spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

stieltjes derivatives and b polynomial spline collocation for volterra integrodifferential equations with singularities in this paper we introduce a new tool the stieltjes derivative and new kinds of meshes which are more practical than the graded meshes based on these we discuss a nonpolynomial spline collocation method for volterra integrodifferential equations with weakly singular kernels under very weak hypotheses which allow the nonsmooth behaviour of the given functions in these equations collocation on such meshes is shown to yield optimal convergence rates an ideal super convergence result is also obtained

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

optimal error estimates of spectral petrov galerkin and collocation methods for initial value problems of fractional differential equations we present optimal error estimates for spectral petrov galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval we also develop laguerre spectral petrov galerkin methods and collocation methods for fractional equations on the half line numerical results confirm the error estimates

numerical solution of volterra fredholm integral equations using the collocation method based on a special form of the muntz legendre polynomials abstract this paper presents a computational technique based on a special family of the muntz legendre polynomials to solve a class of volterra fredholm integral equations the relationship between the jacobi polynomials and muntz legendre polynomials in a particular state are expressed the proposed method reduces the integral equation into algebraic equations via the chebyshev gauss lobatto points so that the system matrix coefficients are obtained by the least squares approximation method the useful properties of the jacobi polynomials are exploited to analyze the approximation error the performance and accuracy of our method are examined with some illustrative 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

taylor collocation method and convergence analysis for the volterra fredholm integral equations in this paper the taylor collocation method is presented for numerically solving the volterra fredholm integral equations in terms of taylor polynomials this method transforms the integral equation to a matrix equation via the collocation points the convergence analysis of the collocation solution on the integral equation is given the method is computationally attractive and applications are demonstrated through illustrative examples

an efficient method to determine the diagonal loading factor using the constant modulus feature the diagonal loading method is a simple and efficient method that improves the robustness of beamformers however determining an ideal diagonal loading factor dlf is not a trivial problem one that still has not been adequately addressed although beamformers are widely used in radar sonar and many other applications in this correspondence we consider a beamformer used only for digital communication applications in digital communications user information is coded into a sequence with the constant modulus cm in the baseband obviously if the beamformer can cancel interference sufficiently well the outputs of the beamformer should satisfy the cm condition more precisely therefore minimizing cm errors can act as a natural criterion for determining dlfs in this correspondence we present a cost function based on the criterion we also propose a direct search algorithm for dlf determination by minimizing the cost function numerical experiments show the effectiveness of the proposed method

convergence analysis of the multistep legendre pseudo spectral method for volterra integral equations with vanishing delays in this paper we extend the single step pseudo spectral method for second kind volterra integral equations with vanishing variable delays to the multistep pseudo spectral method we also analyze the convergence of the hp version of the multistep pseudo spectral method under the l2 norm and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner in addition it is very suitable for long time calculations and large step size situations numerical results show good agreement with the theoretical analysis

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

discrete time orthogonal spline collocation method with application to two dimensional fractional cable equation discrete time orthogonal spline collocation osc methods are presented for the two dimensional fractional cable equation which governs the dynamics of membrane potential in thin and long cylinders such as axons or dendrites in neurons the proposed scheme is based on the osc method for space discretization and finite difference method for time which is proved to be unconditionally stable and convergent with the order o min 2 1 2 2 h r 1 in l 2 norm where h and r are the time step size space step size and polynomial degree respectively and 1 and 2 are two different exponents of fractional derivatives with 0 1 2 1 numerical experiments are presented to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the method described herein and super convergence phenomena at the partition nodes is also exhibited which is a characteristic of the osc methods namely the rates of convergence in the maximum norm at the partition nodes in u x and u y are approximately h r 1 in our numerical experiment

an extension of the spectral tau method for numerical solution of multi order fractional differential equations with convergence analysis the main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations fdes based on a spectral tau method an extension of the operational approach of the tau method with the orthogonal polynomial bases is proposed to convert fdes to its matrix vector multiplication representation the fractional derivatives are described in the caputo sense the spectral rate of convergence for the proposed method is established in the l 2 norm we tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence

a multistep legendre gauss spectral collocation method for nonlinear volterra integral equations we introduce a multistep legendre gauss spectral collocation method for the nonlinear volterra integral equations of the second kind this method is easy to implement and possesses high order accuracy in addition it is very suitable for long time calculations we also derive the optimal convergence of the hp version of the multistep collocation method under the l 2 norm numerical experiments confirm the theoretical expectations

