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

adomian decomposition a tool for solving a system of fractional differential equations abstract adomian decomposition method has been employed to obtain solutions of a system of fractional differential equations convergence of the method has been discussed with some illustrative examples in particular for the initial value problem d 1 y 1 d n y n t a y 1 y n t y i 0 c i i 1 n where a a i j is a real square matrix the solution turns out to be y x e 1 n 1 x 1 a 1 x n a n y 0 where e 1 n 1 denotes multivariate mittag leffler function defined for matrix arguments and a i is the matrix having ith row as a i 1 a i n and all other entries are zero fractional oscillation and bagley torvik equations are solved as illustrative examples

absorbing boundary conditions for the schrodinger equation a large number of differential equation problems which admit traveling waves are usually defined on very large or infinite domains to numerically solve these problems on smaller subdomains of the original domain artificial boundary conditions must be defined for these subdomains one type of artificial boundary condition which can minimize the size of such subdomains is the absorbing boundary condition the imposition of absorbing boundary conditions is a technique used to reduce the necessary spatial domain when numerically solving partial differential equations that admit traveling waves such absorbing boundary conditions have been extensively studied in the context of hyperbolic wave equations in this paper general absorbing boundary conditions will be developed for the schrodinger equation with one spatial dimension using group velocity considerations previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition the well posedness of the initial boundary value problem of the absorbing boundary condition coupled to the interior schrodinger equation will also be discussed extension of the general absorbing boundary condition to higher spatial dimensions will be demonstrated numerical simulations using initial single gaussian double gaussian and a narrow gaussian pulse distributions will be given with comparision to exact solutions to demonstrate the reflectivity properties of various orders of the absorbing boundary condition

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

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

weak ill posedness of spatial discretizations of absorbing boundary conditions for schrodinger type equations when we wish to solve numerically a differential problem defined on an infinite domain it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution these boundary conditions are called absorbing when small reflections to the interior domain are allowed in this paper we develop a general class of absorbing boundary conditions for schrodinger type equations by using rational approximations to the transparent boundary conditions with this approach previous absorbing boundary conditions in the literature are included in this class we use the method of lines for the discretization of the initial boundary value problems obtained this way we show that the ordinary differential systems that arise after the spatial discretization are weakly ill posed explaining a previous conjecture of fevens and jiang the time discretization is carried out with a stable runge kutta methods where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space

adomian decomposition method for solution of differential algebraic equations solutions of differential algebraic equations is considered by adomian decomposition method in e babolian m m hosseini reducing index and spectral methods for differential algebraic equations j appl math comput 140 2003 77 and m m hosseini an index reduction method for linear hessenberg systems j appl math comput in press an efficient technique to reduce index of semi explicit differential algebraic equations has been presented in this paper adomian decomposition method is applied to reduced index problems the scheme is tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

an explicit analytic solution to the thomas fermi equation a new kind of analytic technique namely the homotopy analysis method is employed to give an explicit analytic solution of the thomas fermi equation and the related recurrence formulae of constant coefficients this solution can be regarded as the definition of the exact solution of the thomas fermi equation

asymptotic synchronization of the colpitts oscillator in this paper we deal with the observer based asymptotic synchronization problem for a class of chaotic oscillators some results based on a differential algebraic approach are used in order to determine the algebraic observability of unknown variables the strategy consists of proposing a slave system observer which tends to follow asymptotically the master system the methodology is tested in the real time asymptotic synchronization of the colpitts oscillator by means of a proportional reduced order observer proo of free model type

a high order finite element discretization with local absorbing boundary conditions of the linear schrodinger equation the goal of this paper is to obtain a high order full discretization of the initial value problem for the linear schrodinger equation in a finite computational domain for this we use a high order finite element discretization in space together with an adaptive implementation of local absorbing boundary conditions specifically obtained for linear finite elements and a high order symplectic time integrator the numerical results show that it is possible to obtain simultaneously a very good absorption at the boundary and a very small error in the interior of the computational domain

