Inequalities applicable to retarded Volterra integral equations
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B. G. Pachpatte
2004-12-01
Full Text Available The main objective of this paper is to establish explicit bounds on certain integral inequialities which can be used as tools in the study of certain classes of retarded Volterra integral equations.
Asymptotically periodic solutions of Volterra integral equations
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Muhammad N. Islam
2016-03-01
Full Text Available We study the existence of asymptotically periodic solutions of a nonlinear Volterra integral equation. In the process, we obtain the existence of periodic solutions of an associated nonlinear integral equation with infinite delay. Schauder's fixed point theorem is used in the analysis.
On a Volterra Stieltjes integral equation
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P. T. Vaz
1990-01-01
Full Text Available The paper deals with a study of linear Volterra integral equations involving Lebesgue-Stieltjes integrals in two independent variables. The authors prove an existence theorem using the Banach fixed-point principle. An explicit example is also considered.
Adomian Method for Solving Fuzzy Fredholm-Volterra Integral Equations
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M. Barkhordari Ahmadi
2013-09-01
Full Text Available In this paper, Adomian method has been applied to approximate the solution of fuzzy volterra-fredholm integral equation. That, by using parametric form of fuzzy numbers, a fuzzy volterra-fredholm integral equation has been converted to a system of volterra-fredholm integral equation in crisp case. Finally, the method is explained with illustrative examples.
An approximation scheme for optimal control of Volterra integral equations
Belbas, S. A.
2006-01-01
We present and analyze a new method for solving optimal control problems for Volterra integral equations, based on approximating the controlled Volterra integral equations by a sequence of systems of controlled ordinary differential equations. The resulting approximating problems can then be solved by dynamic programming methods for ODE controlled systems. Other, straightforward versions of dynamic programming, are not applicable to Volterra integral equations. We also derive the connection b...
Algebraic Integrability of Lotka-Volterra equations in three dimensions
Constandinides, Kyriacos
2009-01-01
We examine the algebraic complete integrability of Lotka-Volterra equations in three dimensions. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems. The classification is obtained using Painleve analysis and more specifically by the use of Kowalevski exponents. The imposition of certain integrality conditions on the Kowalevski exponents gives necessary conditions for the algebraic integrability of the corresponding systems. We also show that the conditions are sufficient.
Backward stochastic Volterra integral equations- a brief survey
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YONG Jiong-min
2013-01-01
In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations (BSVIEs, for short). BSVIEs are a natural generalization of backward stochastic diff erential equations (BSDEs, for short). Some interesting motivations of studying BSVIEs are recalled. With proper solution concepts, it is possible to establish the corresponding well-posedness for BSVIEs. We also survey various comparison theorems for solutions to BSVIEs.
Lotka-Volterra type equations and their explicit integration
Gervais, Jean-Loup; Jean-Loup Gervais; Mikhail V Saveliev
1994-01-01
In the present note we give an explicit integration of some two--dimensionalised Lotka--Volterra type equations associated with simple Lie algebras, other than the familiar A_n case, possessing a representation without branching. This allows us, in particular, to treat the first fundamental representations of A_r, B_r, C_r, and G_2 on the same footing.
New homotopy analysis transform algorithm to solve volterra integral equation
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Sunil Kumar
2014-03-01
Full Text Available The main aim of the present work is to propose a new and simple algorithm for Volterra integral equation arising in demography, the study of viscoelastic materials, and in insurance mathematics through the renewal equation by using homotopy analysis transform method. The homotopy analysis transform method is an innovative adjustment in Laplace transform algorithm and makes the calculation much simpler. The solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. The beauty of the paper is coupling of two techniques. Finally, two numerical examples are given to show the accuracy and stability of this method.
Periodic solutions of Volterra integral equations
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M. N. Islam
1988-01-01
Full Text Available Consider the system of equationsx(t=f(t+∫−∞tk(t,sx(sds, (1andx(t=f(t+∫−∞tk(t,sg(s,x(sds. (2Existence of continuous periodic solutions of (1 is shown using the resolvent function of the kernel k. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1 it is necessary that the resolvent of k is integrable in some sense. For a scalar convolution kernel k some explicit conditions are derived to determine whether or not the resolvent of k is integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1 and (2 are btained using the contraction mapping principle as the basic tool.
