strong limit point criteria for a class of singular discrete linear hamiltonian systems this paper is concerned with the limit point case for a class of singular discrete linear hamiltonian systems the limit point case is divided into the strong and the weak limit point cases several sufficient conditions for the strong limit point case are established in consequence two criteria of the strong limit point case for second order formally self adjoint vector difference equations are obtained

analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method the fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials motion of a large thin plate in a newtonian fluid creep and relaxation functions for viscoelastic materials the pi ld m controller for the control of dynamical systems etc phenomena in electromagnetics acoustics viscoelasticity electrochemistry and material science are also described by differential equations of fractional order the solution of the differential equation containing fractional derivative is much involved instead of application of the existing methods an attempt has been made in the present analysis to obtain the solution of nonlinear dynamic system containing fractional derivative ji zeng wang et al coiflets based method in the solution of nonlinear dynamic system containing fractional derivative in the fourth international conference on nonlinear mechanics icnm iv shanghai august 2002 pp 1304 1308 by the relatively new adomian decomposition method the results obtained by this method are then graphically represented and then compared with the exact solution a good agreement of the results is observed

titchmarsh sims weyl theory for complex hamiltonian systems on sturmian time scales abstract we study non self adjoint hamiltonian systems on sturmian time scales defining weyl sims sets which replace the classical weyl circles and a matrix valued m function on suitable cone shaped domains in the complex plane furthermore we characterize realizations of the corresponding dynamic operator and its adjoint and construct their resolvents even order scalar equations and the orr sommerfeld equation on time scales are given as examples illustrating the theory which are new even for difference equations these results unify previous discrete and continuous theories to dynamic equations on sturmian time scales

eigenvalues of second order difference equations with periodic and antiperiodic boundary conditions this paper is concerned with periodic and antiperiodic boundary value problems for self adjoint second order difference equations existence of eigenvalues of these two different boundary value problems is proved numbers of their eigenvalues are calculated and their relationships are obtained in addition a representation of solutions of a nonhomogeneous linear equation with initial conditions is given

continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem abstract this paper is concerned with continuous dependence of the n th eigenvalue of self adjoint discrete sturm liouville problems on the problem the n th eigenvalue is considered as a function on the space of the problems called the n th eigenvalue function a necessary and sufficient condition for all the n th eigenvalue functions to be continuous and several properties of the n th eigenvalue function on a subset of the space of the problems are given they play an important role in the study of continuous dependence of the n th eigenvalue function on the problem continuous dependence of the n th eigenvalue function on the equation and on the boundary condition is studied separately consequently the continuity and discontinuity of the n th eigenvalue function are completely characterized on the whole space of the problems especially asymptotic behavior of the n th eigenvalue function near each discontinuity point is given

an approximate solution based on jacobi polynomials for time fractional convectiondiffusion equation operational matrix of shifted jacobi polynomials is considered solution of time fractional order convectiondiffusion problem is numerically estimated main problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded a system of nonlinear algebraic equations is achieved by approximating the known and unknown functions with the help of shifted jacobi functions problem can be used extensively in science and engineering as in oil reservoir simulations in this article we present a numerical method to numerically solve a time fractional convectiondiffusion equation our method is based on the operational matrices of shifted jacobi polynomials at first problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded then we approximate the known and unknown functions with the help of shifted jacobi functions a system of nonlinear algebraic equations is obtained finally the unknown coefficients are determined by mathematicatm we implemented the proposed method for several examples that they indicate the high accuracy method it should be noted that this method is generalizable to some appropriate problems

sturmian and spectral theory for discrete symplectic systems we consider symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems we establish sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems then using this comparison result we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most n in the last part of the paper we prove the rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences

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

decomposition method for solving fractional riccati differential equations in this article we implement a relatively new analytical technique the adomian decomposition method for solving fractional riccati differential equations the fractional derivatives are described in the caputo sense in this scheme the solution takes the form of a convergent series with easily computable components the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of the solution the corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations some numerical examples are presented to illustrate the efficiency and reliability of the method

comments on five six and seven term karatsuba like formulae for original paper see p l montgomery ibid vol 54 no 3 p 362 369 2005 we show that multiplication complexities of n term karatsuba like formulae of gf 2 x 7 n 19 presented in the above paper can be further improved using the chinese remainder theorem and the construction multiplication modulo x infin w

