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In the present paper a new efficient semi-implicit finite volume method is proposed for the solution of the compressible Euler and Navier - Stokes equations of gas dynamics with general equation of state (EOS). The discrete flow equations lead to a mildly nonlinear system for the pressure, containing a diagonal nonlinearity due to the EOS. The remaining linear part of the system is symmetric and at least positive semi-definite. Mildly nonlinear systems with this particular structure can be very efficiently solved with a nested Newton-type technique.The new numerical method has to obey only a mild CFL condition, which is based on the fluid velocity and not on the sound speed. This makes the scheme particularly interesting for low Mach number flows, because large time steps are permitted. Moreover, being locally and globally conservative, the new method behaves also very well in the presence of shock waves. The proposed algorithm is first validated against the exact solution of a large set of one-dimensional Riemann problems for inviscid flows with three different EOS: the ideal gas law, the van der Waals EOS and the Redlich - Kwong EOS. In the final part of the paper, the method is extended to the two-dimensional viscous case.

This paper is concerned with the formula of mild solutions to impulsive fractional evolution equation. For linear fractional impulsive evolution equations 8-25,27,30,31, described mild solution as integrals over ( t k , t k + 1 ( k = 1 , 2 , ? , m ) and 0, t1. On the other hand, in 26,28,29, their solutions were expressed as integrals over 0,?t. However, it is still not clear what are the correct expressions of solutions to the fractional order impulsive evolution equations. In this paper, firstly, we prove that the solutions obtained in 8-25,27,30,31 are not correct; secondly, we present the right form of the solutions to linear fractional impulsive evolution equations with order 0

This paper presented a precise yet efficient algorithm to determine bifurcation points of periodic solutions in nonlinear dynamical systems.Period doubling and symmetry breaking points were obtained to an accuracy of 10-11.The bifurcation points can be directly determined without a tedious trail and error process. Calculations and analysis on bifurcations of periodic solutions has played a pivotal role in nonlinear dynamics researches. This paper presents an efficient and precise approach, based on the incremental harmonic balance (IHB) method, to exactly determine the critical points at which periodic solutions exhibit bifurcations. The presented method is based on a fact that, some zero harmonic coefficients vary as non-zeros during the bifurcation of a periodic solution. Equating one of these coefficients to an infinitesimal quantity, the bifurcation value can be determined by the IHB method with the control parameter varying. Numerical examples show that, the critical values for the control parameter where a symmetry breaking or period doubling happens can be calculated very accurately. Importantly, each bifurcation value can be directly obtained without a tedious trail and error process.

This paper investigates the problem of the simultaneous fault detection and control (SFDC) for switched linear systems. To meet the control and detection objectives, the time-dependent detection filters and dynamic output feedback controllers are presented in SFDC under a mixed H∞/ H - framework. A mode-dependent average dwell-time (MDADT) approach, which means that each subsystem has its own average dwell time, is adopted in this paper to reduce the conservativeness of the average dwell time method. And the discretized Lyapunov function (DLF) technique is first used to relax the MDADT constraints in SFDC. Some sufficient conditions for designing filters/controllers which satisfy the H∞/ H - performance are given in terms of linear matrix inequalities (LMIs). What's more, a two-step algorithm to solve the SFDC problem is proposed. The effectiveness of the proposed method is illustrated through two simulation examples.

