numerical solutions for systems of fractional differential equations by the decomposition method in this paper we use adomian decomposition method to solve systems of nonlinear fractional differential equations and a linear multi term fractional differential equation by reducing it to a system of fractional equations each of order at most unity we begin by showing how the decomposition method applies to a class of nonlinear fractional differential equations and give two examples to illustrate the efficiency of the method moreover we show how the method can be applied to a general linear multi term equation and solve several applied problems

polynomial nonlinearities in differential equations abstract adomian and his collaborators have found solutions for nonlinear stochastic or deterministic in the limiting case differential equations involving polynomial non linearities using the first author s a n polynomials and recently developed methods of calculating these polynomials it becomes very easy to write solutions for non linear terms such as y m for positive integers m the cases of negative integer m or even decimal powers will appear elsewhere

on the tricomi problem the tricomi equation is solved by the decomposition method 6 8 on the half space x y 0 and also for suitable boundary conditions given on the boundary of the hyperbolic elliptic domain

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

on the control of stochastic systems abstract a general model is available for analysis of control systems involving stochastic time varying parameters in the system to be controlled by the use of the iterative method of the authors or its more recent adaptations for stochastic operator equations it is shown that the statistical separability which is achieved as a result of the method for stochastic operator equations is unaffected by the matrix multiplications in state space equations the method therefore is applicable to the control problem application is made to the state space equation x ax bu c where a b c are stochastic matrices corresponding to stochastic operators i e involving randomly time varying elements e g a ij t a t t f a p s it is emphasized that the processes are arbitrary stochatic processes with known statistics no assumption is made of wiener or markov behavior or of smallness of fluctuations and no closure approximations are necessary the method differs in interesting aspects from volterra series expansions used by wiener and others and has advantages over the other methods because of recent progress in the solution of the nonlinear case it may be possible to generalize the results above to the nonlinear case as well but the linear case is suffcient to show the connections and essential point of separability of ensemble averages

λ difference sequence spaces of fuzzy numbers in this paper using the difference operator of order m and an orlicz function we introduce and examine some classes of sequences of fuzzy numbers we give the relations between the strongly cesaro type convergence and statistical convergence in these spaces furthermore we study some of their properties like completeness solidity symmetricity etc we also give some inclusion relations related to these classes

computing eigenelements of boundary value problems with fractional derivatives variational iteration method has been successfully implemented to handle linear and nonlinear differential equations the main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper first a general framework of the variational iteration method is presented for analytic treatment of differential equations of fractional order where the fractional derivatives are described in caputo sense second the new framework is used to compute approximate eigenvalues and the corresponding eigenfunctions for boundary value problems with fractional derivatives numerical examples are tested to show the pertinent features of this method this approach provides a new way to investigate eigenvalue problems with fractional order derivatives

analytical solution of the linear fractional differential equation by adomian decomposition method in this paper we consider the n term linear fractional order differential equation with constant coefficients and obtain the solution of this kind of fractional differential equations by adomian decomposition method with the equivalent transmutation we show that the solution by adomian decomposition method is the same as the solution by the green s function finally we illustrate our result with some examples

a numerical scheme for the solution of multi order fractional differential equations we present an algorithm for the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 0 combined with suitable initial conditions the fractional derivatives are described in the caputo sense the algorithm is based on adomian s decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components several numerical examples are presented

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

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

nonlinear plasma response abstract the decomposition method g adomian stochastic systems academic press new york london 1983 g adomian stochastic systems ii in press and earlier work of adomian referenced therein allows solution without linearization of problems normally requiring the self consistent field technique the latter technique can provide a sufficiently accurate approximation in weakly coupled plasma but the decomposition method of adomian provides a superior and physically preferable methodology for the general case