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

the non standard finite difference scheme for linear fractional pdes in fluid mechanics a non standard finite difference scheme is developed to solve the linear partial differential equations with time and space fractional derivatives the grunwald letnikov method is used to approximate the fractional derivatives numerical illustrations that include the linear inhomogeneous time fractional equation linear space fractional telegraph equation linear inhomogeneous fractional burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique numerical results are presented graphically and reveal that the non standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order

spectral theory of discrete linear hamiltonian systems this paper is concerned with spectral problems for a class of discrete linear hamiltonian systems with self adjoint boundary conditions where the existence and uniqueness of solutions of initial value problems may not hold a suitable admissible function space and a difference operator are constructed so that the operator is self adjoint in the space then a series of spectral results are obtained the reality of eigenvalues the completeness of the orthogonal normalized eigenfunction system rayleigh s principle the minimax theorem and the dual orthogonality especially the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated

an stable numerical algorithm for identifying the solution of an inverse problem in this paper we consider an inverse parabolic problem with space dependent coefficient mathematical model of the problem consists a parabolic equation in which the condition is unknown at the one of the boundary and to be determined from an overspecified data measured at an interior point inside the body uniqueness of the solution of under study inverse problem will be shown our concern for the numerical procedure for this inverse problem is based on a finite differences scheme stability conditions for numerical solution to inverse problem are stated the approach of proposed method is approximated the unknown function by a set of chebyshev polynomials where the unknown set of expansion coefficients in unknown function are determined from the minimizing the least squares method some numerical examples will be given in the last section

multiple positive solutions for nonlinear dynamical systems on a measure chain in this paper we consider the following dynamical system on a measure chain u 1 t f1 t u1 t u2 t 0 t a b u 2 t f2 t u1 t u2 t 0 t a b with the sturm liouville boundary value conditions ui a u i a 0 ui b ui b 0 for i 1 2 some results are obtained for the existence of three positive solutions of the above problem by using leggett williams fixed point theorem

self adjoint extensions for second order symmetric linear difference equations abstract in this paper self adjoint extensions for second order symmetric linear difference equations with real coefficients are studied by applying the glazman krein naimark theory for hermitian subspaces both self adjoint subspace extensions and self adjoint operator extensions of the corresponding minimal subspaces are completely characterized in terms of boundary conditions where the two endpoints may be regular or singular

finite volume method for solving a one dimensional parabolic inverse problem in this paper finite volume method is used to solve a one dimensional parabolic inverse problem with source term and neumann boundary conditions for the first time some advantages of this approach are developing difference schemes and maintaining certain properties of the physics of the problems especially for the treatment of the source term and the unknown boundary conditions numerical results show that our method is more effective

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

spectral theory of second order vector difference equations abstract this paper is concerned with spectral problems of second order vector difference equations with two point boundary value conditions where the matrix valued coefficient of the leading term may be singular a concept of self adjointness of the boundary value conditions is introduced the self adjointness of the corresponding difference operator is discussed on a suitable admissible function space and fundamental spectral results are obtained the dual orthogonality of eigenfunctions is shown in a special case rayleigh s principles and the minimax theorems in two linear spaces are given as an application a comparison theorem for eigenvalues of two sturm liouville problems is presented

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

arithmetic operations in finite fields of medium prime characteristic using the lagrange representation in this paper we propose a complete set of algorithms for the arithmetic operations in finite fields of prime medium characteristic the elements of the fields ifpk are represented using the newly defined lagrange representation where polynomials are expressed using their values at sufficiently many points our multiplication algorithm which uses a montgomery approach can be implemented in o k multiplications and o k2 log k additions in the base field ifp for the inversion we propose a variant of the extended euclidean gcd algorithm where the inputs are given in the lagrange representation the lagrange representation scheme and the arithmetic algorithms presented in the present work represent an interesting alternative for elliptic curve cryptography