Behind Medicine (M) is Physiology (P), behind Physiology is Physics (P) and behind Physics is always Mathematics (M), for which I expect that the symmetry of the quadruplet MPPM will be compatible with the characteristic bias of hyperbolic partial differential equations, a theme of this paper. I start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension, of which the most prominent example considered here is multiple sclerosis, the most common neurodegenerative, disabling disease in young adults. Other pathologies surveyed here include retinal abnormalities, Transient Global Amnesia, Transient Monocular Blindness, Meniere's disease and Idiopathic Parkinson's disease. It is the hypothesised vascular aspect of these conditions that links medicine to mathematics, through fluid mechanics in very complex networks of moving boundary blood vessels. The second part of this paper is about mathematical modelling of the human cardiovascular system, with particular reference to the venous system and the brain. A review of a recently proposed multi-scale mathematical model then follows, consisting of a one-dimensional hyperbolic description of blood flow in major arteries and veins, coupled to a lumped parameter description of the remaining main components of the human circulation. Derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system, a much neglected aspect of cardiovascular mathematics. Veins, unlike their arterial counterparts, are highly deformable, even collapsible under mild physiological conditions. We address mathematical and numerical challenges. Regarding the numerical analysis of the hyperbolic PDEs, we deploy a modern non-linear finite volume method of arbitrarily high order of accuracy in both space and time, the ADER methodology. In vivo validation examples and brain haemodynamics computations are shown. We also point out two, preliminary but important new findings through the use of mathematical models, namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability.

In this paper we use the notion of the resolvent equation and Lyapunov's method to study boundedness and integrability of the solutions of the nonlinear Volterra integral equation on time scales x ( t ) = a ( t ) - ? t 0 t C ( t , s ) G ( s , x ( s ) ) Δ s , t ? t 0 , ∞ ) ? T . In particular, the existence of bounded solutions with various Lp properties are studied under suitable conditions on the functions involved in the above Volterra integral equation. At the end of the paper we display some examples on different time scales.

In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.

Domination and its variations in graphs are natural models for the location problems in operations research. In this paper, we investigate the rainbow domination number of graphs, which was introduced by Bresar, Henning and Rall. Given a graph G and a positive integer t, a t-rainbow dominating function of G is a function f from vertex set to the set of all subsets of {1, 2, ⋅⋅⋅, t} such that for any vertex v with f ( v ) = ϕ , we have ⋃ u ∈ N ( v ) f ( u ) = { 1 , 2 , ⋯ , t } . The t-rainbow domination problem is to determine the t-rainbow domination number γrt(G) of G, that is the minimum value of ∑v ∈ V(G)|f(v)|, where f runs over all t-rainbow dominating functions of G. The domination number and its variations of generalized Petersen graphs P(n, k) are widely investigated. The exact values of γr2(P(n, 1)) and γr3(P(n, 1)) are already determined in [11, 12]. In this paper, we determine the exact values of γrt(P(n, 1)) for any t ≥ 8 and t = 4 and prove that γ r t ( P ( 2 k , k ) ) = 4 k for t ≥ 6, where P(2k, k) is a special graph.

In this paper, we prove bounds for the unique, positive zero of O G ★ ( z ) : = 1 − O G ( z ) , where OG(z) is the so-called orbit polynomial [1]. The orbit polynomial is based on the multiplicity and cardinalities of the vertex orbits of a graph. In [1] , we have shown that the unique, positive zero δ ≤ 1 of O G ★ ( z ) can serve as a meaningful measure of graph symmetry. In this paper, we study special graph classes with a specified number of orbits and obtain bounds on the value of δ.

Recently, Luo et al. proposed two new codebooks based on the operations of some given sets, which are asymptotically optimal. In the paper, we present new construction of codebooks. We determine maximal cross-correlation amplitude of the constructed codebooks. The constructed codebooks are near optimal to the Welch bound. This new codebooks in the paper generalize Luo et al.′s constructions. The parameters of constructed codebooks are nimble and new under some circumstances.

The simplex integration is a convenient method for the integration over complex domains. It automatically represents the ordinary integral over an arbitrary polyhedron as the algebraic sum of integrals over the oriented simplexes. An existing recursive formula for the integration of monomials over simplex, which was deduced based on special operations of matrices and was presented by the first author of this paper, has significant advantages: not only the computation amount is small, but also the integrals of all the lower order monomials are obtained while computing the integral of the highest order monomial. The extension to a polynomial can be obtained by the linearity of integrals. In this paper, the derivation of the existing recursive formula is detailed. A new recursive formula is emphatically proposed to simplify the existing recursive formula and further increase the speed of computation. The code for computer implementation is also presented. Examples show that the accuracy and efficiency of the recursive formula are higher than numerical integration.

