numerical solution for the variable order linear cable equation with bernstein polynomials abstract in this paper bernstein polynomials method is proposed for the numerical solution of a class of variable order fractional linear cable equation in this paper we adopted bernstein polynomials basis defined on the interval 0 r to solve the equations defined on the section 0 x 0 t the main characteristic behind this approach in this paper is that we derive two kinds of operational matrixes of bernstein polynomials with the operational matrixes the initial equation is transformed into the products of several dependent matrixes which can also be viewed as the system of linear equations after dispersing the variable by solving the linear system of algebraic equations the numerical solutions are acquired only a small number of bernstein polynomials are needed to obtain a satisfactory result numerical examples are provided to show that the method is computationally efficient

fourier series expansion for synchronization of permanent magnet electric motors abstract learning control techniques based on fourier approximation theory are used to solve the tracking control problem via state rotor position and speed and stator currents feedback for permanent magnet synchronous motors performing repetitive tasks smooth rotor position and speed reference signals whose foreknowledge is not assumed and which are periodic of uncertain period are required to be tracked for any motor initial condition the effectiveness of the proposed solution is illustrated by its successfully application to a master slave synchronization problem the presented result illustrates the high potentiality of merging together functional approximation theory and advanced nonlinear control techniques

multicritical dynamics for the j ising model exact dynamical properties are discussed for the j ising model around the multicritical point mcp it is found that the relation of relaxations between the non equilibrium remanent magnetization m t and the equilibrium autocorrelation function q t at the mcp is different from that at the pure critical point the dynamic critical exponent for the ferromagnetic ordering defined by and that for the spin glass ordering defined by become identical at the mcp accurate numerical calculations for them are performed in two and three dimensions using the non equilibrium relaxation analysis the mcp is located at and the exponent is estimated as for the square lattice they are estimated as and for the simple cubic lattice

route to chaos and some properties in the boundary crisis of a generalized logistic mapping abstract a generalization of the logistic map is considered showing two control parameters and that can reproduce different logistic mappings including the traditional second degree logistic map cubic quartic and all other degrees we introduce a parametric perturbation such that the original logistic map control parameter r changes its value periodically according an additional parameter 2 q the value of q gives this period for this system an analytical expression is obtained for the first bifurcation that starts a period doubling cascade and using the feigenbaum universality we found numerically the accumulation point r c where the cascade finishes giving place to chaos in the second part of the paper we study the death of this chaotic behavior due to a boundary crisis at the boundary crisis orbits can reach a maximum value x x max 1 when it occurs the trajectory is mapped to a fixed point at x 0 we show that there exist a general recursive formula for initial conditions that lead to x x max

analysis and optimal control of time varying linear systems via block pulse functions succinct and clear sighted operational properties of block pulse functions are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index piecewise constant solutions equally distributed which are simple in form and convenient for use or implementation are consequently obtained another advantage of this method is that any positive integer can be chosen as the number of sub intervals whereas in the case of walsh function approximation the choice can only be made from 2 4 8 16 32 and so on the algorithms developed in the paper are illustrated by appropriate examples

fractional order legendre operational matrix of fractional integration for solving the riccati equation with fractional order this paper is devoted to both theoretical and numerical study of riccati equation with fractional order a formulation to the fractional order legendre operational matrix of fractional integration is constructed existence and uniqueness results for the considered problem are provided and proved the fractional derivative is described in the caputo sense some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm

numerical solution of multi order fractional differential equations using generalized triangular function operational matrices present article proposes numerical technique for the solution of linear and nonlinear multi order fractional differential equations the proposed method is based on newly computed generalized triangular function operational matrices for riemann liouville fractional order integral theoretical error analysis is performed to estimate the upper bound of absolute error between the exact riemann liouville fractional order integral and its approximation in the triangular functions domain most fractional differential equations do not have closed form solutions development of effective numerical techniques has been an interesting research topic for decades in this context this paper proposes a numerical technique for solving linear and nonlinear multi order fractional differential equations based on newly computed generalized triangular function operational matrices for riemann liouville fractional order integral the orthogonal triangular functions are evolved from a simple dissection of piecewise constant orthogonal block pulse functions theoretical error analysis is performed to estimate the upper bound of absolute error between the exact riemann liouville fractional order integral and its approximation in the triangular functions domain numerical examples are considered for investigating the applicability and effectiveness of proposed technique to solve multi order fractional differential equations the results encourage the use of orthogonal tfs for analysis of real processes exhibiting fractional dynamics