Linear Volterra Integral Equations as the Limit of Discrete Systems
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M. Federson; R.Bianconi; L.Barbanti
2004-01-01
We consider the multidimensional abstract linear integral equation of Volterra typex (t)+(*)∫Rt a (s)x (s)ds =f (t),t∈R,as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions.The functions x,a and f are Banach space-valued de .ned on a compact interval R of R n ,R t is a subinterval of R depending on t∈R and (*)∫denotes either the Bochner-Lebesgue integral or the Henstock integral.The results presented here generalize those in [1]and are in the spirit of [3].As a consequence of our approach,it is possible to study the properties of (1)by transferring the properties of the discrete systems.The Henstock integral setting enables us to consider highly oscillating functions.
Shayma Adil Murad; Hussein Jebrail Zekri; Samir Hadid
2011-01-01
We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.
ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS
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Tao Tang; Xiang Xu; Jin Cheng
2008-01-01
The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is pro-posed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations.
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Shadan Sadigh Behzadi
2011-12-01
Full Text Available In this paper, Adomian decomposition method (ADM and homotopy analysis method (HAM are proposed to solving the fuzzy nonlinear Volterra-Fredholm integral equation of the second kind$(FVFIE-2$. we convert a fuzzy nonlinear Volterra-Fredholm integral equation to a nonlinear system of Volterra-Fredholm integral equation in crisp case. we use ADM , HAM and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy Volterra-Fredholm integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed methods are proved. Examples is given and the results reveal that homotopy analysis method is very effective and simple compared with the Adomian decomposition method.
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Emran Tohidi
2014-01-01
Full Text Available We are concerned with the extension of a Legendre spectral method to the numerical solution of nonlinear systems of Volterra integral equations of the second kind. It is proved theoretically that the proposed method converges exponentially provided that the solution is sufficiently smooth. Also, three biological systems which are known as the systems of Lotka-Volterra equations are approximately solved by the presented method. Numerical results confirm the theoretical prediction of the exponential rate of convergence.
Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM
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Reza Abazari
2013-01-01
Full Text Available The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM, and compared with the differential transform method (DTM. The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences.
Stability and Convergence of Solutions to Volterra Integral Equations on Time Scales
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Eleonora Messina
2015-01-01
Full Text Available We consider Volterra integral equations on time scales and present our study about the long time behavior of their solutions. We provide sufficient conditions for the stability and investigate the convergence properties when the kernel of the equations vanishes at infinity.
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丁皓江; 王惠明; 陈伟球
2004-01-01
The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.
Wang, Tianxiao
2010-01-01
This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present to illustrate the aforementioned optimal control problem. Motivated by the technical skills in solving above problem, a more convenient and briefer method for the unique solvability of M-solution for BSVIEs is proposed. At last, we will investigate a risk minimization problem by means of the maximum principle for FBSVIEs. Closed-form optimal portfolio is obtained in some special cases.
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Shayma Adil Murad
2011-01-01
Full Text Available We study the existence and uniqueness of the solutions of mixed Volterra-Fredholm type integral equations with integral boundary condition in Banach space. Our analysis is based on an application of the Krasnosel'skii fixed-point theorem.
GLOBAL SOLUTIONS OF SYSTEMS OF NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES
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陈芳启; 陈予恕
2001-01-01
The existence of solutions for systems of nonlinear impulsive Volterra integral equations on the infinite interval R+ with an infinite number of moments of impulse effect in Banach spaces is studied. Some existence theorems of extremal solutions are obtained,which extend the related results for this class of equations on a finite interval with a finite number of moments of impulse effect. The results are demonstrated by means of an example of an infinite systems for impulsive integral equations.
Imai, Kenji
2014-02-01
In this paper, a new n-dimensional homogeneous Lotka-Volterra (HLV) equation, which possesses a Lie symmetry, is derived by the extension from a three-dimensional HLV equation. Its integrability is shown from the viewpoint of Lie symmetries. Furthermore, we derive dynamical systems of higher order, which possess the Lie symmetry, using the algebraic structure of this HLV equation.
Existence and Numerical Solution of the Volterra Fractional Integral Equations of the Second Kind
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Abdon Atangana
2013-01-01
Full Text Available This work presents the possible generalization of the Volterra integral equation second kind to the concept of fractional integral. Using the Picard method, we present the existence and the uniqueness of the solution of the generalized integral equation. The numerical solution is obtained via the Simpson 3/8 rule method. The convergence of this scheme is presented together with numerical results.