The nonlocal diffusion model describes the diffusion process of solutes in complex media properly, while the classical theory of partial differential equations can not provide an appropriate description. The purpose of this paper is to provide illustrations from both theoretical and numerical perspectives of the computation of nonlocal diffusion models by Legendre collocation methods. Compared to local numerical methods, Legendre collocation methods can achieve a fixed accuracy with much fewer unknowns whenever the computational domain is regular and the solutions are sufficiently smooth. This paper is a groundwork towards efficient high order methods including spectral and spectral element methods for nonlocal diffusion equations with volume constrained boundary conditions.

This paper is concerned with some of well-known iterative methods in their tensor forms to solve a class of tensor equations via the Einstein product and the associated with least squares problem. Especially, the tensor forms of the LSQR and LSMR methods are presented. The proposed methods use tensor computations with no matricizations involved. We prove that the norm of residual is monotonically decreasing for the tensor form of the LSQR method. The norm of residual of normal equation is also monotonically decreasing for the tensor form of the LSMR method. We also show that the minimum-norm solution (or the minimum-norm least squares solution) of the tensor equation can be obtained by the proposed methods. Numerical examples are provided to illustrate the efficiency of the proposed methods and testify the conclusions suggested in this paper.

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.

This paper introduces sliding mode control for singular Takagi-Sugeno (T-S) fuzzy Markovian jump system with mode-dependent derivative-term coefficient. Firstly, a new continuous switching function is designed. Then sufficient conditions about admissibility of sliding motion under the case of completely known transition rate and uncertain transition rate are presented, respectively. The obtained conditions are strict linear matrix inequalities. Furthermore, a sliding mode controller is presented. Finally, an example is given to illustrate the effectiveness of the obtained results in this paper. (C) 2019 Elsevier Inc. All rights reserved. National Nature Science Foundation of ChinaNational Natural Science Foundation of China (NSFC) [61773108] This work is supported by National Nature Science Foundation of China under Grant 61773108. Authors would like to thank the editors and the reviewers for their helpful comments and suggestions to improve the quality of this paper.

The spectrum of the Laplacian matrices of complex networks is a key factor in network functionality. In this paper, the spectral properties of Supra-Laplacian for partially interdependent networks are investigated. Based on Matrix Perturbation Theory, refined results of the eigenvalue properties of Laplacian matrices are provided, which shows that the size relationship of the first-order approximate solutions of the eigenvalues remains unchanged, even if there is perturbation. Using these results, the theoretical approximate formulae of the minimum non-zero eigenvalue and the maximum eigenvalue of Supra-Laplacian for partially interdependent networks are derived, respectively. The outcomes in this paper are more general and need fewer calculations than that in the literature [13] , [14] . Finally, the simulations and applications of synchronizability and diffusion process verify the feasibility and effectiveness of the proposed solutions. The findings can be instructive for networks when linking to the spectral properties of the Supra-Laplacian.

For any real α ∈ [0, 1], A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) is the Aα-matrix of a graph G, where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the degrees of G. This paper presents some extremal results about the spectral radius λ1(Aα(G)) of Aα(G) that generalize previous results about λ1(A0(G)) and λ 1 ( A 1 2 ( G ) ) . In this paper, we study the behavior of the Aα-spectral radius under some graph transformations for α ∈ [0, 1). As applications, we show that the greedy tree has the maximum Aα-spectral radius in G D when D is a tree degree sequence firstly. Furthermore, we determine that the greedy unicyclic graph has the largest Aα-spectral radius in G D when D is a unicyclic graphic sequence, where G D = { G ∣ G is connected with D as its degree sequence } .