a block pulse operational matrix method for solving two dimensional nonlinear integro differential equations of fractional order abstract in this paper we use two dimensional block pulse functions 2d bpfs and their operational matrix for integration and fractional integration to reduce two dimensional fractional integro differential equations 2d fides to a system of nonlinear algebraic equations the solution is determined by solving this system also by two theorems we prove the convergence of the proposed method finally the numerical solutions guarantee the desired accuracy and efficiency

the sensitivity of a computational l1 approximation technique for integral equations abstract a method of solving integral equations using an l 1 approximation technique has previously been discussed however some questions concerning the sensitivity of such a method were still unanswered in this paper we make investigations towards the important problems of a closeness of parameter and eigenvalue s to one another b noise in the original data and c bifurcations in the solution of the integral equation results are given for these cases illustrating satisfactory behavior of the technique for these so called sensitive problems

laguerre operational matrices for fractional calculus and applications the laguerre operational matrices for the integration and differentiation of a laguerre vector whose elements are laguerre polynomials are generalized to fractional calculus for investigating distributed systems the generalized operational matrices corresponding to 8 1 8 8 82 1 and exp 8 8 1 are derived as examples comparison of the laguerre series approximate inversions of irrational laplace transforms with exact solutions is very satisfactory

on the bifurcation of marotto s map and its application in image encryption abstract the aim of this paper is to address the codimension one bifurcation of marotto s map and its utility in image encryption first of all local stability analysis and local bifurcation analysis of fixed points of the considered map are investigated in details according to the classical bifurcation theory and the center manifold theorem the map exhibits various bifurcation types such as transcritical flip and neimark sacker bifurcations second of all the map is proven to be chaotic in the sense of marotto since image encryption based on chaotic maps is very promising for cryptography marotto s map compound chaos and spatiotemporal chaos are combined to encrypt and decrypt images numerical simulations agree with the analytical framework for the complex dynamics of the map furthermore different test images are used to demonstrate the effectiveness of the method implemented for encryption

solution of nonlinear volterra fredholm hammerstein integral equations via rationalized haar functions abstract rationalized haar functions are developed to approximate of the nonlinear volterra fredholm hammerstein integral equations the properties of rationalized haar functions are first presented and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

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

numerical methods for multi term fractional arbitrary orders differential equations our main concern here is to give a numerical scheme to solve a nonlinear multi term fractional arbitrary orders differential equation

generalized taylor s formula abstract in this paper a new generalized taylor s formula of the kind f x j 0 n a j x a j r n x where a j r x a 0 1 is established such an expression is precisely the classical taylor s formula in case of 1 in addition detailed expressions for r n x and a j that involving caputo fractional derivatives are well constructed some applications including approximation of functions and solutions of fractional differential equations are also given

numerical simulation for coupled systems of nonlinear fractional order integro differential equations via wavelets method abstract in this paper a new method for solving coupled systems of nonlinear fractional order integro differential equations is proposed the idea is to use bernoulli wavelets and operational matrix the main purpose of the technique is to transform the studied systems of fractional order integro differential equations into systems of algebraic equations which can be solved easily illustrative examples and comparisons with haar wavelets and legendre wavelets are included to reveal the effectiveness of the method and the accuracy of the convergence analysis

numerical methods for solving two dimensional nonlinear integral equations of fractional order by using two dimensional block pulse operational matrix in this paper our purpose is to construct a two dimensional fractional integral operational matrix and its use for the numerical solution of two dimensional fractional integral equations we use these operational matrices and properties of two dimensional block pulse functions 2d bpfs to reduce two dimensional fractional integral equations 2d fies to a system of algebraic equations obtained algebraic system based on the original problem can be linear or nonlinear then we show convergence of the proposed methods and we find the error bounds to show the accuracy efficiency and speed of the proposed method linear and nonlinear examples are presented

