construction of high order runge kutta methods which preserve delay dependent stability of ddes this paper is concerned with the construction of some high order runge kutta methods which preserve delay dependent stability of delay differential equations the methods of the first kind are developed by extending the ideas of brugnano et al while the methods of the second kind are developed according to the structure of the stability matrix we show that the derived methods are 0 stable for delay differential equations meanwhile the runge kutta methods can own the same order of the accuracy as the radau methods or gauss methods if the parameters are adequately defined these results not only improve the order of accuracy of the methods investigated by huang but also open an interesting route of finding new 0 stable runge kutta methods finally numerical experiments are proposed to illustrate the theoretical results

parametric instability of a leipholz column under periodic excitation in this paper the parametric instability of a leipholz column under four boundary conditions is examined the study of this prototypical model is intended to provide a basic understanding of the disc brake pad instability the distributed follower type axial load is assumed to be uniform and periodic instability regions are obtained and the existence of combination resonances of the sum and difference types is discussed for each set of boundary conditions it is found that the combination resonance of the sum type exists in all the cases of boundary conditions considered but the difference type exists only in the cases of clamped simply supported and clamped free boundary conditions the combination resonance is shown to be as important as the simple parametric resonance results when compared to a column under a periodic end load show that the instability characteristics of these two columns are considerably different the effect of a constant axial load is to shift the instability regions along the frequency axis

on the gegenbauer transformation the gegenbauer transformation g k is defined for 1 2 k 0 1 2 by where if being the gegenbauer polynomial of index and degree k and l 0 k is the tchebichef polynomial of degree k the transformation is studied on the spaces l p denned by the norm and its boundedness and range on these spaces is determined and inversion formulae are found

analytical and numerical stability of neutral delay integro differential equations and neutral delay partial differential equations this paper is concerned with the analytical and numerical stability of neutral delay integro differential equations ndides and neutral delay partial differential equations ndpdes we study the delay dependent stability of the real coefficient linear test equations for ndides furthermore we prove that the trapezium rule can preserve the delay dependent stability of the test equations considered we also discuss the delay dependent stability of the continuous problems the semi discrete problems and the fully discrete problems of linear ndpdes some numerical experiments are given to confirm the theoretical results

computation of fractional integrals via functions of hypergeometric and bessel type abstract the paper is devoted to computation of the fractional integrals of power exponential functions it is considered a function z defined by z 1 1 1 t 1 1 t e zt d t with positive and complex and z such that re 1 1 and re z 0 the special cases are discussed when z is expressed in terms of the tricomi confluent hypergeometric function a c x and of modified bessel function of the third kind k x representations of these functions via fractional integrals are proved the results obtained apply to compute fractional integrals of power exponential functions in terms of x a c x and k x examples are considered

instability of parametric dynamic systems with non uniform damping abstract a unified way of looking at parametric and combination resonances in systems with periodic coefficients and different amounts of damping in the various modes of vibration is presented the stability boundaries of the coupled mathieu equations are determined by the harmonic balance method fourier series with periods 2t and t being assumed the basic characteristics of the solution are discussed and the method is applied to multiple degree of freedom dynamic systems the destabilizing effect on the combination resonances is obtained by the present method

some analysis methods for rotating systems with periodic coefficients two of the more common procedures for analyzing the stability and forced response of equations with periodic coefficients are reviewed the use of floquet methods and the use of multiblade coordinate and harmonic balance methods the analysis procedures of these periodic coefficient systems are compared with those of the more familiar constant coefficient systems previously announced in star as n82 23702

an approach to investigate the instability of the multiple degree of freedom parametric dynamic systems a new analytical approach to the investigation of the regions of instability of multiple degree of freedom parametric dynamic systems is presented upon assuming a solution as a product of a characteristic component and a vector which has a periodic component expanding this vector into a fourier series and substituting it into the governing equations of motion a system of homogeneous algebraic equations can be obtained by the harmonic balance method then the problem of the stability of the non trivial solutions results in an eigenvalue problem of a non symmetricmatrix numerical results are presented for an undamped and damped mathieu equation whose stability has been investigated by various methods of solution the method has been applied to multiple degree of freedom systems such as lateral bending torsional vibration of a thin beam under parametric excitation and the parametric instability of a column under periodically varying axial loads