stable inversion for nonlinear systems abstract there have been two recent major developments in output tracking for nonlinear systems and our first main contribution is to relate these under appropriate assumptions we show that the bounded solution of the partial differential equation of isidori and byrnes for each trajectory of an exosystem must be given by an integral representation formula of devasia chen and paden under restrictive hypotheses devasia chen and paden develop a picard process that converges to the solution of the integral equation this solution to the integral equation is also a bounded solution to a dynamical equation driven by the desired outputs in aircraft applications our nonlinear systems are perturbations of pure feedback systems with outputs and we find a solution to the stable inversion problem i e finding bounded controls and bounded state trajectories in response to bounded output signals in two steps the first step ignores the perturbation error and computes the major part of the desired control and corresponding state trajectory the second step computes the remaining control and states by finding a noncausal and stable solution to an error driven dynamical equation in other words the method of devasia chen and paden is applied to an error system and not to the full system this two step procedure is the second main contribution of this paper throughout this paper it is assumed that our nonlinear systems have vector relative degree

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

artificial boundary conditions for diffusion equations numerical study abstract recently in 3 a new method has been presented providing a family of artificial boundary conditions for parabolic equations which are local in time and stable this method applies to variable coefficients and curved boundaries but unfortunately there are no error estimates as an alternative we present in this paper numerical simulations showing the efficiency of the method the construction of the artificial boundary conditions depends on the number and values of points at which z 1 2 is interpolated by rational fractions in the complex plane particular attention is paid to the choice of those points

non linear observer design by observer canonical forms this paper studies the non linear observer design problem by observer canonical forms we present necessary and sufficient conditions in terms of the ranks of matrices for the existence of a non linear state coordinate change under which a time variable system is transformed into an observer canonical form these conditions give the multiple versions of li and tao 1986 for the single output case then a comparison is made with the lie algebraic conditions obtained by krener and respondek 1985 and modified by the authors in a recent paper xia and gao 1987

numerical comparison of methods for solving fractional differential algebraic equations fdaes in this paper the variational iteration method vim and the adomian decomposition method adm are implemented to give approximate solutions for fractional differential algebraic equations fdaes both methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations this paper presents a numerical comparison between these two methods and the homotopy analysis method ham for solving fdaes numerical results reveal that the vim and the adm are quite accurate and applicable

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

rigorous absorbing boundary conditions for 3 d one way wave extrapolation absorbing boundary conditions play an important role in one way wave extrapolations by reducing reflections at grid edges clayton and engquist s 2 d absorbing boundary conditions for one way wave extrapolation by depth stepping in the frequency domain are extended to three dimensions using paraxial approximations of the scalar wave equation internal consistency is retained by incorporating the interior extrapolation equation with the absorbing boundary conditions numerical schemes are designed to make the proposed absorbing boundary conditions both mathematically correct and efficient with negligible extra cost synthetic examples illustrate the effectiveness of the algorithm for extrapolation with the 3 d 45 one way wave equation

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

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

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

an algorithm for solving daes with mechanization in this paper by using the theories and methods of mathematical analysis and computer algebra a reliable algorithm of pade approximation method for solving differential algebraic equation systems was established and a new maple procedures mainproc was established too ten examples are presented to illustrate the implementation of the program

nonlinear observers for two classesof perturbed systems in this paper we study two important classes of nonlinear systems with perturbation namely perturbation with output injection and perturbation of pure feedback systems with sufficiently good initial error estimates and sufficiently restricted external inputs and perturbation parameter nonlinear observers for each class of perturbed systems are designed so that the error between the exact states and the estimated states goes to zero our simulation results illustrate the applicability of our technique in building nonlinear observers for these two classes of perturbed systems

iii posedness of absorbing boundary conditions for migration absorbing boundary conditions for wave equation migration were introduced by clayton and engquist we show that one of these boundary conditions the b2 second order condition applied with the 45 third order migration equation is ill posed in fact this boundary condition is subject to two distinct mechanisms of ill posedness a kreiss mode with finite speed at one boundary and another mode of a new kind involving wave propagation at unbounded speed back and forth between two boundaries unlike b2 the third order clayton engquist boundary condition b3 is well posed however we show that it is impossible for any boundary condition of clayton engquist type of order higher than one to be well posed with a migration equation whose order is higher than three

