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• For the biharmonic eigenvalue problem, Wang et al. [Adv. Comput. Math. 45(2019)] first analyzed the mixed DG method (MDG method) under the condition that the eigenfunction u and Δ u. have the same regularity. In this paper, we establish an MDG discretization which is different with that in the existing work and we conduct our analysis under the condition that u and Δ u have different regularities. • The existing literature derived the error estimate of biharmonic eigenvalue by using the estimate of Laplacian eigenvalue problem. In this paper, we establish an identity (see Lemma 3.5) and use it to get the a priori error estimate of biharmonic eigenvalue. We prove that the convergence rate of approximate eigenvalues reaches 2p-2 in h and p − (2 s − 7) in when p ≥ 2 and the eigenfunction u ∈ H S (Ω) (s ≥ p + 1) which is consistent with the estimate in the existing work [Babuska and Osborn, Eigenvalue problems, 1991] for the classical mixed finite element method for biharmonic eigenvalue problem. • Using the identity (see Lemma 3.5) we also discuss the a posteriori error estimate for the biharmonic eigenvalue. The literature on the a posteriori error estimate for Ciarlet-Raviart mixed method is still quite a few. We give an a posteriori error estimator of the approximate eigenvalue and analyze its reliability. The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p ≥ 3 , we derive the a priori error estimates of the approximate eigenfunction in the broken H 1 norm and L 2 norm which are optimal in h and suboptimal in p. When p = 2 , the approximate eigenfunctions converge but with only suboptimal convergence rate. When p ≥ 2 and the eigenfunction u ∈ H s (Ω) (s ≥ p + 1), we prove that the convergence rate of approximate eigenvalues reaches 2 p − 2 in h and p − (2 s − 7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues. [ABSTRACT FROM AUTHOR]

• Compared with [13,16,30] , the time delay is considered on the impulse. Hence, the corresponding main theorems of FTS of CDNs in this paper extend the results of previous related works. In addition, the effects of synchronizing and desynchronizing impulses on the FTS of the CDNs are considered, respectively. • In contrast to [17,21] , the initial value constraint has been removed. The GFTS criteria of CDNs with delayed impulses are derived under synchronizing impulses. Furthermore, a larger settling time bound is estimated when CDNs are subjected to desynchronizing impulses in Theorem 2. This paper mainly studies the finite-time synchronization (FTS) of complex dynamical networks (CDNs) with delayed impulses. By constructing proper Lyapunov function, some sufficient criteria for FTS of CDNs under synchronizing and desynchronizing impulses are established, respectively. Firstly, the existing results of FTS are extended to the case with delayed impulses. Secondly, when the impulses in the CDNs are synchronizing, the existing criterion of local FTS (LFTS) is extended to global FTS (GFTS) and the settling time which depends on the initial values and the impulses is estimated. Furthermore, the bound of settling time is estimated explicitly when the CDNs are subjected to desynchronizing impulses. Finally, three examples are given to illustrate the validity of the obtained results. [ABSTRACT FROM AUTHOR]

• By designing proper barrier functions, the output of every follower is successfully constrained in the desired time-varying set. Compared with most existing constraint control strategies, the dynamic output constraint problem considered in this paper is a more general case. Here, the constraint boundary functions can be asymmetric, or have the same symbol, or even one of the boundary functions is equal to zero. Therefore, the proposed control strategy is more in line with the practical engineering requirements. • Compared with the existed optimal control algorithms with infinite-time convergence, the proposed control protocol can not only achieve the goal of optimization under worst-case perturbations, but also guarantee the stability of MASs in the finite time, which effectively accelerates the convergence rate. • In order to reduce the complexity of the network weights updating algorithm, a new positive function is constructed in the RL algorithm proposed in this paper to simplify the design of adaptive learning rules. Meanwhile, the persistent excitation condition that is difficult to verify online is no longer needed. In this paper, a finite-time optimal consensus control strategy is presented for unknown multi-agent systems (MASs) with the time-varying asymmetric output constraint. Different from existing results, the output constraint problem investigated here eliminates the requirements that constraint boundary functions must be strictly non-zero and have different signs, which is successfully handled by introducing special barrier functions. Moreover, to deal with disturbances well, a reinforcement learning (RL) with the critic-actor-disturbance structure is introduced. Meanwhile, the weights of neural networks are adjusted online by applying the gradient descent method to positive functions newly constructed, which not only significantly simplifies the algorithm but also eliminates the persistent excitation condition. For obtaining a fast convergence rate, the finite-time control technique is embedded into the RL algorithm, and an effective finite-time optimal control scheme is proposed to achieve the consistency of multi-agent system in a finite time. Finally, the effectiveness of the proposed protocol is demonstrated by two simulation examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with the composite convex minimization problem where the cost function consists of a smooth convex function with Lipschitz gradient and a closed proper convex function with non-polyhedral epigraph. For this class of non-polyhedral convex optimization problems, we establish the locally Lipschitzian type error bounds for estimating the distance to the solution set from any feasible point near the solution set, and the globally Lipschitzian type error bounds for the smooth function with a special structure, with the help of the local weak sharpness of minima or the calmness of appropriate multifunctions, and also provide verifiable regularity conditions to guarantee them to hold. Although the derived local error bounds are weaker than the one used in Luo and Tseng (1992,1993) and Wei and Jen (2014), when applying them to an inexact feasible descent method (FDM) and proximal gradient method (PGM), respectively, we still achieve the asymptotic Q -linear convergence and R -linear convergence of the objective value sequence and the iterate sequence, respectively. As an illustration of these results, we obtain the linear convergence of the PGM for the trace norm regularized least squares problem under a regularity condition of the linear operator. [ABSTRACT FROM AUTHOR]