In this paper, the global uniform exponential stability (GUES) for a class of switched nonlinear systems is discussed in both continuous-time and discrete-time contexts. Different from multiple Lyapunov function and average dwell time in the previous work, a general multiple Lyapunov-like function and a modified admissible edge-dependent average dwell time switching scheme based on a directed graph are implemented in this paper. By further allowing the Lyapunov-like function to increase and decrease during the running time of active subsystems, and permitting the admissible transition edge-dependent weights (ATEDWs) to be greater or less than one, the extended stability results for switched systems in a nonlinear situation are first derived. Given a new amalgamation condition between the increasing ratio and decreasing ratio of the Lyapunov-like function at switching instants, the minimal admissible edge-dependent average dwell time (AEDADT) for admissible switching signals are is obtained. By contrasting with the results available on the subject, we allow that all the switching behaviors of the switching systems at switching instants are destabilizing, and the transition weights of admissible transition edge may be less than one1. More especially, even if some of the subsystems are not stable and all the switching behaviors of are destabilizing, the stability property of the switched system can still be preserved. Finally, two numerical examples are illustrated to show the validity of the proposed results.

A certain category of cotangent sums has been proven of importance in the Nyman–Beurling criterion for the Riemann Hypothesis. In previous work ( [12] , [13] ) the authors proved the existence of a unique positive measure μ on R , with respect to which certain normalized cotangent sums, evaluated at rational numbers with fixed denominators are equidistributed. The tools applied in this paper belong to various fields of Mathematics, for instance the relation between the equidistribution mod 1 of the multiples of a number and the Diophantine approximation properties of that number. In this paper we prove an analogous result for the case that the denominator of the rational numbers is a fixed prime number and that the numerator is also prime.

First of all, by characterizing solutions of the quaternion equation a x = x ? b , this paper studies consimilarity of quaternions and some related consequences. For the important role of coneigenvalues in consimilarity tranformations of quaternion matrices, this paper further derives the relations between principle right coneigenvalues of a quaternion matrix and eigenvalues of the corresponding real representation matrix. Then, based on the real representation matrix, an effective algorithm is presented to calculate all coneigenvalues and the associated coneigenvectors of a quaternion matrix. Finally, two numerical examples are given to verify the effectiveness of the proposed algorithm.

Uncertain differential equation is a type of differential equation driven by canonical Liu process. How to obtain the analytic solution of uncertain differential equation has always been a thorny problem. In order to solve uncertain differential equation, early researchers have proposed two numerical algorithms based on Euler method and Runge-Kutta method. This paper will design another numerical algorithm for solving uncertain differential equations via Adams-Simpson method. Meanwhile, some numerical experiments are given to illustrate the efficiency of the proposed numerical algorithm. Furthermore, this paper gives how to calculate the expected value, the inverse uncertainty distributions of the extreme value and the integral of the solution of uncertain differential equation with the aid of Adams-Simpson method.

In this paper, we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations A 1 X = C 1 , Y B 1 = D 1 , A 2 W = C 2 , Z B 2 = D 2 , A 3 V = C 3 , V B 3 = C 4 , A 4 V B 4 = C 5 , A 5 X + Y B 5 + C 6 W + Z D 6 + E 6 V F 6 = G 6 , and constitute an expression of the general solution to the system when it is solvable. The outcomes of this paper encompass some recognized results in the collected works. In addition, we establish an algorithm and a numerical example to illustrate the theory constructed in the paper.

For uncertain differential equations, we cannot always obtain their analytic solutions. Early researchers have described the Euler method and Runge-Kutta method for solving uncertain differential equations. This paper proposes a new numerical method-Adams method to solve uncertain differential equations. Some numerical experiments are given to illustrate the efficiency of our numerical method. Moreover, this paper also gives two numerical methods for calculating the extreme value and the time integral of solutions of uncertain differential equations.

Air pollution models plays a critical role in atmospheric environment research. Chemical kinetic equations is an important component of air pollution models. The chemical equations is numerically sticky because of its stiffness, nonlinearity, coupling and nonnegativity of the exact solutions. Over the past decades, numerous papers about chemical equation solvers have been published. However, these solvers cannot preserve the nonnegativity of the exact solutions. Therefore, in the calculation, the negative numerical concentration values are usually set to zero artificially, which may cause simulation errors. To obtain real nonnegative numerical concentration values, very small step-size has to be adopted. Then enormous amount of CPU time is consumed to solve the chemical equations. In this paper, we revisit this topic and derive a new algorithm. Our algorithm Modified-Backward-Euler (MBE) Method can unconditionally preserve the nonnegativity of the exact solutions. MBE is a simple, robust and efficient solver. It is much faster and more precise than the traditional solvers such as LSODE and QSSA. The numerical results and parameter suggestions are shown at the end of the paper. MBE is based on the P-L structure of the chemical equations and a deeper view into the nature of Euler Methods. It cannot only be used to solve chemical equations, but can also be applied to conquer ordinary differential equations (ODEs) with similar P-L structure.