Al Jarro, Ahmed
2011-09-01
A new predictor-corrector scheme for solving the Volterra integral equation to analyze transient electromagnetic wave interactions with arbitrarily shaped inhomogeneous dielectric bodies is considered. Numerical results demonstrating stability and accuracy of the proposed method are presented. © 2011 IEEE.
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Dishen; Jiabu
2006-01-01
This paper studies the stability and boundedness of the solutions of Volterra integral differential equations with infinite delay in the phase space (Ch, |·|h), the h-uniform stability, h-uniformly asymptotic stability and h-boundedness of solutions are obtained.
Hybrid function method for solving Fredholm and Volterra integral equations of the second kind
Hsiao, Chun-Hui
2009-08-01
Numerical solutions of Fredholm and Volterra integral equations of the second kind via hybrid functions, are proposed in this paper. Based upon some useful properties of hybrid functions, integration of the cross product, a special product matrix and a related coefficient matrix with optimal order, are applied to solve these integral equations. The main characteristic of this technique is to convert an integral equation into an algebraic; hence, the solution procedures are either reduced or simplified accordingly. The advantages of hybrid functions are that the values of n and m are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function, for the solutions of integral equations. We propose that the available optimal values of n and m can minimize the relative errors of the numerical solutions. The high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples. The hybrid function method is superior to other piecewise constant orthogonal functions [W.F. Blyth, R.L. May, P. Widyaningsih, Volterra integral equations solved in Fredholm form using Walsh functions, Anziam J. 45 (E) (2004) C269-C282; M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200 (2007) 12-20] for these problems.
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Sohrab Bazm
2016-11-01
Full Text Available Alternative Legendre polynomials (ALPs are used to approximate the solution of a class of nonlinear Volterra-Hammerstein integral equations. For this purpose, the operational matrices of integration and the product for ALPs are derived. Then, using the collocation method, the considered problem is reduced into a set of nonlinear algebraic equations. The error analysis of the method is given and the efficiency and accuracy are illustrated by applying the method to some examples.
A predictor-corrector scheme for solving the Volterra integral equation
Al Jarro, Ahmed
2011-08-01
The occurrence of late time instabilities is a common problem of almost all time marching methods developed for solving time domain integral equations. Implicit marching algorithms are now considered stable with various efforts that have been developed for removing low and high frequency instabilities. On the other hand, literature on stabilizing explicit schemes, which might be considered more efficient since they do not require a matrix inversion at each time step, is practically non-existent. In this work, a stable but still explicit predictor-corrector scheme is proposed for solving the Volterra integral equation and its efficacy is verified numerically. © 2011 IEEE.
Nonclassical linear Volterra equations of the first kind
Apartsyn, Anatoly S
2003-01-01
This monograph deals with linear integra Volterra equations of the first kind with variable upper and lower limits of integration. Volterra operators of this type are the basic operators for integral models of dynamic systems.
Existence of Solutions to Nonlinear Impulsive Volterra Integral Equations in Banach Spaces
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CHEN Fangqi; TIAN Ruilan
2005-01-01
In this paper, the existence of solutions is studied for nonlinear impulsive Volterra integral equations with infinite moments of impulse effect on the half line R+ in Banach spaces.By the use of a new comparison result and recurrence method, the new existence theorems are achieved under a weaker compactness-type condition, which generalize and improve the related results for this class of equations with finite moments of impulse effect on finite interval and infinite moments of impulse effect on infinite interval.
Wei, Yunxia; Chen, Yanping; Shi, Xiulian; Zhang, Yuanyuan
2016-01-01
We present in this paper the convergence properties of Jacobi spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation. The solution is sufficiently smooth while the source function and the kernel function are smooth. We choose the Jacobi-Gauss points associated with the multidimensional Jacobi weight function [Formula: see text] (d denotes the space dimensions) as the collocation points. The error analysis in [Formula: see text]-norm and [Formula: see text]-norm theoretically justifies the exponential convergence of spectral collocation method in multidimensional space. We give two numerical examples in order to illustrate the validity of the proposed Jacobi spectral collocation method.
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.