high performance parallel numerical methods for volterra equations with weakly singular kernels non stationary discrete time waveform relaxation methods for abel systems of volterra integral equations using fractional linear multistep formulae are introduced fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed a significant expression of the error is proved which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance the numerical experiments confirm the theoretical expectations

meshless method and convergence analysis for 2 dimensional fredholm integral equation with complex factors in this paper the meshless method is presented for numerically solving the 2 dimensional fredholm integral equation with complex factors first the existence and uniqueness of solution are proved by the fixed point theorem second the approximation solution method of the radial basis function interpolation is given the results of convergence analysis and error estimation are also obtained some numerical examples are presented to verify the validity and applicability of the method

solution of weakly singular fractional integro differential equations by using a new operational approach abstract in this study an operational approach based on the shifted jacobi polynomials is presented to solve a class of weakly singular fractional integro differential equations the fractional derivative operators are considered as the caputo sense in addition to finding the operational matrices of integration and product a new operational matrix is derived to be approximated the singular kernels of this class of the functional equations these operational matrices together with the collocation method are applied to convert the main equation into a linear or non linear system of algebraic equations by presenting some theorems conditions of the existence and uniqueness of the solution of the main equation and its corresponding algebraic system are investigated also the stability and convergence of the proposed method are studied and an error bound is obtained in a jacobi weighted sobolev space finally some illustrative examples are provided to demonstrate the efficiency and simplicity of the suggested approach

least squares approximation method for the solution of hammerstein volterra delay integral equations in this paper an efficient numerical method is developed for solving the hammerstein volterra delay integral equations by least squares ls approximation method which is based on a polynomial of degree n to compute an approximation to the solution of hammerstein volterra delay integral equations the convergence analysis of the approximation solution relative to the exact solution of the integral equation is proved and its accuracy is illustrated on two numerical examples the study of this integral equation is important because it has as a particular case the variant of a mathematical model from epidemiology

partial realization for singular systems in standard form abstract the partial realization problem under consideration consists in finding for a given sequence s s k 0 n 1 of blocks matrices a e b c of appropriate size such that s i ce n 1 i a i b and the identity matrix is a linear combination of a and e we discuss the question whether there is always a realization of this form for which the state space dimension is equal to the maximal rank of the underlying hankel matrices we show that this question has an affirmative answer if the block size is less than or equal to 2 and some other cases but not in general the paper strengthens results obtained by manthey et al cf w manthey u helmke d hinrichsen in u helmke et al eds operators systems and linear algebra teubner stuttgart 1997 pp 138 156 the main tools are the results of the authors obtained in connection with vandermonde factorization of block hankel matrices finally an interpretation of the problem in periodic discrete time systems is given

spectral and discontinuous spectral element methods for fractional delay equations we first develop a spectrally accurate petrov galerkin spectral method for fractional delay differential equations fddes this scheme is developed based on a new spectral theory for fractional sturm liouville problems fslps which has been recently presented in m zayernouri and g e karniadakis j comput phys 252 2013 pp 495 517 specifically we obtain solutions to fddes in terms of new nonpolynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of the first kind fslp i correspondingly we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of the second kind fslp ii we prove the wellposedness of the problem and carry out the corresponding stability and error analysis of the pg spectral method in contrast to standard nondelay fractional differential equations the delay character of fddes might induce solutions which are either nonsmooth or piecewise smooth in order to effectively trea

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

recovery of high order accuracy in jacobi spectral collocation methods for fractional terminal value problems with non smooth solutions abstract an open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high order accuracy for non smooth solutions the limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes in this paper we derive and analyze an exponentially accurate jacobi spectral collocation method for the numerical solution of nonlinear terminal value problems involving the caputo fractional derivative of rational order 0 1 the main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate we systematically describe the necessary steps in the implementation process additionally a comprehensive numerical analysis including error estimates under the l and weighted l 2 norms is derived the extensive numerical examples that accompany our analysis confirm our theoretical estimates as well as give additional insights into the convergence behavior of our method for problems with smooth and non smooth solutions