on the asymptotic stability properties of runge kutta methods for delay differential equations in this paper asymptotic stability properties of runge kutta r k methods for delay differential equations ddes are considered with respect to the following test equation begin eqnarray left begin array ccl y t a y t b y t tau 0 3cm y t g t end array right hskip 0 3cm begin array r t 0 0 3cm tau leq t leq 0 end array end eqnarray where a b in bbb r tau 0 and g t is a continuous real valued function in the last few years stability properties of r k methods applied to ddes have been studied by numerous authors who have considered regions of asymptotic stability for any positive delay and thus independent of the specific value of tau

on the similarity of the coleman and lyapunov floquet transformations for modal analysis of bladed rotor structures abstract structures with isotropic bladed rotors can be modally analyzed by eigenvalue analysis of time invariant coleman transformed equations of motion related to the inertial frame or by floquet analysis of the periodic equations of motion the coleman transformation is here shown to be a special case of the lyapunov floquet l f transformation which transforms system equations of structures with anisotropic bladed rotors into a time invariant system using the transition matrix and floquet eigenvectors as a basis the l f transformation is not unique whereby eigensolutions of the time invariant system are not directly related to the modal frequencies and mode shapes observed in the inertial frame this modal frequency indeterminacy is resolved by requiring the periodic mode shapes from the l f approach to be as similar as possible to the mode shapes from the coleman approach for an anisotropic rotor the floquet analysis yields a periodic mode shape that contains harmonics of integer multiples of the rotor speed for inertial state variables these harmonic components show up as resonance frequencies on the sides of the corresponding modal frequency in a computed frequency response function of a simple three bladed turbine with an anisotropic rotor

classical and quantum modes of coupled mathieu equations we expand the solutions of linearly coupled mathieu equations in terms of infinite continued matrix inversions and use it to find the modes which diagonalize the dynamical problem this allows obtaining explicitly the floquet lyapunov transformation to coordinates in which the motion is that of decoupled linear oscillators we use this transformation to solve the heisenberg equations of the corresponding quantum mechanical problem and find the quantum wavefunctions for stable oscillations expressed in configuration space the obtained transformation and quantum solutions can be applied to more general linear systems with periodic coefficients coupled hill equations periodically driven parametric oscillators and to nonlinear systems as a starting point for convenient perturbative treatment of the nonlinearity

a harmonic framework for controllability in linear continuous time periodic systems controllability of linear continuous time periodic systems is dealt with via a harmonic analysis approach for the first time in this paper this approach reveals that controllability of continuous time periodic systems can be connected to necessary and sufficient conditions expressed explicitly with fourier coefficients of the system matrices which can be interpreted in a way similar to what we have seen in linear time invariant cases these controllability conditions shed new light upon structural characteristics of continuous time periodic systems that are hard to know by means of existing time domain controllability criteria in the literature controllability canonical decomposition of linear continuous time periodic systems is revisited through state coordinate transforms of strong analytic features the results are heuristic and significant for examining structural characteristics of continuous time periodic systems and extending controllability related techniques that are widely employed in linear time invariant systems to linear continuous time periodic systems

numerical solution by lmms of stiff delay differential systems modelling an immune response we consider the application of linear multistep methods lmms for the numerical solution of initial value problem for stiff delay differential equations ddes with several constant delays which are used in mathematical modelling of immune response for the approximation of delayed variables the nordsieck s interpolation technique providing an interpolation procedure consistent with the underlying linear multistep formula is used an analysis of the convergence for a variable stepsize and structure of the asymptotic expansion of global error for a fixed stepsize is presented some absolute stability characteristics of the method are examined implementation details of the code difsub dde being a modification of the gear s difsub are given finally an efficiency of the code developed for solution of stiff ddes over a wide range of tolerances is illustrated on biomedical application model