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Farshid Mirzaee
2014-06-01
Full Text Available In this paper, we present a numerical method for solving two-dimensional Fredholm–Volterra integral equations (F-VIE. The method reduces the solution of these integral equations to the solution of a linear system of algebraic equations. The existence and uniqueness of the solution and error analysis of proposed method are discussed. The method is computationally very simple and attractive. Finally, numerical examples illustrate the efficiency and accuracy of the method.
Optimal control of stochastic difference Volterra equations an introduction
Shaikhet, Leonid
2015-01-01
This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equation...
Integrability of some generalized Lotka - Volterra systems
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Bountis, T.C.; Bier, M.; Hijmans, J.
1983-08-08
Several integrable systems of nonlinear ordinary differential equations of the Lotka-Volterra type are identified by the Painleve property and completely integrated. One such integrable case of N first order ode's is found, with N - 2 free parameters and N arbitrary. The concept of integrability of a general dynamical system, not necessarily derived from a hamiltonian, is also discussed.
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DING Hao-jiang; WANG Hui-ming
2005-01-01
The elastodynamic problems of magneto-electro-elastic hollow cylinders in the state ofaxisymmetric plane strain case can be transformed into two Volterra integral equations of the second kind about two functions with respect to time. Interpolation functions were introduced to approximate two unknown functions in each time subinterval and two new recursive formulae are derived. By using the recursive formulae, numerical results were obtained step by step. Under the same time step, the accuracy of the numerical results by the present method is much higher than that by the traditional quadrature method.
Al Jarro, Ahmed
2011-08-01
A hybrid MPI/OpenMP scheme for efficiently parallelizing the explicit marching-on-in-time (MOT)-based solution of the time-domain volume (Volterra) integral equation (TD-VIE) is presented. The proposed scheme equally distributes tested field values and operations pertinent to the computation of tested fields among the nodes using the MPI standard; while the source field values are stored in all nodes. Within each node, OpenMP standard is used to further accelerate the computation of the tested fields. Numerical results demonstrate that the proposed parallelization scheme scales well for problems involving three million or more spatial discretization elements. © 2011 IEEE.
Qualitative permanence of Lotka-Volterra equations.
Hofbauer, Josef; Kon, Ryusuke; Saito, Yasuhisa
2008-12-01
In this paper, we consider permanence of Lotka-Volterra equations. We investigate the sign structure of the interaction matrix that guarantees the permanence of a Lotka-Volterra equation whenever it has a positive equilibrium point. An interaction matrix with this property is said to be qualitatively permanent. Our results provide both necessary and sufficient conditions for qualitative permanence.
Integrable deformations of Lotka-Volterra systems
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Ballesteros, Angel, E-mail: angelb@ubu.es [Departamento de Fisica, Universidad de Burgos, 09001 Burgos (Spain); Blasco, Alfonso, E-mail: ablasco@ubu.es [Departamento de Fisica, Universidad de Burgos, 09001 Burgos (Spain); Musso, Fabio, E-mail: fmusso@ubu.es [Departamento de Fisica, Universidad de Burgos, 09001 Burgos (Spain)
2011-09-05
The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map Δ{sup (2)} that is given by the group multiplication provides the keystone for the explicit construction of a new family of 3N-dimensional integrable systems that, under certain constraints, contain N sets of deformed versions of the 3D LV equations. Moreover, by considering the most generic Poisson-Lie structure on this group, a new two-parametric integrable perturbation of the 3D LV system through polynomial and rational perturbation terms is explicitly found. -- Highlights: → A new Poisson-Lie approach to the integrability of Lotka-Volterra system is given. → New integrable deformations of the 3D Lotka-Volterra system are obtained. → Integrable Lotka-Volterra-type equations in 3N dimensions are deduced.
An integrable symmetric (2+1)-dimensional Lotka-Volterra equation and a family of its solutions
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Hu Xingbiao [Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, PO Box 2719, Beijing 100080 (China); Li Chunxia [Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, PO Box 2719, Beijing 100080 (China); Nimmo, Jonathan J C [Department of Mathematics, University of Glasgow, Glasgow G12 8QW (United Kingdom); Yu Guofu [Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, PO Box 2719, Beijing 100080 (China)
2005-01-07
A symmetric (2+1)-dimensional Lotka-Volterra equation is proposed. By means of a dependent variable transformation, the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables. A bilinear Baecklund transformation is found and then the corresponding Lax pair is derived. Explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric (2+1)-dimensional Lotka-Volterra equation are given. As a special case of the pfaffian solutions, we obtain soliton solutions and dromions.
