temporal direction is discretized using a second order accurate difference scheme then a classic galerkin finite element is employed to obtain a full discrete scheme furthermore for the time discrete and the full discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method finally two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique

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

random iterations of holomorphic contractions in locally convex spaces and of weaker contractions in uniform spaces conditions which guarantee the uniform convergence of random iterations of holomorphic contractions on unbounded domains in complex hausdorff locally convex and sequentially complete topological vector spaces are established also conditions concerning the convergence of random iterations of weaker contractions in hausdorff uniform spaces are given

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

a note on the finite element method for the space fractional advection diffusion equation in this paper a note on the finite element method for the space fractional advection diffusion equation with non homogeneous initial boundary condition is given where the fractional derivative is in the sense of caputo the error estimate is derived and the numerical results presented support the theoretical results

numerical approximation of higher order time fractional telegraph equation by using a combination of a geometric approach and method of line we propose a simple and accurate numerical scheme for solving the time fractional telegraph tft equation within caputo type fractional derivative a fictitious coordinate is imposed onto the problem in order to transform the dependent variable u x t into a new variable with an extra dimension in the new space with the added fictitious dimension a combination of method of line and group preserving scheme gps is proposed to find the approximate solutions this method preserves the geometric structure of the problem power and accuracy of this method has been illustrated through some examples of tft equation

markov finite approximation of frobenius perron operator the problem of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory 3 lasota and yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations s from 0 l to itself 4 later in 6 li and yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping the classical birkhoff s individual ergodic theorem says that if u is an ergodic invariant probability measure under s then for any measurable set a c 0 11 the time average lim 1 n c ax s x which measures the average time spent in a under iteration of s exists to be i for p almost all x where xa is the characteristic function of a 1 on a and 0 off a this seems to suggest that we may use the cesaro sum of the above form to calculate the invariant measure a simple and important example in 5 however shows that the computer round off error can completely dominate the calculation and make the implementation difficult in order to overcome this difficulty li 5 proposed a rigorous numerical procedure which can be implemented on a computer with negligible round off error problem a mapping s 0 l 0 l is called piecewise c2 if there is a partition 0 a a a 1 of 0 l such that for k 1 r the restriction sk of s on uk a is a c2 function which can be extended to the closed interval ak i ak as a c2 function s need not be continuous at the point ak for a c 0 11 we write s a for x s x e a throughout the paper we assume s is a nonsingular measurable transformation i e s is measurable and for any measurable subset a c 0 l with m a 0 m s a 0 here m denotes the lebesgue measure on 0 11 the operator ps l 0 1 l 0 1 defined by