integral fractional pseudospectral methods for solving fractional optimal control problems the main purpose of this work is to provide a unified framework and develop integral fractional pseudospectral methods for solving fractional optimal control problems as a generalization of conventional pseudospectral integration matrices fractional pseudospectral integration matrices fpims and their efficient and stable computation are the key to our new approach in order to achieve this goal we take a special and smart way to compute fpims the essential idea is to transform the fractional integral of lagrange interpolating polynomials through a change of variables into their jacobi weighted integral which can be calculated exactly using the jacobi gauss quadrature this together with the stable barycentric representation of lagrange interpolating polynomials and the explicit barycentric weights for the gauss flipped radau and radau type points corresponding to the jacobi polynomials leads to an exact efficient and stable scheme to compute fpims even at millions of jacobi type points numerical results on two benchmark optimal control problems demonstrate the performance of the proposed pseudospectral methods

a parallel algorithm for large systems of volterra integral equations of abel type a significative number of recent applications require numerical solution of large systems of abel volterra integral equations here we propose a parallel algorithm to numerically solve a class of these systems designed for a distributed memory mimd architecture in order to achieve a good efficiency we employ a fully parallel and fast convergent waveform relaxation wr method and evaluate the lag term by using fft techniques to accelerate the convergence of the wr method and to best exploit the parallel architecture we develop special strategies the performances of the resulting code nswr4 are illustrated on some examples

spectral collocation methods for volterra integro differential equations with noncompact kernels the aim of this paper is to analyze spectral collocation methods for a class of volterra integro differential equations with noncompact kernels the spectral collocation method is proposed to solve the equations after the results on existence uniqueness and regularity of the solutions are established the convergence rates of the methods are proven and a numerical example is provided to verify the theoretical findings

operational tau method for singular system of volterra integro differential equations the legendre spectral tau matrix formulation is proposed to approximate solution of singular system of volterra integro differential equations the existence and uniqueness solution of this system are investigated by means of the smoothing property of a volterra integral operator and some projectors the l 2 convergence of the numerical method is analyzed it is proved theoretically and demonstrated numerically that the proposed method converges exponentially finally two numerical examples illustrate the theoretical results

crank nicolson and legendre spectral collocation methods for a partial integro differential equation with a singular kernel abstract in this paper we present an efficient numerical method for the solution of a partial integro differential equation with a singular kernel in the time direction a crank nicolson finite difference scheme is used to approximate the differential term and the product trapezoidal method is employed to treat the integral term also for space discretization we apply legendre spectral collocation method we discuss the stability and convergence of proposed method and show that the method is unconditionally stable and convergent with order o 3 2 n s where n and s are time step size number of collocation points and regularity of exact solution respectively we compare the numerical results of proposed method with the results of other schemes in the literature in terms of accuracy computational order and cpu time to show the efficiency and applicability of it

convergence analysis of the jacobi collocation method for nonlinear weakly singular volterra integral equations in this work we present an efficient spectral collocation method for numerical solution of a class of nonlinear weakly singular volterra integral equations this type of equations typically has a singular behavior at the left endpoint of the interval of integration for overcoming this non smooth behavior we apply the jacobi collocation method the convergence analysis of the proposed method is investigated in the l and the weighted l2 norms and the results of several numerical experiments are presented which support the theoretical results the computed results are compared wherever possible with those already available in the literature

a new strategy for the numerical solution of nonlinear volterra integral equations with vanishing delays abstract the aim of this paper is to develop a high order algorithm for nonlinear volterra integral equations with vanishing variable delays the algorithm is a non trivial extension of the single step methods this method is based on the variational form so the algorithm is relatively simple this scheme enjoys high order accuracy and can be implemented in a stable and efficient manner we prove the existence and uniqueness of the numerical solution and derive error estimates of the algorithm numerical results show a good agreement with the theoretical analysis

generalized laguerre pseudospectral method based laguerre interpolation abstract in this paper we develop the generalized laguerre pseudospectral method for the second order ordinary differential equation based on the modified generalized laguerre approximation which is very efficient for the long time numerical simulations of dynamical systems the numerical solution possesses the spectral accuracy the global convergence of the proposed algorithm is proved numerical results demonstrate the effectiveness of the proposed method and coincide well with theoretical analysis