identification of nonlinear differential equations via fourier series operational matrix for repeated integration abstract a method for the parameter identification of nonlinear differential equations using fourier series is discussed operational matrix of integration as well as one shot operational matrix for repeated integration is used to reduce the identification problem to that of solving systems of algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

parallel tempering cluster algorithm for computer simulations of critical phenomena in finite size scaling analyses of monte carlo simulations of second order phase transitions one often needs an extended temperature range around the critical point by combining the parallel tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest we introduce a flexible and powerful method for systematic investigations of critical phenomena as a result we gain one to two orders of magnitude in the performance for two and three dimensional ising models in comparison with the recently proposed wang landau recursion for cluster algorithms based on the multibondic algorithm which is already a great improvement over the standard multicanonical variant

on the construction of one dimensional discrete chaos theory based on the improved version of marotto s theorem abstract the purpose of this paper is to design one dimensional chaotic maps with a non zero intercept in the y axis and research the basic dynamic characteristics first by studying the common conditions that make one dimensional discrete dynamical systems chaotic under the modulo operation the general theory of the chaotic behavior of one dimensional nonlinear functions with a non zero origin is presented it is proved that the discrete dynamic system derived from the proposed theory exhibits chaos phenomena in the sense of li yorke utilizing the improved version of marotto s theorem furthermore the part of the discrete dynamical functions satisfying the proposed chaos theory is given the function track bifurcation diagram and lyapunov exponential spectrum are also analyzed numerical simulations agree with the analytical framework for the suggested chaos theory finally given the application of chaos in cryptographic communication two significant corollaries of the original functions as general polynomials are put forward according to the theory of chaos structure

fourier series direct method for variational problems a direct method for solving variational problems using fourier series is presented an operational matrix of integration is first introduced and is utilized to reduce a variational problem to the solution of algebraic equations illustrative examples are also given

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

inverting fractional order transfer functions through laguerre approximation this paper concerns the approximate inversion of fractional order transfer functions the approximating solution is of the form of a weighted sum of laguerre functions the weights of the sum are found from a simple generating expression as case study the inversion of the laplace transform of the abel s equation is considered

effective approach for the identification of boundary conditions in distributed parameter systems this paper is concerned with the identification of boundary conditions in parabolic type distributed systems with boundaries of irregular shape in the present approach finite element discretization in the spatial domain and orthogonal functions expansion in the time domain are adopted to reduce the partial differential equation to a set of algebraic equations the boundary conditions are then estimated by the method of least squares using state observations taken at a few interior points the present approach is very straightforward and at least as shown in illustrative examples the results are in excellent agreement with exact results

analytical approximate solutions for nonlinear fractional differential equations we consider a class of nonlinear fractional differential equations fdes based on the caputo fractional derivative and by extending the application of the adomian decomposition method we derive an analytical solution in the form of a series with easily computable terms for linear equations the method gives exact solution and for non linear equations it provides an approximate solution with good accuracy several examples are discussed

numerical solution of nonlinear volterra integral equations of the second kind by using chebyshev polynomials orthogonal chebyshev polynomials are developed to approximate the solutions of linear and nonlinear volterra integral equations properties of these polynomials and some operational matrices are first presented these properties are then used to reduce the integral equations to a system of linear or nonlinear algebraic equations numerical examples illustrate the pertinent features of the method

numerical solutions of optimal control for time delay systems by hybrid of block pulse functions and legendre polynomials using the operational properties of general block pulse functions and legendre polynomials the linear inverse time systems are transformed into a system of algebraic equations the numerical solutions of the systems are derived moreover applying the results to the linear quadratic optimal control problems the approximate solutions of optimal control of time delay systems are derived