The article of record as published may be found at http://dx.doi.org/10.1016/j.amc.2015.08.039 Under the assumption of the known multiplicity of zeros of nonlinear equations, a class of two-point sextic-order multiple-zero finders and their dynamics are investigated in this paper by means of extensive analysis of modified double-Newton type of methods. Wit the introduction of a bivariate weight function dependent on function-to-function and derivative-to-derivative ratios, higher-order convergence is obtained. Additional investigation is carried out for extraneous fixed points of the iterative maps associated with the proposed methods along with a comparison with typically selected cases. Through a variety of test equations, numerical experiments strongly support the theory developed in this paper. In addition, relevant dynamics of the proposed methods is successfully explored for various polynomials with a number of illustrative basins of attraction. National Research Foundation of Korea Ministry of Education, Science and Technology under the research grant (Project Number: 2015-R1D1A3A-01020808

In this paper, we study a class of damped vibration problems with superquadratic terms at infinity. By using variational methods, we obtain infinitely many nontrivial periodic solutions under conditions weaker than those previously assumed. To the best of our knowledge, there is no published result focusing on this case by the method used in this paper.

Various problems of pure and applied sciences can be studied in the unified frame work of the system of nonlinear equations. In this paper, a new family of iterative methods for solving a system of nonlinear equations is developed by using a new decomposition technique. The convergence of the new methods is proved. Efficiency index of the proposed methods is discussed and compared with some other well-known methods. The upper bounds of the error and the radius of convergence of the methods are also found. For the implementation and performance of the new methods, the combustion problem, streering problem and Van der Pol equation are solved and the results are compared with some existing methods. Several new iterative methods are derived from the general iterative scheme. Using the ideas and techniques of this paper, one may be able to suggest and investigate a wide class of iterative methods for solving the system of nonlinear equations. This is another direction of future research.

In this paper we propose a simple numerical method for approximating solutions of Fredholm-Volterra integral equations of the second kind. The method is based on Picard iteration and uses a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. The paper concludes with numerical examples and a discussion of the approximations proposed.

In this paper, we introduce and analyze two new methods for solving the NP-hard absolute value equations (AVE) A x - | x | = b , where A is an arbitrary n × n real matrix and b ? Rn, in the case, singular value of A exceeds 1. The comparison with other known methods is carried to show the effectiveness of the proposed methods for a variety of randomly generated problems. The ideas and techniques of this paper may stimulate further research.

In our first paper we showed how Gauss's method for determining the initial position and velocity vectors from two inertial position vectors at two times in an idealized Keplerian two-body elliptical orbit can be made more robust and efficient by replacing functional iteration with Newton-Raphson iteration. To do this we split the orbit determination algorithm into two sub-algorithms, the x-iteration to find the zero of the fixed-point function F(x) when the true anomaly angular difference between the two vectors is large and the y-iteration to find the zero of the fixed-point function G(y) when the angular difference is small.We derived necessity and sufficiency conditions for the existence and uniqueness of these zeros and conjectured that both functions were concave. In this paper we first simplify the necessity and sufficiency conditions. We then prove that both F and G are indeed concave.Existence and uniqueness of the zero of G and its concavity then imply quadratic convergence of the Newton-Raphson iterative sequence from an arbitrary starting point in its domain. Quadratic convergence to the zero of F(x) is also the case in its domain except for a small neighborhood of x = 1 , corresponding to the true anomaly difference angle between the position vectors in a small neighborhood of 180?.