the semi explicit volterra integral algebraic equations with weakly singular kernels the numerical treatments this paper deals with some theoretical and numerical results for volterra integral algebraic equations iaes of index 1 with weakly singular kernels this type of equations typically has solutions whose derivatives are unbounded at the left endpoint of the interval of integration for overcoming this non smooth behavior of solutions using the appropriate coordinate transformation the primary system is changed into a new iaes which its solutions have better regularity an effective numerical method based on the chebyshev collocation scheme is designed and its convergence analysis is provided our numerical experiments show that the theoretical results are in good accordance with actual convergence rates obtained by the given algorithm

legendre spectral collocation method for fredholm integro differential difference equation with variable coefficients and mixed conditions in this article the legendre spectral collocation method has been applied to solve fredholm integro differential difference equations with variable coefficients the proposed method is based on the gauss legendre points with the basis functions of lagrange polynomials usually this type of integral equations are very difficult to solve analytically as well as numerically the presented method applied to the integral equation reduces to solve the system of algebraic equations also the numerical results obtained by legendre spectral collocation method have been compared with the results obtained by existing methods illustrative examples have been discussed to demonstrate the validity and applicability of the presented method

stable evaluations of fractional derivative of the muntz legendre polynomials and application to fractional differential equations abstract the aim of this paper is to present efficient and stable methods to compute caputo fractional derivative cfd of the muntz legendre polynomials based on three term recurrence relations and gauss jacobi quadrature rules this approach with collocation method at chebyshev gauss lobatto points has been applied for solving linear and nonlinear fractional multi order differential equations fdes described in caputo sense the main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations in this work for the first time we present the new rates of convergence for projection error which are more accurate than the rate presented by shen and wang in shen and wang 2016 moreover we present convergence rate for spectral collocation method for linear fdes with initial value on a finite interval and endpoint singularities also we propose an error analysis for jacobi gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper finally the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

a multistep legendre pseudo spectral method for volterra integral equations in this paper we extend the single step pseudo spectral method for volterra integral equations of the second kind to the multistep pseudo spectral method we also analyze the convergence of the hp version of the multistep pseudo spectral method under the l2 norm and show that the scheme enjoys high order accuracy and can be implemented in a stable and efficient manner in addition it is very suitable for long time calculations and large step size situations numerical experiments confirm theoretical expectations

superconvergence of legendre projection methods for the eigenvalue problem of a compact integral operator in this paper we consider the galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the legendre polynomials of degree n we prove that the error bounds for eigenvalues are of the order o n 2 r and the gap between the spectral subspaces are of the orders o n r in l 2 norm and o n 1 2 r in the infinity norm where r denotes the smoothness of the kernel by iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence o n 2 r in both l 2 norm and infinity norm we illustrate our results with numerical examples

bernstein series solutions of multidimensional linear and nonlinear volterra integral equations with fractional order weakly singular kernels abstract this paper proposes a quadrature method based on multi variate bernstein polynomials the method is used to solve multidimensional volterra integral equations with weakly singular kernels firstly we use multi variate bernstein polynomials to approximate the unknown function of an equation then a discrete function equation can be obtained by substituting the approximate solution into the equation secondly the discrete function system is transformed into an algebra equation system by using some discrete points we can perform the integral operations without discrete kernel function and the weakly singular integrals can be calculated directly by using quadrature method so the method is easy to implement thirdly we prove the existence and uniqueness of the solution of the approximate equation as well as the error analysis of the proposed method six numerical examples are given to illustrate the efficiency of this method

uniqueness properties of minimal partial realizations abstract we characterize quantities poles and pole chains of the transfer functions of the minimal partial realizations of a given sequence of matrices s s0 s1 sn 1 which are uniquely determined by s furthermore the set of all points at which the transfer functions of all minimal partial realizations are analytic is described

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

numerical solutions for fredholm integral equations of the second kind with weakly singular kernel using spectral collocation method abstract we investigate the spectral collocation for fredholm integral equations of the second kind with weakly singular kernel the jacobi gauss quadrature formula is used to approximate the integral operator in the numerical implementation we obtain the convergence rates for the approximated solution of weakly singular fredholm integral equations which show that the errors of the approximate solution decay exponentially in l norm and weighted l2 norm some numerical examples are given to illustrate the theoretical results