a double expansion method for the frequency response of finite length beams with periodic parameters abstract a double expansion method for the frequency response of finite length beams with periodic distribution parameters is proposed the vibration response of the beam with spatial periodic parameters under harmonic excitations is studied the frequency response of the periodic beam is the function of parametric period and then can be expressed by the series with the product of periodic and non periodic functions the procedure of the double expansion method includes the following two main steps first the frequency response function and periodic parameters are expanded by using identical periodic functions based on the extension of the floquet bloch theorem and the period parametric differential equation for the frequency response is converted into a series of linear differential equations with constant coefficients second the solutions to the linear differential equations are expanded by using modal functions which satisfy the boundary conditions and the linear differential equations are converted into algebraic equations according to the galerkin method the expansion coefficients are obtained by solving the algebraic equations and then the frequency response function is finally determined the proposed double expansion method can uncouple the effects of the periodic expansion and modal expansion so that the expansion terms are determined respectively the modal number considered in the second expansion can be reduced remarkably in comparison with the direct expansion method the proposed double expansion method can be extended and applied to the other structures with periodic distribution parameters for dynamics analysis numerical results on the frequency response of the finite length periodic beam with various parametric wave numbers and wave amplitude ratios are given to illustrate the effective application of the proposed method and the new frequency response characteristics including the parameter excited modal resonance doubling peak frequency response and remarkable reduction of the maximum frequency response for certain parametric wave number and wave amplitude the results have the potential application to structural vibration control

global convergence of successive approximations for partial hadamard integral equations and inclusions abstract this paper deals with the global convergence of successive approximations as well as the uniqueness of solutions for some classes of partial functional hadamard integral equations and inclusions we prove a theorem on the global convergence of successive approximations to the unique solution of our problems

classification and characteristics of floquet factorisations in linear continuous time periodic systems this article has two purposes firstly the article is devoted to collecting basic facts about the floquet theorem and various floquet factorisation algorithms claimed for finite dimensional linear continuous time periodic fdlcp systems secondly the article presents a unified representation framework for various floquet factorisations in fdlcp systems while structural and analytic characteristics about them are examined as well more precisely the following aspects are considered i algorithms for floquet factorisations ii characteristics of floquet factors iii relationships and properties among floquet factorisations most results are reported for the first time while the others are generalised versions of existing ones

numerical solution of constant coefficient linear delay differential equations as abstract cauchy problems summary in this paper we present an approach for the numerical solution of delay differential equations begin equation left begin array l y prime left t right ly left t right my left t tau right t geq 0 y left t right varphi left t right tau leq t leq 0 end array right end equation where tau 0 l m in mathbb c m times m and varphi in c left left tau 0 right mathbb c m right different from the classical step by step method we restate 1 as an abstract cauchy problem and then we discretize it in a system of ordinary differential equations the scheme of discretization is proved to be convergent moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems 1

stability analysis of one step methods for neutral delay differential equations in this paper stability properties of one step methods for neutral functional differential equations are investigate stability regions are characterized for runge kutta methods with respect to the linear test equation begin gathered y left t right ay left t right by left t tau right cy left t tau right t geqq 0 hfill y left t right g left t right tau leqq t leqq 0 hfill end gathered 0 where a b andc are complex parameters in particular it is shown that everya stable collocation method for ordinary differential equations can be extended to a method for neutrals delay differential equations with analogous stability properties the so called np stable method we also investigate how the approximation to the derivative of the solution affects stability properties of numerical methods for neutral equations

a study of symbolic processing and computational aspects in helicopter dynamics even research models of helicopter dynamics often lead to a large number of equations of motion with periodic coefficients and floquet theory is a widely used mathematical tool for dynamic analysis presently three approaches are used in generating the equations of motion these are 1 general purpose symbolic processors such as reduce and macsyma 2 a special purpose symbolic processor dehim dynamic equations for helicopter interpretive models and 3 completely numerical approaches in this paper comparative aspects of the first two purely algebraic approaches are studied by applying reduce and dehim to the same set of problems these problems range from a linear model with one degree of freedom to a mildly non linear multi bladed rotor model with several degrees of freedom further computational issues in applying floquet theory are also studied which refer to 1 the equilibrium solution for periodic forced response together with the transition matrix for perturbations about that response and 2 a small number of eigenvalues and eigenvectors of the unsymmetric transition matrix the study showed the following 1 compared to reduce dehim is far more portable and economical but it is also less user friendly particularly during learning phases 2 the problems of finding the periodic response and eigenvalues are well conditioned

order stars and stability for delay differential equations we consider runge kutta methods applied to delay differential equations y t ay t by t 1 with real a and b if the numerical solution tends to zero whenever the exact solution does the method is called tau 0 stable using the theory of order stars we characterize high order symmetric methods with this property in particular we prove that all gauss methods are tau 0 stable furthermore we present sufficient conditions and we give evidence that also the radau methods are tau 0 stable we conclude this article with some comments on the case where a andb are complex numbers

