stability criteria for certain high even order delay differential equations abstract in this paper we study the asymptotic stability of the zero solution of even order linear delay differential equations of the form y 2 m t j 0 2 m 1 a j y j t j 0 2 m 1 b j y j t where a j and b j are certain constants and m 1 here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

numerical computation of 2d sommerfeld integrals a novel asymptotic extraction technique abstract the accurate and efficient computation of the elements in the impedance matrix is a crucial step in the application of galerkin s method to the analysis of planar structures as was demonstrated in a previous paper it is possible to decompose the angular integral in the polar representation for the 2d sommerfeld integrals in terms of incomplete lipschitz hankel integrals ilhis when piecewise sinusoidal basis functions are employed since bessel series expansions can be used to compute these ilhis a numerical integration of the inner angular integral is not required this technique provides an efficient method for the computation of the inner angular integral however the outer semi infinite integral still converges very slowly when a real axis integration is applied therefore it is very difficult to compute the impedance elements accurately and efficiently in this paper it is shown that this problem can be overcome by using the ilhi representation for the angular integral to develop a novel asymptotic extraction technique for the outer semi infinite integral the usefulness of this asymptotic extraction technique is demonstrated by applying it to the analysis of a printed strip dipole antenna in a layered medium

numerical computation of 2d sommerfeld integrals decomposition of the angular integral abstract spectral domain techniques are frequently used in conjunction with galerkin s method to obtain the current distribution on planar structures when this technique is employed a large percentage of the computation time is spent filling the impedance matrix therefore it is important to develop accurate and efficient numerical techniques for the computation of the impedance elements which can be written as two dimensional 2d sommerfeld integrals once the current distribution has been found then the near zone electric field distribution can be obtained by computing another set of 2d sommerfeld integrals the computational efficiency of the 2d sommerfeld integrals can be improved in two ways the first method which is discussed in this paper involves finding a new way to compute the inner angular integral in the polar representation of these integrals it turns out that the angular integral can be decomposed into a finite number of incomplete lipschitz hankel integrals which in turn can be calculated using series expansions therefore the angular integral can be computed by summing a series instead of applying a standard numerical integration algorithm this new technique is found to be more accurate and efficient when piecewise sinusoidal basis functions are used to analyze a printed strip dipole antenna in a layered medium the incomplete lipschitz hankel integral representation for the angular integral is then used in another paper to develop a novel asymptotic extraction technique for the outer semi infinite integral

on solving systems of multi pantograph equations via spectral tau method abstract the current manuscript focuses on solving systems of multi pantograph equations the spectral tau method is applied for solving systems of multi pantograph equations with shifted jacobi polynomials as basis functions the convergence analysis of the proposed technique is also investigated we introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods

on stability of systems of delay differential equations abstract in this paper we give necessary and sufficient conditions for the asymptotic stability of the zero solution of the system of linear delay differential equations of the form x t ax t 1 ax t where a is an n n matrix 0 is constant and 0 1 we reduce this to systems of first and second order problems our stability results are given in terms of the eigenvalues of a the proof of our results are carried out by an application of pontryagin s criterion for quasi polynomials to the characteristic functions of subsystems of the delay differential equations we also provide four algorithmic stability tests and include several examples

convergence analysis of the chebyshev legendre spectral method for a class of fredholm fractional integro differential equations abstract in this paper we propose and analyze a spectral chebyshev legendre approximation for fractional order integro differential equations of fredholm type the fractional derivative is described in the caputo sense our proposed method is illustrated by considering some examples whose exact solutions are available we prove that the error of the approximate solution decays exponentially in l 2 norm

stability criteria for certain third order delay differential equations in this paper we study the asymptotic stability of the zero solution of third order linear delay differential equations of the form y t p1y t p2y t q1y t q2y t v1y t v2y t where p1 p1 q1 q2 v1 and v2 are certain constants here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

asymptotic and exponential stability of certain third order non linear delayed differential equations frequency domain method we use the frequency domain method to prove that the zero solution of certain third order nonlinear delayed differential equations is asymptotically stable when there is no forcing term we also prove the existence of a bounded solution which is exponentially stable when there is a bounded forcing term the situation for which the non linear term is delayed is also proved

critical delays and polynomial eigenvalue problems in this work we present a new method to compute the delays of delay differential equations ddes such that the dde has a purely imaginary eigenvalue for delay differential equations with multiple delays the critical curves or critical surfaces in delay space that is the set of delays where the dde has a purely imaginary eigenvalue are parameterized we show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value i e commensurate delays the parameterization is done by solving a quadratic eigenvalue problem which is constructed from the vectorization of a matrix equation and hence typically of large size for commensurate delay differential equations the corresponding equation is a polynomial eigenvalue problem as a special case of the proposed method we find a closed form for a parameterization of the critical surface for the scalar case we provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems