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

convergence analysis of spectral collocation methods for a class of weakly singular volterra integral equations abstract in this paper we discuss the application of spectral jacobi collocation methods to a certain class of weakly singular volterra integral equations first we use some function transformations and variable changes to transform the equation into a volterra integral equation defined on the standard interval 1 1 then the jacobi gauss quadrature formula is used to approximate the integral operator for the spectral jacobi collocation method a rigorous error analysis in both the l and weighted l 2 norms is given under the assumption that both the kernel function and the source function are sufficiently smooth finally some numerical examples are provided to illustrate the theoretical results

on exponential convergence of gegenbauer interpolation and spectral differentiation this paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the gegenbauer gauss and gegenbauer gauss lobatto points when the underlying function is analytic on and within an ellipse sharp error estimates in the maximum norm are derived

shifted jacobi spectral galerkin method for solving fractional order initial value problems abstract in this paper the shifted jacobi spectral galerkin method is introduced to deal with fractional order initial value problems fivps in the proposed method the exact solution of the fivp is approximated by using shifted jacobi polynomials on each subinterval of the total time the main advantage of the proposed method is that the rate of convergence in l 2 norm depends on the local smoothness of solution this enables the proposed method to work for nonsmooth solutions the second merit is that the flexibility between the length of sub interval and the degree of polynomial significantly enhances the numerical accuracy we derive the spectral galerkin approximation formula for the volterra integral equation of the second kind which is equivalent to the fivp error analysis in l 2 norm for the case 0 in shifted jacobi polynomials is provided and it justifies the spectral rate of convergence with respect to both the degree and the length of the subinterval when the source function is lipschitz continuous in the second argument numerical illustrations for multi order linear and nonlinear fivps are demonstrated to confirm the convergence rate

spectral collocation method for the time fractional diffusion wave equation and convergence analysis in this paper we consider the numerical solution of the time fractional diffusion wave equation essentially the time fractional diffusion wave equation differs from the standard diffusion wave equation in the time derivative term we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

collocation methods for nonlinear convolution volterra integral equations with multiple proportional delays in this paper we apply the collocation methods to a class of nonlinear convolution volterra integral equations with multiple proportional delays ncviempds we shall present the existence uniqueness and regularity properties of analytic solution for this type equation and then analyze the convergence and superconvergence properties of the collocation solution the numerical results verify our theoretical analysis

further study on robust adaptive beamforming with optimum diagonal loading significant effort has gone into designing robust adaptive beamforming algorithms to improve robustness against uncertainties in array manifold these uncertainties may be caused by uncertainty in direction of arrival doa imperfect array calibration near far effect mutual coupling and other mismatch and modeling errors a diagonal loading technique is obligatory to fulfil the uncertainty constraint where the diagonal loading level is amended to satisfy the constrained value the major drawback of diagonal loading techniques is that it is not clear how to get the optimum value of diagonal loading level based on the recognized level of uncertainty constraint in this paper an alternative realization of the robust adaptive linearly constrained minimum variance beamforming with ellipsoidal uncertainty constraint on the steering vector is developed the diagonal loading technique is integrated into the adaptive update schemes by means of optimum variable loading technique which provides loading on demand mechanism rather than fixed continuous or ad hoc loading we additionally enrich the proposed robust adaptive beamformers by imposing a cooperative quadratic constraint on the weight vector norm to overcome noise enhancement at low snr several numerical simulations with doa mismatch moving jamming and mutual coupling are carried out to explore the performance of the proposed schemes and compare their performance with other traditional and robust beamformers

superconvergence of numerical solutions to weakly singular volterra integro differential equations we discuss the application of a class of spline collocation methods to first order volterra integro differential equations vides which contain a weakly singular kernel t s with 0 1 it will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and graded meshes the grading exponents of graded meshes used are not greater thanm the polynomial degree which is independent of this is in contrast to the theories of spline collocation methods for volterra or fredholm integral equation of the second kind numerical examples are given to illustrate the theoretical results

numerical solution of nonlinear weakly singular partial integro differential equation via operational matrices in this paper we propose and analyze an efficient matrix method based on shifted legendre polynomials for the solution of non linear volterra singular partial integro differential equations pides the operational matrices of integration differentiation and product are used to reduce the solution of volterra singular pides to the system of non linear algebraic equations some useful results concerning the convergence and error estimates associated to the suggested scheme are presented illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method