Abstract: Initial boundary value problems for the heat conduction equation with the constant and piecewise constant coefficients are considered in the paper. The Dirac delta function appears in the initial data. The considered problems are mathematical models of the temperature distribution arising from a pulse point heat source in homogeneous or two composite materials. The Fourier series expansion method is applied for solving these problems. The formal solutions are constructed in the form of the infinite series which are not classical functions. We show that these formal solutions are generalized functions (distributions). We prove that the partial sum of the constructed infinite series is a classical function which can be considered as a regularization of the generalized solution. The method of the computation of this regularized solution is suggested in the paper. The computational examples confirm the robustness of the proposed method. [Copyright &y& Elsevier]

Abstract: Nonic spline is used for the numerical solutions of the eighth order linear special case boundary value problems. The end conditions are derived for the definition of spline. The algorithm developed not only approximates the solutions, but their higher order derivatives as well. The method presented in this paper has also been proved to be second order convergent. Two examples compared with those considered by Inc et al. [M. Inc, D.J. Evans, An efficient approach to approximate solutions of eighth-order boundary-value problems, Int. J. Comput. Math. 81 (6) (2004) 685–692] and Siddiqi et al. [S.S. Siddiqi, E.H. Twizell, Spline solution of linear eighth-order boundary value problems, Comput. Methods Appl. Mech. Eng. 131 (1996) 309–325], show that the method developed in this paper is more efficient. [Copyright &y& Elsevier]

In this paper, we derive approximate analytical solutions of the D-dimension Feynman propagator in presence of the shifted Deng-Fan oscillator by using an appropriate approximation of the centrifugal term. Energy eigenvalues and the corresponding eigenfunctions are obtained for different quantum numbers and some diatomic molecules. In the three-dimensional case, we calculate the energy level values and compare our results with those given in the literature. Our calculated energy levels are in much better agreement with the ones obtained numerically. [ABSTRACT FROM AUTHOR]

The research direction of this paper is passivity and passification of memristive recurrent neural networks (MRNNs) with multi-proportional delays and impulse. Preparing for passive analysis, the model of MRNNs is transformed into the general recurrent neural networks (RNNs) model through the way of non-smooth analysis. Utilizing the proper Lyapunov–Krasovskii functions constructed in this paper and the common matrix inequalities technique, a novel condition is derived which is sufficient to make sure that system is passive. In addition, it relaxes the condition that the symmetric matrices are all required to be positive definite. The final results are presented by linear matrix inequalities (LMIs) and its verification is easy to be carried out by the LMI toolbox. And there are several numerical examples showing the effectiveness and correctness of the derived criteria. [ABSTRACT FROM AUTHOR]

This paper is concerned with nonstationary l 2 − l ∞ filtering for Markov switching repeated scalar nonlinear systems (MSRSNSs) with randomly occurring nonlinearities (RONs), where measurement output is modeled by a mode-dependent random variable that satisfying Bernouli distribution. The new relationship are proposed to depict multiple mutually independent Markov chains between original MMSRSNSs and nonstationary filters. By constructing a proper Lyapnov function, the MSRSNSs is stochastically stable with l 2 − l ∞ performance level is guaranteed. Accordingly, the nonstationary filters are designed, where filters are characterised by a two-layer structure. The paper provides a numerical example verifying the efficacy of established technique. [ABSTRACT FROM AUTHOR]

In this paper, the solution of Benjamin–Bona–Mahony–Burgers (BBMB) equation is approximated by using collocation method. Stability of the method is discussed. Also the convergence of the method is proved. The method is used on some examples, and the numerical results have been obtained and compared with the exact solutions. [ABSTRACT FROM AUTHOR]