multistability and stable asynchronous periodic oscillations in a multiple delayed neural system abstract we consider a network of three identical neurons with multiple discrete signal transmission delays the model for such a network is a system of nonlinear delay differential equations after some consideration of the absolute synchronization of the system and the global attractivity of the zero solution we present a detailed discussion about the boundaries of the stability region of the trivial solution this allows us to determine the possible codimension one bifurcations which occur in the system in particular we show the existence of standard hopf bifurcations giving rise to synchronized periodic solutions and of d 3 equivariant hopf bifurcations giving rise to three types of periodic solutions phase locked mirror reflecting and standing waves hopf hopf and hopf steady state bifurcations interactions are shown to exist and give rise to coexistence of stable synchronized and desynchronized solutions perturbation techniques coupled with the floquet theory are used to determine the stability of the phase locked oscillations

stability criteria for certain second order delay differential equations with mixed coefficients in this paper we study asymptotic stability of the zero solution of the second order linear delay differential equation y t p1y t p2y t q1y t q2y t where p1p2 q1 and q2 are constants with p1p2 0 and q1q2 0 is a constant in proving our results we make use of pontryagin s theory of quasi polynomials

a novel asymptotic extraction technique for the efficient evaluation of a class of double sommerfeld integrals when the galerkin method of moments mom is used in the numerical analysis of microwave microstrip networks elements of a network impedance matrix are required two dimensional 2d sommerfeld spectral integrals must be evaluated in such cases when semi infinite circuit feed lines are used certain specific types of sommerfeld integrals arise whose integrands oscillate rapidly and present serious numerical difficulties for their evaluation in this paper a novel accurate and much more efficient asymptotic extraction technique aet is developed that involves extracting an inner asymptotic angular integral which is then evaluated analytically before evaluating the outer integral numerically

asymptotic stability of a mechanical robotics model with damping and delay in this paper we study the asymptotic stability of a mechanical robotics model with damping and delay this model yields a certain linear third order delay differential equation in proving our results we make use of pontryagin s theory for quasi polynomials

stability criteria for certain high odd order delay differential equations in this paper we study the asymptotic stability of the zero solution of odd order linear delay differential equations of the formy 2 m 1 t j 02ma jy j t j 02mb jy j t t where a j and b j are certain constants and m 1 here t 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical 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 solution of semidifferential equations by collocation method we study the numerical solution of semidifferential equation of order 4 by collocation spline method we derive a system of equations that characterizing the numerical solution some numerical examples are also presented to illustrate our results

stability analysis of the zero solution for two class evolution equations abstract by constructing some suitable lyapunov type functionals and applying the theory of the definite quadratic form we obtain the stability of the zero solution of two classes of evolution equations with delays including reaction diffusion equations and damped wave equations our criteria depend on the derivatives of delays consequently when the delays are constants these criteria are independent of the magnitudes of the delays so the delays are harmless for the stability of the zero solution

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

comparative study of three numerical schemes for fractional integro differential equations this paper presents a comparative study of three numerical schemes such as linear quadratic and quadratic linear scheme for the fractional integro differential equations defined in terms of the caputo fractional derivatives the error estimates of the respective approximations are also established numerical tests of the discussed schemes show that all schemes work well and when the number of terms approximating the solution is increased the desired solution is achieved the accuracy of the numerical schemes with respect to the step size h is analysed and illustrated through various tables finally comparative performances of the schemes are discussed comparative study of three schemes such as linear quadratic and quadratic linear schemes is presented for the fractional integro differential equations the convergences of the schemes are also established the performances of the schemes are compared through the test examples

collocation method with convergence for generalized fractional integro differential equations abstract in this paper we study a numerical approach for some class of generalized fractional integro differential equations gfides defined in terms of the b operators presented recently we develop collocation method for linear and nonlinear forms of gfides the numerical approach uses the idea of collocation methods for solving integral equations legendre polynomials are used to approximate the solution in finite dimensional space with convergence analysis the obtained approximate solution recovers the solution of the fractional integro differential equation fide defined using caputo derivatives in a special case fides containing convolution type kernels appear in diverse area of science and engineering applications therefore some test examples varying the kernel in the b operator are considered to perform the numerical investigations the numerical results validate the presented scheme and provide good accuracy using few legendre basis functions

oscillation of third order trinomial delay differential equations the purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equation y t p t y t q t y tau t 0 new comparison th

numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis in this paper a system of nonlinear fractional differential equations fdes are considered they have been solved by legendre wavelets method combining with its operational matrix however there are no articles about solving this system using wavelets method the main purpose of this technique is to transform the initial equations into a nonlinear system of algebraic equations which can be solved easily the convergence and error analysis are presented to show the correctness and feasibility of method proposed for solving the above mentioned problem finally the applicability and efficiency of the mentioned approach are demonstrated by three numerical examples