stability of discretized partial element equivalent efie circuit models the instabilities associated with integral equation techniques for the solution of electromagnetic problems in the time domain are well known instabilities may be due to either the numerical technique used for the time integration or problems created by the discrete representation for the numerical solution of the problem in this paper we concentrate on the discretization issue the stability problem occurs for various discretizations and formulations here we use the partial element equivalent circuit peec formulation of the electric field integral equation efie in the circuit domain this leads to a better understanding of the issues at hand we show why the discretized model can be unstable and we suggest a circuit motivated technique to stabilize the solution

on the range and invertibility of a class of mellin multiplier transforms i abstract we study the range and invertibility in weighted versions of lp 0 of the operators nm and ml whose mellin multipliers are s m and s l respectively we deduce corresponding results for the laplace transform which generalise those of widder the paper is the first of a series concerning operators whose multipliers involve products and quotients of gamma functions and which include as special cases operators with g function kernels studied by rooney

spectral properties and their applications of frequency response operators in linear continuous time periodic systems article history received 25 july 2008 available online 11 december 2008 submitted by h zwart spectral properties related to the frequency response operators of finite dimensional linear continuous time periodic fdlcp systems are examined rigorously and thoroughly in the paper as applications of the spectral features positive realness of fdlcp systems is scrutinized and then a harmonic hamiltonian criterion is derived for the h norm of fdlcp systems in particular positive realness of fdlcp systems is interpreted in term of toeplitz operators for the first time in this study together with a testing algorithm deriving the harmonic hamiltonian criterion with a frequency domain approach bridges the spectra of the frequency response operators in fdlcp systems with their time domain behaviors

stability of the radau ia and lobatto iiic methods for delay differential equations recently we have proved that the radau ia and lobatto iiic methods are p stable i e they have an analogous stability property to a stability with respect to scalar delay differential equations ddes in this paper we study stability of those methods applied to multidimensional ddes we show that they have a similar property to p stability with respect to multidimensional equations which satisfy certain conditions for asymptotic stability of the zero solutions the conditions are closely related to stability criteria for ddes considered in systems theory

stability and accuracy of time discretizations for initial value problems this paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems here the treatment is extended to cover also implicit schemes and by placing the accuracy of the schemes into a more central position in the discussion general method free statements are again obtained more specialized results are additionally given for linear multistep methods and for the taylor series method

the test problem class of volterra functional differential equations in banach space abstract in this paper we introduce the test problem classes which are entitled d 1 2 and d 1 2 respectively with respect to the initial value problems of nonlinear volterra functional differential equations in banach spaces a series of stability results of the analytic solution are obtained and a condition estimate for the class d 0 1 2 which based on logarithmic matrix norm is also obtained the above results extend the existing results for ordinary differential equations

instability of runge kutta methods when applied to linear systems of delay differential equations this paper investigates the stability of runge kutta methods when they are applied to the complex linear system of delay differential equations y prime left t right ly left t right my left t 1 right where l m in mathbb c d times d we prove that no runge kutta method preserves asymptotic stability

mellin transforms of generalized fractional integrals and derivatives we obtain the mellin transforms of the generalized fractional integrals and derivatives that generalize the riemann liouville and the hadamard fractional integrals and derivatives we also obtain interesting results which combine generalized r m operators with generalized stirling numbers and lah numbers for example we show that 1 1 corresponds to the stirling numbers of the 2nd kind and 2 1 corresponds to the unsigned lah numbers further we show that the two operators r m and m r r m n generate the same sequence given by the recurrence relation s n k i 0 r m m r n 2 k i 1 r i r i s n 1 k i 0 k n with s 0 0 1 and s n 0 s n k 0 for n 0 and 1 min r m n 1 k or k 0 finally we define a new class of sequences for r 1 3 1 4 1 5 1 6 and in turn show that 1 2 1 corresponds to the generalized laguerre polynomials

regional observability for hadamard caputo time fractional distributed parameter systems abstract in this paper regional gradient exact and approximate observability problems are studied on hadamard caputo time fractional distributed parameter systems without any knowledge of the initial vector and its gradient several equivalent criteria are first provided to achieve the regional observability based on these characterizations for both strategic and gradient strategic zone sensors are developed then by employing the hilbert uniqueness method hum we explicitly reconstruct the initial vector and its gradient respectively a one dimension example is finally included to illustrate our results