This paper focuses on the convergence of a class of collocation methods for Volterra integral equations of the second kind with highly oscillatory Bessel functions. Compared to existing theoretical results, sharper frequency-related convergence rates of these methods are established by exploring the asymptotic expansions of solutions and solving error equations. Theoretical results in this paper show the direct Filon method and continuous linear collocation method share the same convergence rate. Both of them admit a better convergence rate compared to the piecewise constant collocation method in solving Volterra integral equations with highly oscillatory Bessel kernels. These results are verified by numerical experiments.

In this paper, we will consider a class of neutral stochastic functional differential equations with Levy processes. Levy processes contain a number of very important processes as special cases such as Brownian motion, the Poisson process, stable and self-decomposable processes and subordinators, and so on. But its sample paths are discontinuity, which makes the analysis more difficult. In this paper, we try to get over this difficulty. The contributions of this paper are as follows: (a) we will use Lyapunov functional method to study the pth moment asymptotic stability and almost sure asymptotic stability of neutral stochastic functional differential equations with Levy processes; (b) under the result of (a), we will investigate two types of continuity of the solution: continuous in the pth moment and continuous in probability. Finally, we provide an example to illustrate the usefulness of the obtained results.

The purpose of this paper is to investigate the problem of robust passivity analysis for delayed stochastic impulsive neural networks with leakage and additive time-varying delays. The novel contribution of this paper lies in the consideration of a new integral inequality proved to be well-known Jensen's inequality and takes fully the relationship between the terms in the Leibniz-Newton formula within the framework of linear matrix inequalities (LMIs). By constructing a suitable Lyapunov-Krasovskii functional with triple and four integral terms using Jensen's inequality, integral inequality technique and LMI frame work, which guarantees stability for the passivity of addressed neural networks. This LMI can be easily solved via convex optimization techniques. Finally, two interesting numerical examples are given to show the effectiveness of the theoretical results.

This paper is interested in numerical simulations of the interaction of the fluid flow with an airfoil, particularly the problem of the aeroelastic response of the airfoil to a sudden gust is considered. The main attention is paid to the finite element approximations of the incompressible viscous flow over a flexibly supported airfoil. The gust is modelled using the time dependent boundary condition. The structure vibration is governed by the nonlinear system of ordinary differential equations. The flow is described using the Reynolds averaged Navier-Stokes equations, the system is enclosed by the two equation k-ω turbulence model together with the transition model based on the intermittency equation. Modelling of this laminar - turbulence transition of the flow on the airfoil surface is the main novelty of the paper. The motion of the computational domain is treated with the aid of the arbitrary Lagrangian-Eulerian method. The solution of the nonlinear coupled problem is discussed and numerically tested using a stabilized finite element method.

In this paper we introduce a discontinuous Galerkin method on polygonal meshes. This method arises from the discontinuous Galerkin composite finite element method (DGFEM) for source problems on domains with micro-structures. In the context of the present paper, the flexibility of DGFEM is applied to handle polygonal meshes. We prove the a priori convergence of the method for both eigenvalues and eigenfunctions for elliptic eigenvalue problems. Numerical experiments highlighting the performance of the proposed method for problems with discontinuous coefficients and on convex and non-convex polygonal meshes are presented.

This paper is concerned with the existence and uniqueness of solutions for a coupled system of Caputo type sequential fractional differential equations supplemented with nonlocal Riemann-Liouville integral boundary conditions. The existence of solutions is derived by applying Leray-Schauder's alternative, while the uniqueness of solution is established via Banach's contraction principle. An illustrative example is also included. The paper concludes with some interesting observations.

This paper is concerned with the numerical simulation of the interaction of compressible viscous flow with a nonlinear elastic structure. The flow is described by the compressible Navier-Stokes equations written in the arbitrary Lagrangian-Eulerian (ALE) form. For the elastic deformation the St. Venant-Kirchhoff model is used. In the space discretization the discontinuous Galerkin finite element method (DGM) is applied both for the flow problem in a time-dependent domain and for the dynamic nonlinear elasticity system. We show that the DGM is applicable to the discretization of both problems. As a new result we particularly present the application of the DGM to the discretization of the dynamic nonlinear elasticity problem and the DGM solution of the fluid-structure interaction (FSI). The applicability of the developed technique is demonstrated by several numerical experiments. The main novelty of the paper is the application of the DGM to the FSI problem using the model of compressible flow coupled with nonlinear elasto-dynamic system.