a generalized fractional order legendre wavelet tau method for solving fractional differential equations abstract a new fractional order wavelet basis as generalization of the classical legendre wavelet is defined the operational matrices both for derivative and fractional derivative in the sense of caputo for this fractional order wavelet are derived then a numerical scheme based on these operational matrices and the typical tau method is proposed for solving some nonlinear fractional differential equations illustrative examples show that the present wavelet tau method is numerically effective and convenient for solving fractional differential equations moreover the obtained results confirm that in comparison with the classical legendre wavelet method the fractional order wavelet basis is more efficient and accurate for solving fractional differential equations error analysis and convergence of the proposed wavelet method are also provided

solutions of convolution integral and fredholm integral equations via double fourier series double fourier series are developed to approximate the solutions of the convolution integral and fredholm integral equations properties of fourier series are first briefly presented and the operational matrix of integration together with the product operational matrix is utilized to reduce the computation of integral equations to a set of simultaneous linear algebraic equations the method is computationally very attractive and applications are demonstrated through illustrative examples

direct walsh wavelet packet method for variational problems a direct ritz method for solving variational problems is demonstrated walsh wavelet packet bases are used as the basis functions it is shown how the form of the operational matrix of integration affects the accuracy of the obtained approximate solutions one or two approximation fields are formulated and the method of lagrange multipliers is used to solve illustrative problems with free and constrained ends

direct approach for the optimal control of linear time delay systems via shifted legendre polynomials the present paper proposes a direct approach to the optimal control of linear time delay systems in the first step both control and state are approximated by shifted legendre polynomials and then the legendre coefficients vector of states can be expressed in terms of the legend re coefficients vector of controls in the second step the relation between legendre coefficients vectors of states and controls which satisfies the state equations approximately is introduced into the performance index next the variational principle is applied to find the legendre coefficients vector of near optimal control that minimizes the performance index directly in the present approach the legendre coefficients vector of sub optimal control can be obtained directly from a set of algebraic equations thus both difficulties in solving simultaneous partial differential equations of large dimension or in solving n sets of ordinary differential equations are avoided

an application of rationalized haar functions for variational problems a direct method for solving variational problems using rationalized haar is presented an operational matrix of integration and the cross product of two rationalized haar function vectors are utilized to reduce a variational problem to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

analysis and optimal control of time varying systems via chebyshev polynomials abstract the operational properties of the forward and backward integration and the product of chebyshev polynomials are derived these properties are applied to the analysis and optimal control of time varying linear systems by the approximation of time functions by truncated chebyshev series due to the unique property of the chebyshev polynomials excellent results are obtained as demonstrated by examples

numerical solution of integral equations system of the second kind by block pulse functions abstract this paper endeavors to formulate block pulse functions to propose solutions for the fredholm integral equations system to begin with we describe the characteristic of block pulse functions and will go on to indicate that through this method a system of fredholm integral equations can be reduced to an algebraic equation numerical examples presented to illustrate the accuracy of the method

comparative study of selected parallel tempering methods we review several parallel tempering schemes and examine their main ingredients for accuracy and efficiency the present study covers two selection methods of temperatures and several choices for the exchange of replicas including a recent novel all pair exchange method we compare the resulting schemes and measure specific heat errors and efficiency using the two dimensional 2d ising model our tests suggest that an earlier proposal for using numbers of local moves related to the canonical correlation times is one of the key ingredients for increasing efficiency and protocols using cluster algorithms are found to be very effective some of the protocols are also tested for efficiency and ground state production in 3d spin glass models where we find that a simple nearest neighbor approach using a local n fold way algorithm is the most effective finally we present evidence that the asymptotic limits of the ground state energy for the isotropic case and for an anisotropic case of the 3d spin glass model are very close and may even coincide

analytical solution of a fractional diffusion equation by variational iteration method in the present paper the analytical approximate solution of a fractional diffusion equation is deduced with the help of powerful variational iteration method by using an initial value the explicit solutions of the equation for different cases have been derived which accelerate the rapid convergence of the series solution the present method performs extremely well in terms of efficiency and simplicity numerical results for different particular cases of the problem are presented graphically