highly accurate absorbing boundary conditions for wide angle wave equations we develop a new class of absorbing boundary conditions abcs to prevent unwanted artifacts and wraparounds associated with aperture truncation in migration modeling using high order one way wave equations the fundamental approach behind the proposed development is the efficient discretization of the half space beyond the boundary of interest using midpoint integrated imaginary finite elements an idea recently utilized in the development of effective one way wave equations the proposed absorbing boundary conditions essentially add absorbing layers at the aperture truncation points we derive the absorbing boundary conditions analyze their properties and develop a stable explicit finite difference scheme to solve the downward continuation problem modified by these boundary conditions with the help of numerical examples we conclude that with as few as three absorbing layers i e two additional gridpoints the waves can be absorbed completely thus preventing associated artifacts

nonlinear observer design by observer error linearization this paper studies the observer design problem by the observer error linearization approach for nonlinear systems with and without inputs necessary and sufficient conditions for the existence of the linearization transformation are derived for nonlinear systems without inputs the conditions are shown to be corrections to an existing result a computation procedure and a different set of necessary and sufficient conditions based on the computation procedure are presented

canonical form observer design for non linear time variable systems an observer of canonical phase variable form for non linear time variable systems is introduced the development of this non linear time variable form requires regularity of the non linear time variable observability matrix of the system from the relationships derived during the development it follows that a non linear time variable observer can be dimensioned by an eigenvalue assignment with respect to the canonical state coordinates if a linearization of system non linearities about the reconstructed state trajectories is permissible this is an assumption similar to that of the extended kalman filter based on a linearization about the current estimate

a canonical expansion for nonlinear systems the importance of differential geometry in particular lie brackets of vector fields in the study of nonlinear systems is well established under very mild assumptions we show that a real analytic nonlinear system has an expansion in which the coefficients are computed in terms of lie brackets this expansion occurs in a special coordinate system we also explain the concept of a pure feedback system for control design involving a nonlinear system one approach is to put the system in its canonical expansion and approximate by that part having only feedback paths

stability of the method of lines it is well known that a necessary condition for the lax stability of the method of lines is that the eigenvalues of the spatial discretization operator scaled by the time stepk lie within a distanceo k of the stability region of the time integration formula ask 0 in this paper we show that a necessary and sufficient condition for stability except for an algebraic factor is that the pseudo eigenvalues of the same operator lie within a distanceo o k of the stability region ask 0 our results generalize those of an earlier paper by considering a runge kutta and other one step formulas b implicit as well as explicit linear multistep formulas c weighted norms d algebraic stability e finite and infinite time intervals and f stability regions with cusps in summary the theory presented in this paper amounts to a transplantation of the kreiss matrix theorem from the unit disk for simple power iterations to an arbitrary stability region for method of lines calculations

numerical approach to differential equations of fractional order in this paper the variational iteration method and the adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order the two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations in these schemes the solution takes the form of a convergent series with easily computable components this paper presents a numerical comparison between the two methods for solving systems of fractional differential equations numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order

discrete transparent boundary conditions for the numerical solution of fresnel s equation the paper presents a construction scheme of deriving transparent i e reflection free boundary conditions for the numerical solution of fresnel s equation being formally equivalent to schrodinger s equation in contrast to previous suggestions the method advocated here treasts the discrete problem after discretization of the time like variable i e in a rothe method which leads to a sequence of coupled boundary value problems the thus obtained boundary conditions appear to be of a nonlocal cauchy type as it turns out each kind of linear implicit discretization induces its own discrete transparent boundary conditions numerical experiments on technologically relevant examples from integrated optics are included