variable order fractional numerical differentiation for noisy signals by wavelet denoising in this paper a numerical method is proposed to estimate the variable order fractional derivatives of an unknown signal in noisy environment firstly the wavelet denoising process is adopted to reduce the noise effect for the signal secondly polynomials are constructed to fit the denoised signal in a set of overlapped subintervals of a considered interval thirdly the variable order fractional derivatives of these fitting polynomials are used as the estimations of the unknown ones where the values obtained near the boundaries of each subinterval are ignored in the overlapped parts finally numerical examples are presented to demonstrate the efficiency and robustness of the proposed method an efficient method is proposed to numerically estimate the variable order fractional derivatives of a noisy signal wavelet denoising is adopted to reduce the noise effect in a signal polynomials are constructed to fit the denoised signal in a set of overlapped subintervals of the considered interval the variable order fractional derivatives of a noisy signal are numerically obtained based on fitting polynomials numerical examples of four cases are considered to demonstrate the efficiency and robustness of the proposed method

on legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations the variational iteration method and the homotopy analysis method as alternative methods have been widely used to handle linear and nonlinear models the main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently this paper deals with the numerical solutions of nonlinear fractional differential equations where the fractional derivatives are considered in caputo sense the main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations in these algorithms legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work the proposed algorithms are capable of reducing the size of calculations improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals numerical examples are examined to show the efficiency of the algorithms

nonlinear differential equations with negative power nonlinearities abstract differential equations involving the term y m where m is a positive integer are solved by the decomposition method 1 3 441 452

variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains abstract in this paper a variational formulation of the equivalent eigenstrain method is established a functional of the hashin shtrikman type is proposed such that the solution of the equivalent eigenstrain equation is a unique minimizer of the functional moreover it is also shown that the equivalent eigenstrain equation is the euler lagrange equation of the potential energy of the inclusions an approximate solution of the equivalent eigenstrain equation is then found as a minimizer of the functional on a finite dimensional span of basic eigenstrains special attention is paid to possible symmetries of the problem the variational formulation is illustrated by determination of effective linear elastic properties in particular material with a simple cubic microstructure is considered in detail a solution for the polynomial radial basic eigenstrains approximation is found in particular for the homogeneous eigenstrain approximation the effective moduli are derived in an exact closed form

a reliable technique for solving the wave equation in an infinite one dimensional medium in this paper the initial value problem of the one dimensional wave equation where the domain of the space variable is unbounded will be handled by using the reliable decomposition method the solution is obtained in the form of a rapid convergent power series with elegantly computable terms comparing the decomposition method we used with several other methods that have been advanced for solving this model shows that the new technique is reliable powerful and promising

application of the decomposition method to inversion of matrices abstract the decomposition method g adomian stochastic systems academic press new york london 1983 is applied to inversion of matrices by approximation

an efficient method for solving systems of fractional integro differential equations in this study a decomposition method for approximating the solution of systems of fractional integro differential equations is implemented the fractional derivative is considered in the caputo sense the approximate solutions are calculated in the form of a convergent series with easily computable components numerical results show that this approach is easy to implement and accurate when applied to integro differential equations

a reliable method for the numerical solution of the kinetics problems this paper deals with the implementation of the adomian s decomposition method in chemical applications one often encounters systems of ordinary differential equations this method is intended to solve the approximate solutions of the linear and nonlinear kinetic problems without usual restrictive assumptions a new approach to differential problems is particularly valuable as a tool for scientists and applied mathematicians because it provides immediate and visible symbolic terms of analytic solution as well as its numerical approximate solution to both linear and nonlinear problems without linearization adomian solving frontier problems of physics the decomposition method kluwer academic publishers boston 1994 j math anal appl 135 1988 501 it does also not require discretization and consequently massive computation we also give a comparison between a reliable decomposition and a conventional methods such as the taylor series and the fourth order runge kutta method for system of ordinary differential equations the numerical results demonstrate that the new method is quite accurate and readily implemented

numerical solution of fractional pantograph differential equations by using generalized fractional order bernoulli wavelet in the current study new functions called generalized fractional order bernoulli wavelet functions gfbwfs based on the bernoulli wavelets are defined to obtain the numerical solution of fractional order pantograph differential equations in a large interval for the concept of fractional derivative we will use caputo sense by using riemann liouville fractional integral operator first the generalized fractional order bernoulli wavelets are constructed then these functions and their properties are employed to derive the gfbwfs operational matrices of fractional integration and pantograph the operational matrices of integral and pantograph are utilized to reduce the problem to a set of algebraic equations finally some examples are included for demonstrating the validity and applicability of our method