numerical solution of the mixed volterra fredholm integro differential multi term equations of fractional order abstract this paper aims to introduce two effective numerical approaches for solving the mixed volterra fredholm integro differential equation of fractional order and multi order with initial values despite the fact that the second approach transfers the integro differential equations into a system of algebraic equations through the usage of operational matrices these problems can be transferred to a system of algebraic equations by expanding the solution s highest order derivative through the block pulse functions bpfs this is done by using the generalized operational matrices of bpfs for differentiation along with the fractional calculus properties in the first scheme in which convergence of the solution obtained has been shown in the following the accuracy and applicability of two proposed approaches will be compared by some relevant numerical examples for which the exact solution is known

double shifted chebyshev series for convolution integral and integral equations double shifted chebyshev polynomials are developed in this study to approximate the solutions of the convolution integral volterra integral equation and fredholm integral equation this method simplifies the computations of integral equations to the successive solutions of a linear algebraic equation in matrix form in addition the computational complexity can be reduced remarkably three examples are illustrated it is seen that the proposed approach is straightforward and convenient and converges faster in finding approximations than other existing orthogonal function methods

spatiotemporal chaos in the one dimensional complex ginzburg landau equation the dynamical behavior of a large one dimensional system obeying the cubic complex ginzburg landau equation is studied numerically as a function of parameters near a supercritical bifurcation two types of chaotic behavior can be distinguished beyond the benjamin feir instability a phase turbulence regime with a conserved phase winding number and no phase dislocations space time defects and a defect regime with a nonzero density of defects the transition between the two can either be continuous or discontinuous hysteretic depending on parameters the spatial decay of the phase correlation function is inferred to be exponential in both regimes with a sharp decrease of the correlation length upon entering the defect phase the temporal decay of correlations is exponential in the defect regime

legendre series approach to identification and analysis of linear systems in this paper the legendre operational matrix of integration is introduced and it is subsequently used for the identification and analysis of linear systems the algorithms proposed are easily implementable in a digital computer and they are analogous to those already derived for other types of orthogonal functions

analysis and optimal control of time varying linear systems via shifted legendre polynomials this paper extends the application of shifted legendre polynomial expansion to time varying systems the extension is achieved through representing the product of two shifted legendre series in a new shifted legendre series with this treatment of the product of two time functions the operational properties of the shifted legendre polynomials are fully applied to the analysis and optimal control of time varying linear systems with quadratic performance index

numerical solution of fractional differential equations using the generalized block pulse operational matrix the riemann liouville fractional integral for repeated fractional integration is expanded in block pulse functions to yield the block pulse operational matrices for the fractional order integration also the generalized block pulse operational matrices of differentiation are derived based on the above results we propose a way to solve the fractional differential equations the method is computationally attractive and applications are demonstrated through illustrative examples

analysis of linear time varying systems and bilinear systems via fourier series a method for finding an approximate solution of linear time varying systems and bilinear systems via fourier series is proposed properties of fourier series are first briefly presented and the operational matrix for integration together with the product operational matrix are utilized to reduce the dynamical equation of a linear time varying system or bilinear system to a set of simultaneous linear algebraic equations illustrative examples are also given

direct method to solve volterra integral equation of the first kind using operational matrix with block pulse functions this article proposes a simple efficient direct method for solving volterra integral equation of the first kind by using block pulse functions and their operational matrix of integration first kind integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution some examples are presented to illustrate efficiency and accuracy of the proposed method

application of operational matrices for solving system of linear stratonovich volterra integral equation abstract in this paper we give a brief review on block pulse functions bpfs and obtain operational matrix and stochastic operational matrix of integration based on bpfs then these operational matrices are used to solve system of linear stratonovich volterra integral equations by applying proposed method the system of linear stratonovich volterra integral equations reduce to system of linear algebraic equations which can be solved by a convenient numerical method also the error analysis is proved under several mild conditions we show the rate of convergence is o h finally by applying this method on two examples we demonstrate accuracy and efficiency of the proposed method all of the numerical calculation are performed on computer using a program written in matlab