delay dependent stability of high order runge kutta methods this paper is concerned with the study of the delay dependent stability of runge kutta methods for delay differential equations first a new sufficient and necessary condition is given for the asymptotic stability of analytical solution then based on this condition we establish a relationship between 0 stability and the boundary locus of the stability region of numerical methods for ordinary differential equations consequently a class of high order runge kutta methods are proved to be 0 stable in particular the 0 stability of the radau iia methods is proved

compositions of hadamard type fractional integration operators and the semigroup property abstract this paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of hadamard in the space xcp of functions f on r 0 such that 0 u c f u p du u ess sup u 0 u c f u for c r in particular in the space lp 0 1 p the semigroup property and its generalizations are established for the generalized hadamard type fractional integration operators under consideration conditions are also given for the boundedness in xcp of these operators they involve kummer confluent hypergeometric functions as kernels

a modified bessel type integral transform and its compositions with fractional calculus operators on spaces f p m and f p m abstract the paper is devoted to study the integral transform l f x 0 xt f t d t x 0 with the kernel z 1 1 1 t 1 1 t e zt d t for 0 re 1 1 r re z 0 which is a generalization of the modified bessel function of the third kind or macdonald function k z properties of z are investigated and compositions of the operator l with the left and right sided liouville fractional integrals and derivatives are proved

asymptotic stability in the numerical solution of linear pure delay differential equations as abstract cauchy problems abstract in bellen and maset preprint an approach is presented for solving the delay differential equation dde y t ly t my t t 0 y t t t 0 where 0 l m c m m and c 0 c m passing through a reformulation of as an abstract cauchy problem and its discretization in a system of ordinary differential equations odes in this paper we investigate the asymptotic stability of this approach on the class of ddes with l 0

asymptotic representations for hypergeometric bessel type function and fractional integrals the paper is devoted to the study of asymptotic relations for the function z 1 1 1 t 1 1 t e ztdt generalising tricomi confluent hypergeometric function and modified bessel function of the third kind the full asymptotic representations for z at zero and infinity are established applications are given to obtain full asymptotic expansions near zero and infinity for the liouville fractional integral i f x 1 x f t dt t x 1 x 0 c re 0 and for the erdelyi kober type fractional intergral i f x x x t 1 1 f t dt t x 1 x 0 c re 0 with 0 and c of power exponential function f t and for three other fractional integrals

infinite eigenvalue method for stability analyses of canonical linear systems with periodic coefficients this paper presents a technique that provides both necessary and sufficient conditions for stability of canonical linear differential equations with periodic coefficients and is well suited for numerical application existing methods of stability analyses not subject to small parameter restrictions such as the widely used floquet numerical integration method fnim and the infinite determinant methods provide necessary conditions but are impractical for providing sufficient conditions for this class of systems with multiple degrees of freedom furthermore the infinite determinant methods are restricted to only certain classes of linear periodic systems whereas the fnim is subject to numerical problems when dealing with canonical systems the method proposed in this paper referred to as the infinite eigenvalue method iem is based on floquet theory and the infinite determinant methods it is applicable to the general class of multiple degree of freedom linear canonical systems and is not subject to small parameter restrictions a brief review of floquet theory and the fnim is given the iem is then presented and applied to a spacecraft dynamics problem of current interest the fnim is applied to the same problem and the two methods are compared

on the range of the hankel and extended hankel transforms abstract the range of the hankel and extended hankel transforms on some spaces of functions is described the paley wiener theorem for the hankel transform is also obtained

a cutting plane algorithm for robustness analysis of periodically time varying systems an algorithm for robustness analysis of periodic systems is derived the system under consideration consists of a linear periodically time varying plant in feedback interconnection with a structured uncertainty conditions for robust stability and robust performance can be formulated in terms of periodic integral quadratic constraints iqcs in this way the robustness analysis becomes a problem of optimizing the parameters of the iqc a cutting plane algorithm is suggested for solving this infinite dimensional optimization problem