In this paper, we present a new modified method for solving multi-level multi objective linear fractional programming problems (ML-MOLFPPs) based on fuzzy goal programming (FGP) approach with some modifications in the algorithm suggested by Baky (2010) 18 which dealt with multi-level multi objective linear programming problem (ML-MOLPP). In proposed modified approach, numerator and denominator function of each objective at each level are individually transformed into fuzzy goals and their aspiration levels are determined using individual best solutions. Different linear membership functions are defined for numerator and denominator function of each objective function. Then highest degree of each of these membership goals is achieved by minimising the sum of negative deviational variables. The proposed algorithm simplifies the ML-MOLFPP by eliminating solution preferences by the decision makers at each level, thereby avoiding difficulties associate with multi-level programming problems and decision deadlock situations. The aim of this paper is to present simple and efficient technique to obtain compromise optimal solution of ML-MOLFP problems. Numerical examples are illustrated in order to support the proposed modified FGP technique.

In this paper, to analyze the effect of the cross patch infection between different groups to the spread of gonorrhea in a community, we establish the complete global dynamics of a multi-group SIS epidemic model with varying total population size by a threshold parameter. In the proof, we use special Lyapunov functional techniques, not only one proposed by the paper Pruss et?al., 2006, but also the other one for a varying total population size with some ideas specified to our model and no longer need a grouping technique derived from the graph theory which is commonly used for the global stability analysis of multi-group epidemic models.

In this paper, the underlying general linear methods (GLMs) are adapted to linear neutral multidelay-integro-differential equations (NMIDEs). In order to obtain stability criteria of the extended GLMs, the corresponding results in paper of Zhang and Vandewalle (2008) are generalized. Based on the concepts of A(α)-stability and A-stability of the underlying GLMs, a serial of asymptotic stability criteria of the extended GLMs are obtained.

The purpose of this paper is to introduce a new class of the generalized mixed equilibrium problems with a new definition of the relaxed monotonicity for bi-functions in topological vector spaces. By employing the KKM technique and under some appropriate assumptions on the considering nonlinear mappings, we obtain the existence of a solution for the generalized mixed equilibrium problems with the new concept of the relaxed monotonicity and coercivity condition(in order to relax the compactness of the domains of the nonlinear mappings) in the setting of topological vector spaces. Moreover, the compactness and convexness of the solution set are investigated. The results in the paper extend and generalize the corresponding results, especially Sintunavarat (2013) 20 in this area by providing mild assumptions in order to guarantee the existence of a solution for the generalized mixed equilibrium problem.

In this paper, we analyze the discontinuous Galerkin (DG) finite element method for the steady two-dimensional transport-reaction equation on Cartesian grids. We prove the L2 stability and optimal L2 error estimates for the DG scheme. We identify a special numerical flux for which the L2-norm of the solution is of order p + 1, when tensor product polynomials of degree at most p are used. We further prove superconvergence towards a particular projection of the directional derivative. The order of superconvergence is proved to be p + 1/2. We also provide a very simple derivative recovery formula which is O ( h p + 1 ) superconvergent approximation to the directional derivative. Moreover, we establish an O ( h 2 p + 1 ) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element. Finally, we prove that the proposed a posteriori DG error estimates converge to the true errors in the L2-norm at O ( h p + 1 ) rate and that the global effectivity index converges to unity at O ( h ) rate. Our results are valid without the flow condition restrictions. We perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal.