application of the collocation method for solving nonlinear fractional integro differential equations in this paper using the collocation method we solve the nonlinear fractional integro differential equations nfide of the form f t y t a cd t a 0y t a cd t a ry t email protected t y t a tk t s f s y s ds y k a d k k 0 1 m 0 1 we study the convergence and the stability analysis of this method for f t y t a cd t a 0y t a cd t a ry t y t email protected j 0 rb j a cd t a jy t g t some numerical examples are given to show the efficiency of the presented method

on the functional integral equations of mixed type and integro differential equations of fractional orders the existence of monotonic solution for the functional integral equation of the mixed typex t g t email protected 1 0k 1 t s f 1s 0 sk 2 s t f 2 t x t email protected be proved in l 1 0 1 such kind of equations plays a significant role in many branches of mathematical physics engineering and economics on the other hand our considered equation are the general form of hammerstein and volterra integral equations see 4 and 5 as an application we prove the existence of solution for the initial value problem of the fractional order integro differential equation dx t dt g t email protected 1 0k 1 t s f 1 s d bx s ds email protected 0 1

numerical computation of the incomplete lipschitz hankel integral je o a z abstract two factorial neumann series expansions are derived for the incomplete lipschitz hankel integral je 0 a z these expansions are used together with the neumann series expansion given by agrest in an algorithm which efficiently computes je 0 a z to a user defined number of significant digits other expansions for je 0 a z which are found in the literature are also discussed but these expansions are found to offer no significant computational advantages when compared with the expansions used in the algorithm

properties of the third order trinomial differential equations with delay argument abstract in this paper we study properties of the third order trinomial delay differential equation y t p t y t g t y t 0 by transforming this equation to binomial second third order differential equation employing suitable comparison theorems we establish new results on asymptotic behavior of solutions of the studied equation

stabilization role of inhibitory self connections in a delayed neural network abstract in a delayed hopfield neural network that is strongly connected with non inhibitory interconnections fast and inhibitory self connections lead to global convergence to a unique equilibrium of the network by applying monotone dynamical systems theory and an embedding technique we prove that this conclusion remains true without the requirement of strong connectivity or non inhibitory interconnections

the chebyshev legendre method implementing legendre methods on chebyshev points a new collocation method for the numerical solution of partial differential equations is presented this method uses the chebyshev collocation points but because of the way the boundary conditions are implemented it has all the advantages of the legendre methods in particular l 2 estimates can be easily obtained for hyperbolic and parabolic problems

a new prediction scheme for input delay compensation in restricted feedback linearizable systems the input output inversion of a system under the effect of input delays typically relies on the ability to predict the future of the system s state indeed if the latter is known ahead of time one can cope with the input delay by using a prediction of the state instead of the state itself such methods are efficient when the plant is stable but become numerically unstable otherwise we present a new method to compensate input delays our approach relies on imposing a desired error dynamics which is designed to be linear and asymptotically stable at the origin then the state prediction is computed from the state reference trajectory and the predicted error dynamics in this technical note we concentrate on the case study of systems in strict feedback form and present a simple backstepping procedure

reconstruction of exponentially rate of convergence to legendre collocation solution of a class of fractional integro differential equations in this paper legendre collocation method an easy to use variant of the spectral methods for the numerical solution of a class of fractional integro differential equations fide s is researched in order to obtain high order accuracy for the approximation the integral term in the resulting equation is approximated by using legendre gauss quadrature formula an efficient convergence analysis of the proposed method is given and rate of convergence is established in the l 2 norm due to the fact that the solutions of fide s usually have a weak singularity at origin we use a variable transformation to change the original equation into a new equation with a smooth solution we prove that after this regularization technique numerical solution of the new equation by adopting the legendre collocation method has exponentially rate of convergence numerical results are presented which clarify the high accuracy of the proposed method

nonlinear galerkin method using chebyshev and legendre polynomials i the one dimensional case a new strategy stemming from the nonlinear galerkin method m marion and r temam siam j numer anal 26 1989 pp 1139 1157 for solving linear elliptic and nonlinear dissipative evolution equations by using chebyshev and legendre polynomials is presented the essential idea is to decompose the solution into a low mode part and a high mode part and to treat them separately the robustness of the method over the classical galerkin method is substantiated by rigorous error estimates and preliminary numerical experiments

the variational lyapunov function and strict stability theory for differential systems abstract in this paper we investigate strict stability of differential systems by variational lyapunov function we obtain some sufficient conditions and comparison theorems