Sidorov, Denis
2012-01-01
Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution's regular part refinement is proposed using the successive approximations method.
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Emiko ISHIWATA; Yoshiaki MUROYA
2009-01-01
To compute long term integrations for the pantograph differential equation with proportional delay qt,0 ＜ q ≤1:y'(t) = ay(t) + by(qt) +f(t),y(0) = Yo,we offer two kinds of numerical methods using special mesh distributions,that is,a rational approximant with 'quasi-uniform meshes'(see E.Ishiwata and Y.Muroya [Appl.Math.Comput.,2007,187:741-747]) and a Gauss collocation method with 'quasi-constrained meshes'.If we apply these meshes to rational approximant and Gauss collocation method,respectively,then we obtain useful numerical methods of order p* = 2m for computing long term integrations. Numerical investigations for these methods are also presented.
An existence theorem for Volterra integrodifferential equations with infinite delay
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Ferenc Izsak
2003-01-01
Full Text Available Using Schauder's fixed point theorem, we prove an existence theorem for Volterra integrodifferential equations with infinite delay. As an appplication, we consider an $n$ species Lotka-Volterra competitive system.
Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations
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E. Messina
2008-01-01
Full Text Available We consider nonlinear difference equations of unbounded order of the form xi=bi−∑j=0iai,jfi−j(xj, i=0,1,2,…, where fj(x (j=0,…,i are suitable functions. We establish sufficient conditions for the boundedness and the convergence of xi as i→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.
Time Reversal of Volterra Processes Driven Stochastic Differential Equations
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L. Decreusefond
2013-01-01
Full Text Available We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past-dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.
On stochastic fractional Volterra equations in Hilbert space
Karczewska, Anna; Lizama, Carlos
2006-01-01
In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties of stochastic convolutions are given. The paper provides a sufficient condition for a stochastic convolution to be a strong solution to a stochastic Volterra equation.
On various integrable discretizations of a general two-component Volterra system
Babalic, Corina N.; Carstea, A. S.
2013-04-01
We present two integrable discretizations of a general differential-difference bicomponent Volterra system. The results are obtained by discretizing directly the corresponding Hirota bilinear equations in two different ways. Multisoliton solutions are presented together with a new discrete form of Lotka-Volterra equation obtained by an alternative bilinearization.
Calculation of Volterra kernels for solutions of nonlinear differential equations
van Hemmen, JL; Kistler, WM; Thomas, EGF
2000-01-01
We consider vector-valued autonomous differential equations of the form x' = f(x) + phi with analytic f and investigate the nonanticipative solution operator phi bar right arrow A(phi) in terms of its Volterra series. We show that Volterra kernels of order > 1 occurring in the series expansion of
Calculation of Volterra kernels for solutions of nonlinear differential equations
van Hemmen, JL; Kistler, WM; Thomas, EGF
2000-01-01
We consider vector-valued autonomous differential equations of the form x' = f(x) + phi with analytic f and investigate the nonanticipative solution operator phi bar right arrow A(phi) in terms of its Volterra series. We show that Volterra kernels of order > 1 occurring in the series expansion of th
POSITIVE PERIODIC SOLUTIONS OFIMPULSIVE LATKA-VOLTERRA EQUATIONS
Institute of Scientific and Technical Information of China (English)
LiJianli; ShenJianhua
2005-01-01
By using the continuation of coincidence degree theory, we study the periodic Lotka-Volterra equations with impulses, and some sufficient onditions for the existence of positive periodic solutions are obtained.
Food Web Assembly Rules for Generalized Lotka-Volterra Equations
DEFF Research Database (Denmark)
Härter, Jan Olaf Mirko; Mitarai, Namiko; Sneppen, Kim
2016-01-01
In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. For the generalized Lotka-Volterra equations, sustainable coexistence necessitates nonzero determinant of the interaction matrix. Here we show that this......In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. For the generalized Lotka-Volterra equations, sustainable coexistence necessitates nonzero determinant of the interaction matrix. Here we show...