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

analytical solution of a time fractional navier stokes equation by adomian decomposition method abstract the aim of the present analysis is to apply adomian decomposition method for the solution of a time fractional navier stokes equation in a tube by using an initial value the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically the present method performs extremely well in terms of efficiency and simplicity

algebraic equations with exponential terms abstract the decomposition method is applied to algebraic equations containing exponential terms the n term approximation n is rapidly damped as n increases yielding an oscillating convergence of superior accuracy n 5 and n 9 for k 2

on the problem of linear elasticity for an infinite region containing a finite number of non intersecting spherical inhomogeneities abstract the problem of linear elasticity for an infinite number of non intersecting spherical and more generally ellipsoidal inhomogeneities is attacked the approach taken does not misrepresent geometry of inhomogeneities although the continuity conditions at the intefaces are only approximately satisfied the principal idea of the approach is to extend the method of kachanov 1985 int j fracture28 r11 r19 1987 int j solids structures23 23 43 for interacting cracks to the realm of the eshelby equivalent inclusion method the application to a test problem for two spherical cavities suggests that the approach can be useful for predictions of the overall response of composite materials and interfacial stress concentrations

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

statistical limit points of sequences of fuzzy numbers in the current work we extend the concepts of statistical limit and cluster points of a sequence of fuzzy numbers corresponding to the definitions in proc am math soc 4 1993 1187 for the sequences of real numbers later we discuss the relations among sets of ordinary limit points statistical limit points and statistical cluster points of sequences of fuzzy numbers

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

on adomian s decomposition method and some comparisons with picard s iterative scheme abstract we consider the general mathematical framework of adomian s decomposition method g adomian stochastic systems academic press new york 1983 for a large class of nonlinear operator equations picard s iterative scheme is considered for the same class for equations and compared with the decomposition method the paper identifies carefully all substantial differences between the two methods and shows that various advantages exist for the decomposition method

on a class of nonlinear stochastic dynamical systems analysis of the transient behaviour abstract a class of nonlinear dynamical systems with random parameters and initial conditions is considered the transient time evolution of the mean value of the dependent variable is analyzed both by means of the stochastic averaging method and by adomian s decomposition method a proof of the equivalence of the decomposition method to the time series approximation of the dynamical response is provided an application is made to the van der pol equation and finally a comparison between the two previously discussed methods is given

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

a comparison between adomian decomposition method and taylor series method in the series solutions in this work we will compare the performance of the adomian decomposition method and the taylor series method applied to the solution of linear and nonlinear ordinary differential equations the comparison shows that the decomposition method is reliable efficient and easy to use from a computational viewpoint although the two methods provide the solution in an infinite series the decomposition method provides a fast convergent series of easily computable components and eliminates cumbersome computational work needed by taylor series method

application of legendre wavelets for solving fractional differential equations in this paper we develop a framework to obtain approximate numerical solutions to ordinary differential equations odes involving fractional order derivatives using legendre wavelet approximations the properties of legendre wavelets are first presented these properties are then utilized to reduce the fractional ordinary differential equations fodes to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution

solutions to the polya szego conjecture and the weak eshelby conjecture eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform he then conjectured that the converse is true that is that if the field inside an inclusion is uniform for all uniform loadings then the inclusion is of elliptic or ellipsoidal shape we call this the weak eshelby conjecture in this paper we prove this conjecture in three dimensions in two dimensions a stronger conjecture which we call the strong eshelby conjecture has been proved if the field inside an inclusion is uniform for a single uniform loading then the inclusion is of elliptic shape we give an alternative proof of eshelby s conjecture in two dimensions using a hodographic transformation as a consequence of the weak eshelby s conjecture we prove in two and three dimensions a conjecture of polya and szego on the isoperimetric inequalities for the polarization tensors pts the polya szego conjecture asserts that the inclusion whose electrical pt has the minimal trace takes the shape of a disk or a ball

a convenient computational form for the adomian polynomials abstract recent important generalizations by g adomian stochastic systems academic press 1983 have extended the scope of his decomposition method for nonlinear stochastic operator equations see also iterative method inverse operator method symmetrized method or stochastic green s function method very considerably so that they are now applicable to differential partial differential delay and coupled equations which may be strongly nonlinear and or strongly stochastic or linear or deterministic as subcases thus for equations modeling physical problems solutions are obtained rapidly easily and accurately the methodology involves an analytic parametrization in which certain polynomials an dependent on the nonlinearity are derived this paper establishes simple symmetry rules which yield adomian s polynomials quickly to high orders