error estimates of generalized spectral iterative methods with accurate convergence rates for solving systems of fractional two point boundary value problems abstract the main purpose of this study is to provide an efficient spectral iterative method based on fractional interpolants for solving a class of linear and nonlinear fractional two point boundary value problems involving left and right fractional derivatives in order to achieve this goal by composition of iterative method and spectral method in each step the unknown solution of system is expanded by using two classes of generalized jacobi functions to obtain numerically coefficients for this system of fractional differential equations on the interval 1 1 the singularities of the solution are considered of types 1 t and 1 x where 0 l 2 norm finally some numerical results are given to demonstrate the effectiveness and applicability of the proposed method and accuracy of the presented convergence rates

vandermonde factorization and canonical representations of block hankel matrices abstract we study to which extent well known facts concerning vandermonde factorization or canonical representation of scalar hankel matrices transfer to block hankel matrices with p q blocks it is shown that nonsingular block hankel matrices can be factored like in the scalar case into nonconfluent vandermonde matrices and that the theorem on full rank factorization of arbitrary hankel matrices transfers in a weak version to the 2 2 block case but not to larger block sizes in general the minimal rank of a vandermonde factorization both with finite nodes and affine is described in terms of the hankel matrix the main tools are realization partial realization and moebius transformations

error estimate in fractional differential equations using multiquadratic radial basis functions the purpose of this paper is to investigate the collocation method based on multiquadric mq radial basis functions rbfs for fractional differential equations fdes in the process of doing this a scattered data rbf interpolation is considered a posterior error bound for the smooth solutions of fdes in the proposed method is established in the l 2 norm using an alternate characterization of the native space numerical results are presented which confirm the theoretical prediction of the convergence behavior of the proposed method

legendre spectral collocation method for the solution of the model describing biological species living together in this article numerical techniques have been developed to solve a system of two nonlinear integro differential equations which arises in biological model this model has been solved by legendre spectral collocation method and bernstein polynomials the present methods reduce this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by newton s method also the results obtained by present methods have been compared with that of by some existing methods available in literature some illustrative examples have been discussed to show the applicability and accuracy of the present methods

an efficient and fast parallel method for volterra integral equations of abel type in this paper we present an efficient and fast parallel waveform relaxation method for volterra integral equations of abel type obtained by reformulating a nonstationary waveform relaxation method for systems of equations with linear coefficient constant kernel to this aim we consider the laplace transform of the equation and here we apply the recurrence relation given by the chebyshev polynomial acceleration for algebraic linear systems back in the time domain we obtain a three term recursion which requires at each iteration the evaluation of convolution integrals where only the laplace transform of the kernel is known for this calculation we can use a fast convolution algorithm numerical experiments have been done also on problems where it is not possible to use the original nonstationary method obtaining good results in terms of improvement of the rate of convergence with respect the stationary method

partial realization of descriptor systems abstract we address the partial realization problem for linear descriptor systems a general solution to this problem using the markov parameters of the system defined via its laurent series is provided for proper descriptor systems we also discuss numerically feasible algorithms for computing a partial minimal realization based on the unsymmetric lanczos and arnoldi processes applications to model reduction for examples from computational fluid dynamics and mechanical systems with holonomic constraints are also given

a generalized block by block method for solving linear volterra integral equations abstract one of the numerical methods for solving linear volterra integral equations is block by block method which is explained in l m delves j l mohamed computational methods for integral equations cambridge university press 1985 l m delves j walsh numerical solution of integral equations oxford university press 1974 and p k kyte p puri computational methods for linear integral equations birkhauser boston 2002 in this article we explain a general method for constructing block by block systems for solving volterra integral equations then we deduce some of the special cases especially the linz s block by block method which explained in these references

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

parallel in time algorithm with spectral subdomain enhancement for volterra integral equations nsf of china 11201077 11071203 91130002 nsf of fujian province 2012j01007 fuzhou university 0460022456 hong kong research grant council gif grants hong kong baptist university frg grants

application of fibonacci collocation method for solving volterra fredholm integral equations in this paper a new matrix method based on fibonacci polynomials and collocation points is proposed for numerically solving the volterra fredholm integral equations in fact the approximate solution of the problem in the truncated fibonacci series form is obtained by this method also convergence analysis of the proposed method is provided under several mild conditions the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments

a spectral collocation method for multidimensional nonlinear weakly singular volterra integral equation abstract this paper is concerned with the convergence properties of chebyshev spectral collocation method when used to approximate the solution of multidimensional nonlinear volterra integral equation of the second kind with a weakly singular kernel we consider the case that the underlying solution is sufficiently smooth the chebyshev collocation discretization is proposed for this equation in the present paper we provide a rigorous error analysis which justifies that the errors of approximate solution decay exponentially in weighted l 2 norm and l norm numerical results are presented to demonstrate the effectiveness of the spectral method