Sidorov, Denis
2011-01-01
The Volterra integral equations of the first kind with piecewise smooth kernel are considered. Such equations appear in the theory of optimal control of the evolving systems. The existence theorems are proved. The method for constructing approximations of parametric families of solutions of such equations is suggested. The parametric family of solutions is constructed in terms of a logarithmic-power asymptotics.
Existence Theorem for Integral and Functional Integral Equations with Discontinuous Kernels
2012-01-01
Existence of extremal solutions of nonlinear discontinuous integral equations of Volterra type is proved. This result is extended herein to functional Volterra integral equations (FVIEs) and to a system of discontinuous VIEs as well.
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Hermann BRUNNER
2009-01-01
The aims of this paper are (i) to present a survey of recent advances in the analysis of superconvergence of collocation solutions for linear Volterra-type functional integral and integro-differential equations with delay functions θ(t) vanishing at the initial point of the interval of integration (with θ(t) = qt (0 ＜ q ＜ 1,t ≥0) being an important special case),and (ii) to point,by means of a list of open problems,to areas in the numerical analysis of such Volterra functional equations where more research needs to be carried out.
Some stability conditions for scalar Volterra difference equations
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Leonid Berezansky
2016-01-01
Full Text Available New explicit stability results are obtained for the following scalar linear difference equation \\[x(n+1-x(n=-a(nx(n+\\sum_{k=1}^n A(n,kx(k+f(n\\] and for some nonlinear Volterra difference equations.
Xu, Run; Ma, Xiangting
2017-01-01
In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations.
Hungry Volterra equation, multi boson KP hierarchy and Two Matrix Models
Hisakado, M
1998-01-01
We consider the hungry Volterra hierarchy from the view point of the multi boson KP hierarchy. We construct the hungry Volterra equation as the ``fractional '' BT. We also study the relations between the (discrete time) hungry Volterra equation and two matrix models. From this point of view we study the reduction from (discrete time) 2d Toda lattice to the (discrete time) hungry Volterra equation.
The Asymptotic Behavior for Numerical Solution of a Volterra Equation
Institute of Scientific and Technical Information of China (English)
Da Xu
2003-01-01
Long-time asymptotic stability and convergence properties for the numerical solution of a Volterra equation of parabolic type are studied. The methods are based on the first-second order backward difference methods. The memory term is approximated by the convolution quadrature and the interpolant quadrature. Discretization of the spatial partial differential operators by the finite element method is also considered.
Splitting methods for partial Volterra integro-differential equations
Brunner, H.; Houwen, P.J. van der; Sommeijer, B.P.
1999-01-01
The spatial discretization of initial-value problems for (nonlinear) parabolic or hyperbolic PDEs with memory terms leads to (large) systems of Volterra integro-differential equations (VIDEs). In this paper we study the efficient numerical solution of such systems by methods based on linear multiste
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Ammar Ali Neamah
2014-01-01
Full Text Available The paper uses the Local fractional variational Iteration Method for solving the second kind Volterra integro-differential equations within the local fractional integral operators. The analytical solutions within the non-differential terms are discussed. Some illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems for the integral equations.
Volterra equation for pricing and hedging in a regime switching market
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Anindya Goswami
2014-12-01
Full Text Available It is known that the risk minimizing price of European options in Markov-modulated market satisfies a system of coupled PDE, known as generalized B–S–M PDE. In this paper, another system of equations, which can be categorized as a Volterra integral equations of second kind, are considered. It is shown that this system of integral equations has smooth solution and the solution solves the generalized B–S–M PDE. Apart from showing existence and uniqueness of the PDE, this IE representation helps to develop a new computational method. It enables to compute the European option price and corresponding optimal hedging strategy by using quadrature method.
A mathematical model on fractional Lotka-Volterra equations.
Das, S; Gupta, P K
2011-05-21
The article presents the solutions of Lotka-Volterra equations of fractional-order time derivatives with the help of analytical method of nonlinear problem called the homotopy perturbation method (HPM). By using initial values, the explicit solutions of predator and prey populations for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical biological model. Copyright © 2011 Elsevier Ltd. All rights reserved.
One-step block method for solving Volterra integro-differential equations
Mohamed, Nurul Atikah binti; Majid, Zanariah Abdul
2015-10-01
One-step block method for solving linear Volterra integro-differential equations (VIDEs) is presented in this paper. In VIDEs, the unknown function appears in the form of derivative and under the integral sign. The popular methods for solving VIDEs are the method of quadrature or quadrature method combined with numerical method. The proposed block method will solve the ordinary differential equations (ODEs) part and Newton-Cotes quadrature rule is applied to calculate the integral part of VIDEs. Numerical problems are presented to illustrate the performance of the proposed method.