The computation of the eigenvalues of a Sturm–Liouville problem is a difficult task, when high-index eigenvalues are computed. In most previous methods, it can be seen that the uncertainty of the results increases as the estimated eigenvalues grow larger. This paper is to present some new methods in which, not only the error of calculating the higher eigenvalues does not grow, but it also vanishes as eigenvalues tend to infinity. Moreover, the proposed method gives good estimates of eigenfunctions corresponding to high eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper we study an inverse problem for weighted second order Sturm–Liouville equations. We show that the zeros of any subsequence of eigenfunctions, or a dense set of nodes, are enough to determine the weight. We impose weaker hypotheses for positive weights, and we generalize the proof to include indefinite weights. Moreover, the parameters in the boundary conditions can be determined numerically by using a shooting method. [ABSTRACT FROM AUTHOR]

• For the Stokes eigenvalue problem, we give a new expression for the a priori error estimate of eigenfunction, which shows clearly that ∥ u h − u ∥ 0 is a small quantity of higher order compared with ∥ u h − u ∥ h + ∥ σ h − σ ∥ 0. • For the Stokes eigenvalue problem, we present a direct analysis, which does not involve the Helmholtz decomposition, for estimating the reliability bound. We give an a posteriori error formula in Rd (d = 2 , 3) which contains the residual term and the consistent error term, and we use two technique lemmas (Lemma 3.1 in [1] for the CR element and Lemma 5.6.4 in [37] for the ECR element) to estimate the consistent term and prove the reliability of the estimator for the problem. • We give adaptive finite element algorithm based on direct solution (direct-AFEM) and the adaptive finite element algorithm based on the shifted inverse iteration (shifted inverse-AFEM) for the Stokes eigenvalue problem. We provide numerical experiments both in two and three-dimensional domains. Numerical results indicate that the numerical eigenvalues obtained by the above two methods achieve the optimal convergence order O (dof − 2 / d) and approximate the exact ones from below. In this paper, we discuss the a posteriori error estimates and adaptive algorithm of non-conforming mixed finite elements including the Crouzeix–Raviart element and the enriched Crouzeix–Raviart element for the Stokes eigenvalue problem in R d (d = 2 , 3). We give the a posteriori error estimators and prove their reliability and efficiency. Based on the a posteriori error estimators we built two adaptive algorithms, the direct AFEM and the shifted-inverse AFEM. Numerical experiments and theoretical analysis are consistent, which indicates that the numerical eigenvalues obtained by the above two adaptive algorithms achieve the optimal convergence order O(d o f − 2 d ) and approximate the exact ones from below. [ABSTRACT FROM AUTHOR]

In this paper, the Chen-Lee-Liu (C-L-L) equation is investigated by the Darboux transformation (DT) method. A specific construction of the N-fold DT for C-L-L equation is derived in a simple way. The form of the N-fold DT is a matrix polynomial and each element of the matrix can be expressed by a ratio of two determinants. Furthermore, by choosing suitable eigenvalues and eigenfunctions under the reduction conditions, we can obtain the determinant solution of the reduced C-L-L equation. Moreover, the generalized DT (gDT) for C-L-L equation is also constructed through the limiting technique. As applications of the gDT method, the first and the second-order rational solitons with vanished background (VBC) and non-vanished background (NVBC) from different seeds are calculated for C-L-L equation. We hope our results can be realized by experiments in plasma physics and optical fibers. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce a q-version of the prolate spheroidal wave functions which is the eigenfunction of the “concentration problem” related to sampling and interpolation of q-Hankel bandlimited functions. As application we give a q-sampling theorem which extrapolates functions in space with as sampling points. [Copyright &y& Elsevier]

Abstract: In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains . This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as those of a finite linear operator. It is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements, instead of using the ubiquitous Rayleigh quotient of finite linear algebra. Also, an alternative expression for the Rayleigh quotient in the associated infinite dimensional Sobolev space which avoids the integration of gradients is introduced and shown to be more efficient. The method can also be used in order to produce the spectral decomposition of any given function . [Copyright &y& Elsevier]