absorbing boundary conditions for diffusion equations summary we construct and analyse a family of absorbing boundary conditions for diffusion equations with variable coefficients curved artifical boundary and arbitrary convection it relies on the geometric identification of the dirichlet to neumann map and rational interpolation of z 1 2 in the complex plane the boundary conditions are stable accurate and practical for computations

the decomposition method for initial value problems abstract in this paper initial value problems for the heat wave and plate equations in which the vectorial space variable varies in the whole r d will be handled using the reliable decomposition method the solutions are obtained in the form of rapidly convergent power series with elegantly computable terms showing that the new technique is reliable powerful and promising

adomian decomposition method for solution of nonlinear differential algebraic equations in m m hosseini adomian decomposition method with chebyshev polynomials appl math comput in press an efficient modification of the adomian decomposition method was presented by using chebyshev polynomials also in m m hosseini adomian decomposition method for solution of differential algebraic equations j comput appl math in press solution of linear differential algebraic equations was considered by adomian decomposition method in this paper standard and modified adomian decomposition methods are applied to non linear differential algebraic equations the schemes are tested for some examples and the results demonstrate reliability and efficiency of the proposed methods

observers for nonlinear systems in steady state after constructing a nonlinear observer and applying state feedback depending on the estimated states to asymptotically stabilize the system we introduce new inputs of bounded norm to the system we then show that the parameters for the observer with sufficiently good initial error estimates can be chosen so that the error between states and estimates goes to zero particular remarks are appropriate for the class of global feedback linearizable systems where the bounds on the norm of the initial states and the amplitude of the new inputs can be as large as desired

on the homotopy analysis method for nonlinear problems a powerful easy to use analytic tool for nonlinear problems in general namely the homotopy analysis method is further improved and systematically described through a typical nonlinear problem i e the algebraically decaying viscous boundary layer flow due to a moving sheet two rules the rule of solution expression and the rule of coefficient ergodicity are proposed which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering an explicit analytic solution is given for the first time with recursive formulas for coefficients this analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem

the decomposition method for cauchy advection diffusion problems in this paper the solution of cauchy problems for the advection diffusion equation is obtained using the decomposition method in the case when the flow velocity is constant an analytical solution can be derived whilst for variable flow velocity symbolic numerical computations need to be performed

on the numerical solution of differential algebraic equations with index 3 in this study numerical solution of differential algebraic equations daes with index 3 has been presented by pade approximation an index reduction for differential algebraic equation with index 3 is suggested first index 3 system is reduced to index 1 system then we have applied the method for solving differential algebraic equations with index 3 numerically

discrete transparent boundary conditions for schrodinger type equations we present a general technique for constructing nonlocal transparent boundary conditions for one dimensional schrodinger type equations our method supplies boundary conditions for the family of implicit one step discretizations of schrodinger s equation in time the use of mikusi ski s operator approach in time avoids direct and inverse transforms between time and frequency domains and thus implements the boundary conditions in a direct manner

solving a system of nonlinear fractional differential equations using adomian decomposition adomian decomposition method has been employed to obtain solutions of a system of nonlinear fractional differential equations dxiyi x ni x y1 yn yi k 0 cki 0 k i 1 i n and dxi denotes caputo fractional derivative some examples are solved as illustrations using symbolic computation

artificial boundary conditions for 2d problems in geophysics abstract the numerical solution of a class of two dimensional problems in geophysics is considered the earth is modeled as a semi infinite acoustic or elastic medium in order to make the computational domain finite a semi circular artificial boundary ob is introduced thus truncating the original semi infinite domain an exact nonlocal relation is then derived on ob this relation is used as a boundary condition on ob which is combined with the finite element scheme employed in the computational domain the approach is applied to the helmholtz equation modeling time harmonic acoustic waves and to problems in elastostatics numerical examples validate the method and demonstrate its performance

numerical solution of differential algebraic equations with index 2 abstract in this paper numerical solution of differential algebraic equations daes with index 2 has been presented using pade approximation method an index reduction for differential algebraic equation with index 2 is suggested first index 2 system is reduced to index 1 system then we have applied the method for solving differential algebraic equations with index 2 numerically