fractional calculus in the mellin setting and hadamard type fractional integrals abstract the purpose of this paper and some to follow is to present a new approach to fractional integration and differentiation on the half axis r 0 in terms of mellin analysis the natural operator of fractional integration in this setting is not the classical liouville fractional integral i 0 f but j 0 c f x 1 0 x u x c log x u 1 f u du u x 0 for 0 c r the mellin transform of this operator is simply c s m f s for s c it c t r the mellin transform of the associated fractional differentiation operator d 0 c f is similar c s m f s the operator d 0 c f may even be represented as a series in terms of xkf k x k n 0 the coefficients being certain generalized stirling functions sc k of second kind it turns out that the new fractional integral j 0 c f and three further related ones are not the classical fractional integrals of hadamard j mat pure appl ser 4 8 1892 101 186 but far reaching generalizations and modifications of these these four new integral operators are first studied in detail in this paper more specifically conditions will be given for these four operators to be bounded in the space xcp of lebesgue measurable functions f on 0 for c such that 0 u c f u p du u for 1 p ess sup u 0 u c f u for p in particular in the space lp 0 for 1 p connections of these operators with the liouville fractional integration operators are discussed the mellin convolution product in the above spaces plays an important role

a wavelet galerkin procedure to investigate time periodic systems transient vibration and stability analysis abstract a wavelet galerkin procedure is introduced in order to obtain transient and periodic solutions of multi degree of freedom d o f s dynamical systems with time periodic coefficients numerical comparisons achieved with a runge kutta method emphasize that the wavelet based procedure is reliable even in the case of problems involving both smooth or non smooth parametric excitations and a relatively large number of degrees of freedom the procedure is then applied to study the vibrations of some theoretical parametrically excited systems since problems of stability analysis of non linear systems are often reduced after linearization to problems involving linear differential systems with time varying coefficients the method is shown to be effective for the computation of the floquet exponents that characterize stable unstable parameters areas and consequently allows estimators for stability instability levels to be provided stability diagrams of some theoretical examples including a single degree of freedom mathieu oscillator and a two degree of freedom parametrically excited system illustrate the relevance of the method finally future studies are outlined for the extension of the wavelet method to the non linear case

mellin transform analysis and integration by parts for hadamard type fractional integrals abstract this paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of hadamard in the space xpc of lebesgue measurable functions f on r 0 such that 0 u c f u p du u ess sup u 0 u c f u for c r in particular in the space lp 0 1 p formulas for the mellin transforms of the four hadamard type fractional integral operators are established as well as relations of fractional integration by parts for them

weak delay dependent stability of runge kutta methods for differential systems with distributed delays abstract this paper discusses the stability of runge kutta methods for differential systems with distributed delays weak delay dependent stability of pouzet type runge kutta methods is studied in this paper it is revealed that under suitable conditions runge kutta methods preserve weak delay dependent stability an algorithm is employed to check weak delay dependent stability of numerical solutions and the effectiveness of the derived theoretical results is demonstrated with numerical examples

stability of explicit and diagonal implicit runge kutta methods for nonlinear volterra functional differential equations in banach spaces this paper is concerned with the stability properties of explicit and diagonal implicit runge kutta methods when applied to initial value problems of nonlinear volterra functional differential equations in banach spaces a series of new stability results of explicit and diagonal implicit runge kutta methods are obtained

robustness analysis of linear periodic time varying systems subject to structured uncertainty in this paper we show how floquet theory may be combined with a technique known as lifting to cast a linear periodically time varying system subject to structured linear time invariant uncertainty in the form of a linear fractional transformation the stability and performance robustness of the resulting system may then be analysed using standard analysis methods a significant advantage of the proposed approach is that it allows the computation of a worst case destabilising uncertainty combination which may be used to estimate the conservatism of the computed robustness margin an example is given to illustrate the application of the proposed approach

on integral transformations with g function kernels integral transformations t of the formal form where g is a meijer g function are studied on the spaces r of functions f such that results are obtained on the boundedness representation and range of the transformations and on when they have an inverse of the same formal form

stability analysis of parametrically excited systems using spectral collocation abstract the spectral collocation method is used to determine the stability of parametrically excited systems and compared with the traditional transition matrix approach results from a series of test problems demonstrate that spectral collocation converges rapidly in addition the spectral collocation method preserves the sparsity of the underlying system matrices a property not shared by the transition matrix approach as a result spectral collocation can be used for very large systems and can utilize sparse eigensolvers to reduce computational memory and time for the large scale system studied up to 40 degrees of freedom the spectral collocation method was on average an order of magnitude faster than the transition matrix approach using matlab this computational advantage is implementation specific in a c implementation of the algorithm the transition matrix method is faster than the spectral collocation overall the method proves to be simple efficient reliable and generally competitive with the transition matrix method