Abstract: In this paper, vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales. Based on classical laminated beam theory, the governing equations of motion for a time-variant axial speed are obtained using Newton’s second law of motion and constitutive relations. The method of multiple scales, an approximate analytical method, is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained. The stability boundaries of the system near resonance are determined via the Routh–Hurwitz criterion. Finally, a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response, natural frequencies and stability boundaries of the system. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the problem of laminar forced convection between two parallel plates. We present an unified numerical approach for some problems related to this case: the problem of viscous dissipation with Dirichlet and Neumann boundary conditions and the Graetz problem. The solutions of these problems are obtained by a series expansion of the complete eigenfunctions system of some Sturm–Liouville problems. The eigenfunctions and eigenvalues of this Sturm–Liouville problem are obtained by using Galerkin’s method. Numerical examples are given for viscous fluids with various Brinkman numbers. [Copyright &y& Elsevier]

Abstract: In this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme. This new numerical technique preserves the qualitative properties of the analytic solution due to the Sturm–Liouville structure of the underlying discrete linear boundary-value problem and has computational stability advantages vs other methods. Illustrative examples are included. [Copyright &y& Elsevier]

Abstract: This paper deals with the singularly perturbed boundary value problem for a linear second order differential–difference equation of the convection–diffusion type with small delay parameter of whose solution has a boundary layer. The fitted mesh technique is employed to generate a piecewise-uniform mesh, condensed in the neighborhood of the boundary layers. B-spline collocation method is used with fitted mesh. Parameter-uniform convergence analysis of the method is discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small delay on boundary layer has also been discussed. Several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution. [Copyright &y& Elsevier]

Abstract: In this paper an efficient method based on the Adomian decomposition for computing the eigenelements of fourth-order Sturm-Liouville boundary value problems is developed. Numerical examples show that the method proposed is easy to implement and produces accurate results. [Copyright &y& Elsevier]

Abstract: This paper considers the calculation of eigenvalues for a class of crosswise vibration equation of the beam by Galerkin’s method and obtains the estimation of the errors. The results are very useful for studies of the eigenvalue problems. Some numerical examples are given. [Copyright &y& Elsevier]

In this paper we show that ifwhere D is a bounded region in RN; 1, and h is a function which changes sign in D, has a positive solution for a function f it continues to have a positive solution provided that f is suitably perturbed. [Copyright &y& Elsevier]

In this paper we evaluate the asymptotic formulas for the eigenvalues, the eigenfunctions and the normalization numbers of some version of a Dirichlet problem in a finite interval with explosive factor. [Copyright &y& Elsevier]

In the present paper, we derive the biorthogonality condition for Stokes flow in wedge-shaped trenches bounded by radial lines and concentric circles centered at the vertex of the wedge. This condition, is a property satisfied by the eigenfunctions and adjoint eigenfunctions needed to compute the coefficients of the eigenfunction expansion solution of the corresponding Stokes flow problem. [Copyright &y& Elsevier]

This paper deals with eigenvalues and normalized eigenfunctions for a Sturm–Liouville eigenvalue problem. The technique we have used is based on applying a differential transform to our eigenvalue problem. We have compared our results with some known analytical results. [Copyright &y& Elsevier]

Several quaternion Fourier transforms have received considerable attention in the last years. In this paper, based on the symplectic decomposition of quaternions, we prove that the Hermite functions in the spherical basis are the eigenfunctions of the QFTs. The relationship with Radon transform, the real and complex Paley-Wiener theorem, as well as the sampling formula for the 2-sided QFT are established in an easy way. Moreover, a two parameter fractional Fourier transform is introduced based on a new direct sum decomposition of the L 2 space. The explicit formula and bound of the kernel are obtained. Compared with the Clifford-Fourier transform defined using eigenfunctions, the main adventure of this transform is that the integral kernel has a uniform bound for all dimensions. [ABSTRACT FROM AUTHOR]

In this article, we propose some substantial spectral data for Sturm–Liouville problem with separated boundary conditions in frame of newly defined truncated M -derivative which contains truncated Mittag-Leffler function. The proposed such spectral data like representation of solution for Sturm-Liouville problem subjected to the both boundary and initial conditions, asymptotic formulas of the eigenfunctions and eigenvalues and their normalized counterparts are implemented in terms of underlying efficient limit-based local derivative. We establish all these mentioned data for two different representation of solution under different conditions. In order to obtain these crucial results, certain tools like integration by parts formula, Leibniz rule, variation of parameters method and so forth are offered in view of the truncated M -derivative. Moreover, uniqueness of the solution of the Sturm–Liouville equation is leadingly showed. Substantially, the main advantage of this potent derivative allowing arbitrary change of order is that it contains truncated Mittag–Leffler function enabling us to deal better with the behavior of the matter in hand and to consider it as a generalized version of other extensive local derivatives in literature owing to the additional parameter inside definition. Thereby the objective of this paper is pivotally dedicated to observe the behavior of the spectral structure of Sturm–Liouville problem by supporting with the simulation analysis. [ABSTRACT FROM AUTHOR]