an h p version of the continuous petrov galerkin finite element method for volterra integro differential equations with smooth and nonsmooth kernels we present an h p version of the continuous petrov galerkin cpg finite element method for linear volterra integro differential equations with smooth and nonsmooth kernels we establish a priori error estimates in the l 2 h 1 and l infty norms that are completely explicit with respect to the local discretization and regularity parameters for singular solutions caused by the weakly singular kernels we prove that algebraic convergence rates can be achieved for the h p version of the cpg method with quasi uniform meshes moreover we show that exponential rates of convergence can be achieved for solutions with start up singularities by using geometric time partitions and linearly increasing polynomial degrees numerical experiments are provided to illustrate the theoretical results

spectral approximation methods and error estimates for caputo fractional derivative with applications to initial value problems abstract in this paper we revisit two spectral approximations including truncated approximation and interpolation for caputo fractional derivative the two approaches have been studied to approximate riemann liouville r l fractional derivative by chen et al and zayernouri et al respectively in their most recent work for truncated approximation the reconsideration partly arises from the difference between fractional derivative in r l sense and caputo sense caputo fractional derivative requires higher regularity of the unknown than r l version another reason for the reconsideration is that we distinguish the differential order of the unknown with the index of jacobi polynomials which is not presented in the previous work also we provide a way to choose the index when facing multi order problems by using generalized hardy s inequality the gap between the weighted sobolev space involving caputo fractional derivative and the classical weighted space is bridged then the optimal projection error is derived in the non uniformly jacobi weighted sobolev space and the maximum absolute error is presented as well for the interpolation analysis of interpolation error was not given in their work in this paper we build the interpolation error in non uniformly jacobi weighted sobolev space by constructing fractional inverse inequality with combining collocation method the approximation technique is applied to solve fractional initial value problems fivps numerical examples are also provided to illustrate the effectiveness of this algorithm

legendre multi projection methods for solving eigenvalue problems for a compact integral operator in this paper we consider the m galerkin and m collocation methods for solving the eigenvalue problem for a compact integral operator with smooth kernels using legendre polynomial bases we obtain error bounds for the eigenvalues and the gap between the spectral subspaces in both legendre m galerkin and legendre m collocation methods we also obtain superconvergence results for the eigenvalues and iterated eigenvectors in both l 2 and infinity norm we illustrate our results with numerical examples

kernel structure of block hankel and toeplitz matrices and partial realization abstract the kernel structure of block hankel and toeplitz matrices is studied this leads to the concept of a fundamental system which is basic in the theory and provides in particular a natural background for the partial realization problem

on recursiveness and related topics in linear systems a decomposition of matrix formal power series linear systems is developed the interconnections between formal power series realization pade approximation recursiveness berlekamp massey algorithm feedback interconnections matrix linear fractional representations matrix continued fractions and the matrix euclidean algorithm are thereby revealed

collocation methods for second kind volterra integral equations with weakly singular kernels in this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of volterra integral equations vies are smooth if the exact solutions are not smooth however suitable transformations can be made so that the new vies possess smooth solutions spline collocation methods with uniform meshes applied to these new vies are then shown to be able to yield optimal global convergence rates the general theory is applied to a typical case i e the integral kernels consisting of the singular term t s

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

on minimality in the partial realization problem abstract a compression algorithm is constructed which allows one to reduce an arbitrary realization of a finite sequence of matrices m 1 m r to a minimal partial realization of m 1 m r furthermore a state space criterion of minimality of a partial realization and a formula for the minimal state space dimension are obtained

equivalence of adaptive array diagonal loading and omnidirectional jamming diagonal loading where a diagonal matrix is added to the estimate of the covariance matrix is often used in adaptive arrays to minimize the array s responsiveness to random fluctuations in the estimate of the interference environment poor estimates of the covariance matrix often result in high random sidelobes in the sample matrix inversion smi adapted pattern it is shown in this communication that an omnidirectional spread of uniform jammers arriving at a linear array of half wavelength spaced elements with cosine element patterns also produces a diagonal matrix this equivalence of form is useful in visualizing the effects of diagonal loading and understanding the resulting patterns

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method