timely communication efficient algorithms for solving a fourth order equation with the spectral galerkin method we show that one can derive an o n 3 spectral galerkin method for fourth order biharmonic type elliptic equations based on the use of chebyshev polynomials the use of chebyshev polynomials provides a fast transform between physical and spectral space which is advantageous when a sequence of problems must be solved e g as part of a nonlinear iteration this improves the result of shen siam j sci comput 16 1995 pp 74 87 which reported an o n 4 algorithm inferior to the o n 3 method developed earlier shen siam j sci comput 15 1994 pp 1440 1451 based on legendre polynomials but less practical in the case of multiple problems we further compare our method with an improved implementation of the legendre galerkin method based on the same approach

application of the fractional differential transform method to fractional order integro differential equations with nonlocal boundary conditions in this paper the fractional differential transform method is developed to solve fractional integro differential equations with nonlocal boundary conditions the method is described and illustrated with numerical examples the results reveal that the method is accurate and easy to apply

exponentially accurate spectral and spectral element methods for fractional odes current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy here we present different methods for fractional odes that lead to exponentially fast decay of the error first we develop a petrov galerkin pg spectral method for fractional initial value problems fivps of the form d t 0 u t f t and fractional final value problems ffvps d t t u t g t where 0 1 subject to dirichlet initial final conditions these schemes are developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently developed in 1 specifically we obtain solutions to fivps and ffvps in terms of the new fractional non polynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind fslp ii subsequently we develop a discontinuous spectral method dsm of petrov galerkin sense for the aforementioned fivps and ffvps where the basis functions do not satisfy the initial final conditions finally we extend the dsm scheme to a discontinuous spectral element method dsem for efficient longer time integration and adaptive refinement in these discontinuous schemes we employ the asymptotic eigensolutions to fslp i ii which are of jacobi polynomial forms as basis and test functions our numerical tests confirm the exponential algebraic convergence respectively in p and h refinements for various test cases with integer and fractional order solutions

cubic b spline approximation for linear stochastic integro differential equation of fractional order abstract in this paper the cubic b spline collocation method is used for solving the stochastic integro differential equation of fractional order we show that stochastic integro differential equation of fractional order is equivalent to a modified stochastic integral equation then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation using this method the problem solving turns into a linear system solution of equations also the convergence analysis of this numerical approach has been discussed in the end examples are given to test the accuracy and the implementation of the method the results are compared with the results obtained by other methods to verify that this method is accurate and efficient

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

stability theorems for nonlinear scalar delay differential equations in this paper the stability on nonlinear scalar delay differential equations is studied and improved 3 2 stability theorems are presented

numerical solution of fractional integro differential equations by collocation method abstract this paper deals with the numerical solution of fractional integro differential equations of the type d q y t p t y t f t 0 t k t s y s d s t i 0 1 by polynomial spline functions we derive a system of equations that characterizing the numerical solution some numerical examples are also provided to illustrated our results

oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations the equation to be considered iswhere pi t 0 i n and q t are continuous and positive on some half line a it is known that always has strictly monotone nonoscillatory solutions defined on a so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of in this paper sufficient conditions are given for this situation to hold for the structure of the solution space of is also studied

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

a novel method for solving second order fractional eigenvalue problems the paper presents a new numerical method for solving eigenvalue problems for fractional differential equations it combines two techniques the method of external excitation mee and the backward substitution method bsm the first one is a mathematical model of physical measurements when a mechanical electrical or acoustic system is excited by some source and resonant frequencies can be determined by using the growth of the amplitude of oscillations near these frequencies the bsm consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters these free parameters are determined by the use of the collocation procedure some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results the numerical results show that the proposed method is of a high accuracy and is efficient for solving of a wide class of eigenvalue problems

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

instability results for the damped wave equation in unbounded domains abstract we extend some previous results for the damped wave equation in bounded domains in r d to the unbounded case in particular we show that if the damping term is of the form a with bounded a taking on negative values on a set of positive measure then there will always exist unbounded solutions for sufficiently large positive in order to prove these results we generalize some existing results on the asymptotic behaviour of eigencurves of one parameter families of schrodinger operators to the unbounded case which we believe to be of interest in their own right

variational iteration method and homotopy perturbation method for fourth order fractional integro differential equations in this article linear and nonlinear boundary value problems for fourth order fractional integro differential equations are solved by variational iteration method and homotopy perturbation method the fractional derivatives are described in the caputo sense the solutions of both problems are derived by infinite convergent series which are easily computable and then graphical representation shows that both methods are most effective and convenient one to solve linear and nonlinear boundary value problems for fourth order fractional integro differential equations

approximate solution of fractional integro differential equations by taylor expansion method in this paper taylor expansion approach is presented for solving approximately a class of linear fractional integro differential equations including those of fredholm and of volterra types by means of the mth order taylor expansion of the unknown function at an arbitrary point the linear fractional integro differential equation can be converted approximately to a system of equations for the unknown function itself and its m derivatives under initial conditions this method gives a simple and closed form solution for a linear fractional integro differential equation in addition illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method