Stochastic Volterra Equation Driven by Wiener Process and Fractional Brownian Motion
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Zhi Wang
2013-01-01
Full Text Available For a mixed stochastic Volterra equation driven by Wiener process and fractional Brownian motion with Hurst parameter H>1/2, we prove an existence and uniqueness result for this equation under suitable assumptions.
Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW
Zhu, Li; Fan, Qibin
2013-05-01
Fractional calculus is an extension of derivatives and integrals to non-integer orders and has been widely used to model scientific and engineering problems. In this paper, we describe the fractional derivative in the Caputo sense and give the second kind Chebyshev wavelet (SCW) operational matrix of fractional integration. Then based on above results we propose the SCW operational matrix method to solve a kind of nonlinear fractional-order Volterra integro-differential equations. The main characteristic of this approach is that it reduces the integro-differential equations into a nonlinear system of algebraic equations. Thus, it can simplify the problem of fractional order equation solving. The obtained numerical results indicate that the proposed method is efficient and accurate for this kind equations.
Bi-Hamiltonian systems and Lotka-Volterra equations: A three dimensional classification
Plank, Manfred
1995-01-01
We study three dimensional bi-Hamiltonian systems in general and use the obtained results to classify all three dimensional Lotka-Volterra equations, which admit a bi-Hamiltonian representation. In der vorliegenden Arbeit studieren wir drei-dimensionale bi-Hamiltonsche Systeme und klassifizieren alle drei-dimensionalen Lotka-Volterra Gleichungen, welche eine bi-Hamiltonsche Darstellung zulassen.
Food Web Assembly Rules for Generalized Lotka-Volterra Equations
DEFF Research Database (Denmark)
Härter, Jan Olaf Mirko; Mitarai, Namiko; Sneppen, Kim
2016-01-01
apparent competition. In agreement with data, the assembly rules predict high species numbers at intermediate levels and thinning at the top and bottom. Using comprehensive food web data, we demonstrate how omnivores or parasites with hosts at multiple trophic levels can loosen the constraints and help......In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. For the generalized Lotka-Volterra equations, sustainable coexistence necessitates nonzero determinant of the interaction matrix. Here we show...... that this requirement is equivalent to demanding that each species be part of a non-overlapping pairing, which substantially constrains the food web structure. We demonstrate that a stable food web can always be obtained if a non-overlapping pairing exists. If it does not, the matrix rank can be used to quantify...
Direct and inverse solver for the 3D optoacoustic Volterra equation
Stritzel, J; Wollweber, M; Roth, B
2016-01-01
The direct problem of optoacoustic signal generation in biological media consists of solving the inhomogeneous optoacoustic wave equation for an initial acoustic stress profile. In contrast, the mathematically challenging inverse problem requires the reconstruction of the initial stress profile from a proper set of observed signals. In this article, we consider the particular case of a Gaussian transverse irradiation source profile in the paraxial approximation of the wave equation, for which the direct problem along the beam axis can be cast into a linear Volterra integral equation of the second kind. This integral equation can be used in two ways: as a forward solver to predict optoacoustic signals in terms of the direct problem, and as an inverse solver for which we here devise highly efficient numerical schemes used for the reconstruction of initial pressure profiles from observed signals, constituting a methodical progress of computational aspects of optoacoustics. In this regard, we explore the validity...
Local Integrability and Linearizability of Three-dimensional Lotka-Volterra Systems
Aziz, Waleed
2011-01-01
We investigate the local integrability and linearizability of three dimensional Lotka-Volterra equations at the origin. Necessary and sufficient conditions for both integrability and linearizability are obtained for (1,-1,1), (2,-1,1) and (1,-2,1)-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable.
Nonstandard numerical integrations of a Lotka-Volterra system
Bhowmik, S.K.
2009-01-01
In this article, we consider a three dimensional Lotka-Volterra system. We have developed some nonstandard numerical integrations of the model which preserve all properties of real solutions, and they are consistent. We have shown some numerical results to support this methods.