a discrete analogue of lyapunov type inequalities for nonlinear difference systems in this paper we establish several new lyapunov type inequalities for the nonlinear difference system dx n email protected n x n 1 email protected n y n m 2y n dy n email protected n x n 1 n 2x n 1 email protected n y n when the end points are not necessarily usual zeros but rather generalized zeros our results improve almost all related existing ones

numerical methods for fourth order fractional integro differential equations this paper outlines a reliable strategy for solving linear and nonlinear boundary value problems for fourth order fractional integro differential equations the fractional derivative is considered in the caputo sense the adomian decomposition method is found to be fast and accurate numerical examples are presented to illustrate the efficiency and reliability of the method

application of sinc collocation method for solving an inverse problem in this article the identification of an unknown time dependent source term in an inverse problem of parabolic type with nonlocal boundary conditions is considered the main approach is to change the inverse problem to a system of volterra integral equations the resulting integral equations are convolution type which by using sinc collocation method are replaced by a system of linear algebraic equations the convergence analysis is included and it is shown that the error in the approximate solution is bounded in the infinity norm by the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system to show the efficiency of the present method an example is presented the method is easy to implement and yields very accurate results

application of the operational matrix of fractional order legendre functions for solving the time fractional convection diffusion equation the operational matrix of fractional order legendre functions method are considered the fractional order convection diffusion problem is solved the problem can be used extensively in science and engineering as in oil reservoir simulations problems which an initially discontinuous profile is propagated by diffusion and convection in this paper the application of the operational matrix of fractional order legendre functions flfs to solve the time fractional convection diffusion equation has been investigated fractional calculus has been applied to model the engineering and physical processes which are best described with other mathematical tools the time variable of the time fractional convection diffusion equation and its space variable have been approximated by flfs and shifted legendre polynomials respectively the fractional derivatives together with product matrices of flfs are employed to convert the solution of this problem to the solution of a system of algebraic equations

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

subquadratic space complexity binary field multiplier using double polynomial representation this paper deals with binary field multiplication we use the bivariate representation of binary field called double polynomial system dps presented in this concept generalizes the composite field representation to every finite field as shown in the main interest of dps representation is that it enables to use lagrange approach for multiplication and in the best case fast fourier transform approach which optimizes lagrange approach we use here a different strategy from to perform reduction and we also propose in this paper some new approaches for constructing dps we focus on dps which provides a simpler and more efficient method for coefficient reduction this enables us to avoid a multiplication required in the montgomery reduction approach of and thus to improve the complexity of the dps multiplier the resulting algorithm proposed in the present paper is subquadratic in space o n1 31 and logarithmic in time the space complexity is 33 percent better than in and 18 percent faster it is asymptotically more efficient than the best known method specifiably more efficient than when n 3 000 furthermore our proposal is available for every n and not only for n a power of two or three

discrete linear hamiltonian eigenvalue problems abstract this paper introduces general discrete linear hamiltonian eigenvalue problems and characterizes the eigenvalues assumptions are given among them the new notion of strict controllability of a discrete system that imply isolatedness and lower boundedness of the eigenvalues due to the quite general assumptions discrete sturm liouville eigenvalue problems of higher order are included in the presented theory

analytical approximate solutions of systems of fractional algebraic differential equations by homotopy analysis method abstract in this paper we develop a framework to obtain approximate solutions to systems of algebraic differential equations of fractional order by employing the homotopy analysis method ham the study highlights the efficiency of the method and its dependence on the auxiliary parameter numerical examples are examined to highlight the significant features of the ham method the method shows improvements over existing analytical techniques

an inverse heat conduction problem with a nonlinear source term abstract in this paper the nonlinear problem of an inhomogeneous heat equation with linear boundary conditions will be considered the surface heat flux history of a heated conducting body will be identified the approach of the proposed method is to approximate the unknown function using linear polynomial pieces which are determined consecutively from the solution of the minimization problem on the basis of overspecified data some numerical examples will also be presented