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

method of lines approximations of delay differential equations we consider a special type of numerical methods for delay differential equations ddes by introducing a new independent variable an initial value problem for ddes is converted into an initial boundary value problem for the convection equation thus it is also possible to get an approximate solution to the dde problem by solving the initial boundary value problem with a suitable numerical method instead of solving the original problem in this paper we study a family of method of lines mol approximations to the problem which is obtained by applying runge kutta rk methods for space discretization and prove their convergence under the assumption that the rk methods satisfy a condition known as an algebraic characterization of a stability the result is also confirmed by numerical experiments moreover we show that the condition derives several stability properties of the mol approximations

stability and asymptotic stability of θ methods for nonlinear stiff volterra functional differential equations in banach spaces this paper is concerned with the stability and asymptotic stability of q methods for the initial value problems of nonlinear stiff volterra functional differential equations in banach spaces a series of new stability and asymptotic stability results of q methods are obtained

asymptotic stability barriers for natural runge kutta processes for delay equations this paper investigates the asymptotic stability properties of a class of numerical methods for delay differential equations ddes the so called natural runge kutta methods we first examine the behavior of these methods when applied to the neutral model equation y t a y t b y t 1 c y t 1 with a b c in mathbb r we also consider the case when a b c in mathbb c and provide a suitable geometric characterization of their asymptotic stability regions then by means of the obtained results in conjunction with those given in n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 we are able to give a final answer concerning the possible preservation of asymptotic stability of the considered class of methods when applied to systems of linear ddes of the form y t l y t m y t 1 with l m in mathbb r d times d d 1 we are interested here in methods that produce stable numerical solutions for all values of the parameters a b and c in the first equation and l and m in the second one for which the exact solution tends to zero to this aim we direct our attention to a novel stability setting recently introduced and investigated for the scalar nonneutral case see n guglielmi numer math 77 1997 pp 467 485 n guglielmi ima j numer anal 18 1998 pp 399 418 n guglielmi and e hairer numer math 83 1999 pp 371 383 n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 s maset numer math 87 2000 pp 355 371

stability of runge kutta methods for linear delay differential equations this paper investigates the stability of runge kutta methods when they are applied to the complex linear scalar delay differential equation y prime left t right ay left t right by left t 1 right this kind of stability is called tau stability we give a characterization of tau stable runge kutta methods and then we prove that implicit euler method is tau stable

delay dependent stability of symmetric schemes in boundary value methods for ddes we consider the symmetric schemes in boundary value methods bvms applied to delay differential equations y t ay t by t t with real coefficients a and b if the numerical solution tends to zero whenever the exact solution does the symmetric scheme with k 1 m k 2 boundary conditions is called t k 1 k 2 0 stable three families of symmetric schemes namely the extended trapezoidal rules of first etrs and second etr 2s kind and the top order methods toms are considered in this paper by using the boundary locus technology the delay dependent stability region of the symmetric schemes are analyzed and their boundaries are found then by using a necessary and sufficient condition the considered symmetric schemes are proved to be t n n 1 0 stable

new approach to a generalized fractional integral abstract the paper presents a new fractional integration which generalizes the riemann liouville and hadamard fractional integrals into a single form conditions are given for such a fractional integration operator to be bounded in an extended lebesgue measurable space semigroup property for the above operator is also proved we give a general definition of the fractional derivatives and give some examples

an analysis of delay dependent stability for ordinary and partial differential equations with fixed and distributed delays this paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays and with the study of the stability of discretizations of such differential equations we start with a delay dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients we study the exact stability region of the continuous problem as a function of the parameters of the model next it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems in the second part of the paper we study delay partial differential equations the stability region of the fully continuous problem is analyzed first then a semidiscretization in space is applied it is shown that the spatial discretization leads to a reduction of the stability region when the standard second order central difference operator is employed to approximate the diffusion operator finally we consider the delay dependent stability of the fully discrete problem where the partial differential equation is discretized both in space and in time some numerical examples and further discussions are given