Recently, Bai, Parlett and Wang presented a class of parameterized inexact Uzawa (PIU) methods for solving saddle point problems (Bai et?al., 2005). In this paper, we develop a new generalized PIU method for solving both nonsingular and singular saddle point problems. The necessary and sufficient conditions of the convergence (semi-convergence) for solving nonsingular (singular) saddle point problems are derived. Meanwhile, the characteristic of eigenvalues of the iteration matrix corresponding to the above iteration method is discussed. We further show that the generalized PIU-type method proposed in this paper has a wider convergence (semi-convergence) region than some classical Uzawa methods, such as the inexact Uzawa method, the SOR-like method, the GSOR method and so on. Finally, numerical examples are given to illustrate the feasibility and efficiency of this method.

In this paper, we study a suspension bridge equation with memory effects. For the suspension bridge equation without memory, there are many classical results. Existing results mainly devoted to existence and uniqueness of a weak solution, energy decay of solution and existence of global attractors. However the existence of global attractors for the suspension bridge equation with memory was no yet considered. The object of the present paper is to provide some results on the well-posedness and long-time behavior to the suspension bridge equation when the unique damping mechanism is given by the memory term.

In this paper we study an asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form y Δ ( t ) = f ( t , y ( t ) ) , where f : T × R n ? R n , and T is a time scale. For a given set ? ? T × R n , we formulate conditions for function f which guarantee that at least one solution y of the above system stays in ?. Unlike previous papers the set ? is considered in more general form, i.e., the time section ?t is an arbitrary closed bounded set homeomorphic to the disk (for every t ? T ) and the boundary ? T ? does not contain only egress points. Thanks to this, we can investigate a substantially wider range of equations with various types of bounded solutions. A relevant example is considered.The results are new also for non-autonomous systems of difference equations and the systems of impulsive differential equations.

The paper deals with the solution of systems of IVPs with discontinuous right-hand side functions f. In our previous work we considered IVPs with continuous f?'s having discontinuous derivatives. We assumed the global Lipschitz continuity of f, and the analysis was heavily based on that assumption. In the present paper we skip the continuity assumption. We assume that f is a piecewise regular function that satisfies the Lipschitz condition only in some regions of the domain. (By the regular function we mean a function belonging to a Holder class.) A hypersurface where the regularity of f breaks down is not known. We define an algorithm DISC-IVP for solving such problems, and analyze its error and cost. The error analysis in the presence of discontinuities of f is essentially different than that in the continuous case. We show in a subclass of f?'s that the worst-case error of the algorithm (measured in the conservative supremum norm over the interval of integration) is of the same order as for globally regular functions. The algorithm and the analysis do not use heuristic arguments. In particular, we are able to bound the cost of the algorithm, which leads to the bound on the e-complexity of the problem (the minimal cost of computing an e-approximation to the solution). The e-complexity in the discontinuous case described above turns out to be of the same order as that in the globally regular case. In the other words, the solution of discontinuous problems is no more difficult than the solution of globally regular ones. Furthermore, the algorithm DISC-IVP is error and cost optimal.

A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theoretical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.

In this paper, the issue of admissibility for discrete singular systems with time-varying delay is addressed. The main contribution of this paper is to reduce the conservatism of the considered system by adopting an improved reciprocally convex combination approach to bound the forward difference of the double summation term, and utilizing the reciprocally convex combination approach to bound the forward difference of the triple summation term in the Lyapunov function. Without employing decomposition and equivalent transformation of the considered system, a strict delay-dependent linear matrix inequality (LMI) criterion is built to guarantee the considered system to be regular, causal and stable. A numerical example is given to illustrate the effectiveness of this proposed method.

In the present paper we apply a sinc-Gaussian technique to compute approximate values of the eigenvalues of Dirac systems and Dirac systems with eigenvalue parameter in one or two boundary conditions. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc-method. Numerical worked examples with tables and illustrative figures are given at the end of the paper showing that this method gives us better results in comparison with the classical sinc-method in Annaby and Tharwat (2007, 2012) 5,6].

This paper aims to provide a natural generalization and unification of a series of multiple integral representations for special classes of hypergeometric polynomials recently obtained by several authors. This generalization is obtained by considering a very large family of hypergeometric polynomials. The multiple integral representations given in this paper may be viewed as linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.