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

a new high order adi numerical difference formula for time fractional convection diffusion equation abstract based on exponential transformation quadratic interpolation polynomial and pade approximation a new high order finite difference scheme is proposed for solving the two dimensional 2d time fractional convection dominated diffusion equation of order 0 1 the resulting scheme is of 3 order accuracy in time and fourth order accuracy in space in order to reduce the amount of computation the alternating direction implicit adi scheme is further developed numerical experiments are given to demonstrate the high accuracy and robustness of our new scheme

multiple positive solutions of bvps for third order discrete difference systems this paper is concerned with the following email protected 3u 1 k f 1 k u 1 k u 2 k 0 email protected 0 t d 3u 2 k f 2 k u 1 k u 2 k 0 email protected 0 t with the dirichlet boundary conditionu 1 0 u 1 1 u 1 t 3 0 u 2 0 u 2 1 u 2 t 3 by using leggett williams fixed point theorem sufficient conditions are obtained for the existence of three positive solutions to the above system

d type anticipatory iterative learning control for a class of inhomogeneous heat equations in this paper a d type anticipatory iterative learning control ilc scheme is applied to the boundary control of a class of inhomogeneous heat equations where the heat flux at one side is the control input while the temperature measurement at the other side is the control output by transforming the inhomogeneous heat equation into its integral form and exploiting the properties of the embedded jacobi theta functions the learning convergence of ilc is guaranteed through rigorous analysis without any simplification or discretization of the 3d dynamics in the time space as well as iteration domains the adopted ilc scheme makes full use of the process repetition and deals with state independent or state dependent uncertainties meanwhile due to the feedforward characteristic of ilc the proposed scheme not only makes anticipatory compensation possible to overcome the heat conduction delay in boundary output tracking but also eliminates the gain margin limitation encountered in feedback control in the end an illustrative example is presented to demonstrate the performance of the proposed ilc scheme

oscillation theorems for discrete symplectic systems with nonlinear dependence in spectral parameter abstract in this paper we open a new direction in the study of discrete symplectic systems and sturm liouville difference equations by introducing nonlinear dependence in the spectral parameter we develop the notions of finite eigenvalues and finite eigenfunctions and their multiplicities and prove the corresponding oscillation theorem for dirichlet boundary conditions the present theory generalizes several known results for discrete symplectic systems which depend linearly on the spectral parameter our results are new even for special discrete symplectic systems namely for sturm liouville difference equations symmetric three term recurrence equations and linear hamiltonian difference systems

existence of positive solutions of bvps for second order nonlinear difference systems this paper is concerned with the following system d 2u i k f i k u 1 k u 2 k u n k 0 k 0 t i 1 2 nwith the dirichlet boundary conditionu i 0 u i t 2 0 i 1 2 n some new results are obtained for the existence and multiplicity of positive solutions to the above system by using nonlinear alternative of leray schauder type krasnosel skii s fixed point theorem in a cone and leggett williams fixed point theorem in particular it proves that the above system has n positive solutions under suitable conditions where n is an arbitrary integer

residual minimization least squares method for inverse heat conduction abstract a numerical method is systematically developed for resolving an inverse heat conduction problem in the presence of noisy discrete data this paper illustrates the effect of imposing constraints on the unknown function of interest a volterra integral equation of the first kind is derived and used as the starting point for residual minimization least squares methodology symbolic manipulation is exploited for purposes of augmenting the computational methodology preliminary indications suggest that the imposition of physical constraints on the system drastically reduces the level of mathematical sophistication needed for accurately approximating the unknown function of interest these constraints are actually available in many design studies or from models which are derived by physical processes

application jacobi spectral method for solving the time fractional differential equation abstract our aim from this article is to propose a spectral method for numerically solving a general form of time fractional differential equation with boundary conditions our method is based on the shifted jacobi polynomials and has four steps as follows at first the problem is converted to a new problem with homogeneous conditions step 1 then an integro differential equation is obtained which is equivalent with the new problem step 2 next all the known and unknown functions which are existing in the integro differential equation are approximated via the shifted jacobi polynomials step 3 utilizing operational matrices of shifted jacobi polynomials and collocation method a system of nonlinear algebraic equations is achieved which is solved by newton s iterative method step 4 in order to show and verify the accuracy and performance of the proposed method we examine our method on some illustrative examples