a comparison between adomian s decomposition methods and perturbation techniques for nonlinear random differential equations abstract comparison of the adomian decomposition method and regular perturbation techniques applied to the solution of nonlinear vector random differential equations shows that the decomposition method is superior and generally applicable even for the cases where perturbation is applicable the comparison shows that the decomposition method is easier to compute and supplies quantitatively reliable results

picard iteration algorithm combined with gauss seidel technique for initial value problems abstract in this paper we introduce and implement a relatively new improvement for the well known picard s method for studying linear and nonlinear systems of ordinary differential equations in this method the solution takes the form of a rapidly convergent series with easily computable components a mathematica computer algebra system mathematica 5 1 is used to introduce a code which generates the series different illustrative examples have been considered as well as different comparison criterion between the proposed method and modern reliable and conventional methods the deeply mathematical roots of the method fixed point theorem and the correspondence with the theory of algebraic systems makes it is very interesting theoretical considerations are being discussed and convergence of the method is addressed other extensions and applications for further work are mentioned

the decomposition method for approximate solution of the goursat problem abstract in this paper the adomian s decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the goursat partial differential equation linear and nonlinear goursat examples are examined and the method supplied quantitatively reliable results for these types of problems the accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the goursat problem

nonlinear reaction with fractional dynamics in this paper we consider a system of three fractional differential equations describing a nonlinear reaction our analysis includes both analytical technique and numerical simulation this allows us to control the efficiency of the numerical method we combine the numerical approximation of fractional integral with finding zeros of a function of one variable the variable gives a desired solution the solution has a peak its position in time depends on the order of fractional derivative the two other solutions of this system have a completely monotonic character

a reliable modification of adomian decomposition method in this paper we propose a powerful modification of adomian decomposition method that will accelerate the rapid convergence of the series solution the validity of the modified technique is verified through illustrative examples in all cases of differential and integral equations we obtained excellent performance from the modified algorithm the modification could lead to a promising approach for many applications in applied sciences

semilinear stochastic systems analysis with the method of the stochastic green s function and application in mechanics abstract this work deals with the analysis of a class of semilinear vector stochastic operator equations the method of the stochastic green s function due to adomian is extended to the above mentioned class of equations some perturbation techniques are also developed and a pertinent application in nonlinear mechanics is considered

solutions to the eshelby conjectures we present solutions to the eshelby conjectures based on a variational inequality we first discuss the meanings of the original eshelby s statement by fourier analysis we establish the connection between the homogeneous eshelby inclusion problem and the classic newtonian potential problem we then proceed to the solutions of the eshelby conjectures under some hypothesis on the material properties and restricted to connected inclusions with lipschitz boundary we show that one version of the eshelby conjectures is valid in all dimensions and the other version is valid in two dimensions we also show the existence of multiply connected inclusions in all dimensions and the existence of non ellipsoidal connected inclusions in three and higher dimensions such that in physical terms and in the context of elasticity some uniform eigenstress of the inclusion induces uninform strain on the inclusion we numerically calculate these special inclusions based on the finite element method

coupled nonlinear stochastic differential equations abstract systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co workers examples include two coupled riccati equations coupled linear equations stochastic coupled equations with product terms and n coupled stochastic differential equations

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

solving nonlinear differential equations with decimal power nonlinearities abstract the decomposition method stochastic systems academic press new york 1983 nonlinear stochastic operator equations academic press new york in press is applied to the solution of nonlinear differential equations of the form ly ny x t where l is a linear differential operator and ny is a nonlinear term of the form y with a decimal number

blow up for solutions of some linear wave equations with mixed nonlinear boundary conditions this paper deals with the blow up in solutions of some linear wave equations with mixed nonlinear boundary conditions the finite time blow up in solutions are caused from the nonlinear boundary conditions even if the initial data are smooth and well defined for all times the pointwise blow up in exact solution and blow up in energy solution are investigated