solving a multi order fractional differential equation using adomian decomposition abstract an algorithm has been developed to convert the multi order fractional differential equation d y t f t y t d 1 y t d n y t y k 0 c k k 0 m where m d denotes caputo fractional derivative of order into a system of fractional differential equations further adomian decomposition method has been employed to solve the system of fractional differential equations some illustrative examples are presented

fractional order legendre laguerre functions and their applications in fractional partial differential equations abstract in this paper we consider a new fractional function based on legendre and laguerre polynomials for solving a class of linear and nonlinear time space fractional partial differential equations with variable coefficients the concept of the fractional derivative is utilized in the caputo sense the idea of solving these problems is based on operational and pseudo operational matrices of integer and fractional order integration with collocation method we convert the problem to a system of algebraic equations by applying the operational matrices pseudo operational matrices and collocation method also we calculate the upper bound for the error of integral operational matrix of the fractional order we illustrated the efficiency and the applicability of the approach by considering several numerical examples in the format of table and graph we also describe the physical application of some examples

spectral theory of left definite difference operators this paper is concerned with the spectral theory for the second order left definite difference boundary value problems existence of eigenvalues of boundary value problems is proved numbers of their eigenvalues are calculated and fundamental spectral results are obtained

numerical solutions for fractional kdv burgers equation by adomian decomposition method abstract based on the symbolic computation system maple adomian decomposition method developed for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional kdv burgers equation the fractional derivatives are described in the caputo sense according to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional kdv burgers equation with time and space fractional derivatives finally the solutions of our model equation are calculated in the form of convergent series with easily computable components

eigenvalues of second order difference equations with coupled boundary conditions abstract this paper is concerned with coupled boundary value problems for self adjoint second order difference equations existence of eigenvalues is proved numbers of eigenvalues are calculated and relationships between the eigenvalues of a self adjoint second order difference equation with three different coupled boundary conditions are established these results extend the relevant existing results of periodic and antiperiodic boundary value problems

lyapunov inequalities and stability for discrete linear hamiltonian systems abstract in this paper we establish several new lyapunov type inequalities for discrete linear hamiltonian systems when the end points are not necessarily usual zeros but rather generalized zeros which generalize and improve almost all related existing ones applying these inequalities an optimal stability criterion is obtained

the glazman krein naimark theory for a class of discrete hamiltonian systems abstract in this paper the glazman krein naimark theory for a class of discrete hamiltonian systems is developed a minimal and a maximal operators gkn sets and a boundary space for the system are introduced algebraic characterizations of the domains of self adjoint extensions of the minimal operator are given a close relationship between the domains of self adjoint extensions and the gkn sets is established it is shown that there exist one to one correspondences among the set of all the self adjoint extensions the set of all the d dimensional lagrangian subspaces of the boundary space and the set of all the complete lagrangian subspaces of the boundary space

weyl titchmarsh theory for a class of discrete linear hamiltonian systems abstract this paper is concerned with establishing the weyl titchmarsh theory for a class of discrete linear hamiltonian systems over a half line fundamental properties of solutions regular spectral problems and the corresponding maximal and minimal operators are first studied matrix disks are constructed and proved to be nested and converge to a limiting set some precise relationships among the rank of the matrix radius of the limiting set the number of linearly independent square summable solutions and the defect indices of the minimal operator are established based on the above results a classification of singular discrete linear hamiltonian systems is given in terms of the defect indices of the minimal operator and several equivalent conditions on the cases of limit point and limit circle are obtained respectively especially several problems in the limit point case are more carefully investigated including fundamental properties of square summable solutions properties of the weyl function which is the unique element in the limiting set in this case and inhomogeneous boundary problems self adjointness of the corresponding hamiltonian operator relationship between the spectrum of the hamiltonian operator and the analyticity of the weyl function as well as the dependence of the spectrum on the boundary data in which some interesting separation results for the spectrum are obtained finally another set of four equivalent conditions on the limit point case are established

limit point and limit circle criteria for singular second order linear difference equations with complex coefficients this paper is concerned with limit point and limit circle criteria of singular second order linear difference equations with complex coefficients formally self adjoint second order linear difference operators are formulated several sufficient conditions and sufficient and necessary conditions for the limit point and limit circle cases are established these results extend some relevant existing results

homotopy analysis method for higher order fractional integro differential equations in this paper we present the homotopy analysis method shortly ham for obtaining the numerical solutions of higher order fractional integro differential equations with boundary conditions the series solution is developed and the recurrence relations are given explicitly the initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary conditions the comparison of the results obtained by the ham with the exact solutions is made the results reveal that the ham is very effective and simple the ham contains the auxiliary parameter h which provides us with a simple way to adjust and control the convergence region of series solution

application of the restrained optimal perturbation method to study the backward heat conduction problem in this paper a restrained optimal perturbation method is firstly proposed to solve the backward heat conduction problem the initial temperature distribution will be identified from the overspecified data a regularization term is introduced in the objective functional for overcoming the ill posedness of this problem spectral projected gradient algorithm is used to solve the optimal problem and we give the sensitivity analysis of the initial value the results of numerical experiments are also presented

a jacobi gauss lobatto collocation method for solving generalized fitzhugh nagumo equation with time dependent coefficients in this paper we propose a new jacobi gauss lobatto collocation method for solving the generalized fitzhugh nagumo equation the jacobi gauss lobatto points are used as collocation nodes for spatial derivatives the proposed method has the advantage of obtaining the solution in terms of the jacobi parameters a and b in addition the problem is reduced to a system of ordinary differential equations in time this system can be solved by any standard numerical techniques numerical solutions obtained by this method when compared with the exact solution reveal that the obtained solution produces high accurate results numerical results show that the proposed method is of high accuracy and is efficient for solving the generalized fitzhugh nagumo equation also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

error estimate of eigenvalues of perturbed second order discrete sturm liouville problems abstract this paper is concerned with eigenvalues of perturbed second order vector discrete sturm liouville problems by some variational properties of eigenvalues of discrete sturm liouville problems error estimates of eigenvalues of perturbed problems sufficiently close to a given sturm liouville problem are given under a certain non singularity condition perturbations of the coefficient functions of the difference equation the weight function and the coefficients of the boundary condition are all considered this together with higher dimension involved results in a certain complexity of the problem and difficulty of study as a direct consequence continuous dependence of eigenvalues on problems is obtained under the non singularity condition in addition an example is presented to illustrate the necessity of the non singularity condition

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

jacobi collocation method for the approximate solution of some fractional order riccati differential equations with variable coefficients abstract this paper presents a computational method for the approximate solution of arbitrary order non linear fractional riccati differential equations with variable coefficients proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the jacobi polynomials convergence analysis of the proposed method is provided numerical results for different fractional orders of the riccati differential equations are discussed figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of jacobi polynomials such as the legendre polynomials the chebyshev polynomials of the second kind the chebyshev polynomials of the third kind the chebyshev polynomial of the fourth kind and the gegenbauer or ultraspherical polynomials numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods it is observed that the results from the proposed method are more accurate maximum absolute error and the root mean square error tables are given for all five kinds of polynomials for comparison purposes

an efficient jacobi pseudospectral approximation for nonlinear complex generalized zakharov system in this paper we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized zakharov system gzs subject to initial boundary conditions the jacobi pseudospectral approximation is investigated for spatial approximation of the gzs it possesses the spectral accuracy in space the jacobi gauss lobatto quadrature rule is established to treat the boundary conditions and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable this scheme has the advantage of allowing us to obtain the spectral solution in terms of the jacobi parameters and which therefore means that the algorithm holds a number of collocation methods as special cases finally two illustrative examples are implemented to assess the efficiency and high accuracy of the jacobi pseudo spectral scheme

numerical solution of inverse heat conduction problem with nonstationary measurements the inverse heat conduction problem involves the calculation of surface heat flux and or temperature histories from transient measured temperatures inside solid in this paper the temperature is not specified at x 0 in an inverse heat conduction problem we choose an additional moving boundary condition this problem may be subdivided into two separate problems one of these problems is a moving boundary problem which is solved exactly by means of fundamental solution of heat equation the second problem is an inverse moving boundary problem which is solved with the residual minimization least squares method by using numerical examples we show the accuracy of our method