Integrability of Lotka-Volterra Planar Complex Cubic Systems
Dukarić, Maša; Giné, Jaume
In this paper, we study the Lotka-Volterra complex cubic systems. We obtain necessary conditions of integrability for these systems with some restriction on the parameters. The sufficiency is proved for all conditions, except one which remains open, using different methods.
Nonstandard numerical integrations of a Lotka-Volterra system
Bhowmik, S.K.
2009-01-01
In this article, we consider a three dimensional Lotka-Volterra system. We have developed some nonstandard numerical integrations of the model which preserve all properties of real solutions, and they are consistent. We have shown some numerical results to support this methods.
Existence and Boundedness of Solutions for Nonlinear Volterra Difference Equations in Banach Spaces
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Rigoberto Medina
2016-01-01
Full Text Available We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated.
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Ali Konuralp
2014-01-01
Full Text Available Application process of variational iteration method is presented in order to solve the Volterra functional integrodifferential equations which have multi terms and vanishing delays where the delay function θ(t vanishes inside the integral limits such that θ(t=qt for 0
Shakurov, I R; Asadullin, R M
2014-01-01
In this article we study the inverse problem of finding coefficients of Lotka-Volterra's equations on one given solution. The conditions of the uniqueness and existence of the inverse problem are found.
Food Web Assembly Rules for Generalized Lotka-Volterra Equations.
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Jan O Haerter
2016-02-01
Full Text Available In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. For the generalized Lotka-Volterra equations, sustainable coexistence necessitates nonzero determinant of the interaction matrix. Here we show that this requirement is equivalent to demanding that each species be part of a non-overlapping pairing, which substantially constrains the food web structure. We demonstrate that a stable food web can always be obtained if a non-overlapping pairing exists. If it does not, the matrix rank can be used to quantify the lack of niches, corresponding to unpaired species. For the species richness at each trophic level, we derive the food web assembly rules, which specify sustainable combinations. In neighboring levels, these rules allow the higher level to avert competitive exclusion at the lower, thereby incorporating apparent competition. In agreement with data, the assembly rules predict high species numbers at intermediate levels and thinning at the top and bottom. Using comprehensive food web data, we demonstrate how omnivores or parasites with hosts at multiple trophic levels can loosen the constraints and help obtain coexistence in food webs. Hence, omnivory may be the glue that keeps communities intact even under extinction or ecological release of species.
Food Web Assembly Rules for Generalized Lotka-Volterra Equations.
Haerter, Jan O; Mitarai, Namiko; Sneppen, Kim
2016-02-01
In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. For the generalized Lotka-Volterra equations, sustainable coexistence necessitates nonzero determinant of the interaction matrix. Here we show that this requirement is equivalent to demanding that each species be part of a non-overlapping pairing, which substantially constrains the food web structure. We demonstrate that a stable food web can always be obtained if a non-overlapping pairing exists. If it does not, the matrix rank can be used to quantify the lack of niches, corresponding to unpaired species. For the species richness at each trophic level, we derive the food web assembly rules, which specify sustainable combinations. In neighboring levels, these rules allow the higher level to avert competitive exclusion at the lower, thereby incorporating apparent competition. In agreement with data, the assembly rules predict high species numbers at intermediate levels and thinning at the top and bottom. Using comprehensive food web data, we demonstrate how omnivores or parasites with hosts at multiple trophic levels can loosen the constraints and help obtain coexistence in food webs. Hence, omnivory may be the glue that keeps communities intact even under extinction or ecological release of species.
Institute of Scientific and Technical Information of China (English)
娄梅枝
2003-01-01
In this paper, A.B.Mingarelli's result is generalized to General Volterra-Stieltjes Integro-differential Equations. Comparison theorem and equivalence condition of non-oscillation are obtained. Classical Sturm comparison theorem and some conclusions are generalized.
Generalized Bihari Type Integral Inequalities and the Corresponding Integral Equations
2009-01-01
We study some special nonlinear integral inequalities and the corresponding integral equations in measure spaces. They are significant generalizations of Bihari type integral inequalities and Volterra and Fredholm type integral equations. The kernels of the integral operators are determined by concave functions. Explicit upper bounds are given for the solutions of the integral inequalities. The integral equations are investigated with regard to the existence of a minimal and a maximal soluti...
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Berenguer MI
2010-01-01
Full Text Available This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces and .
Moiseiwitsch, B L
2005-01-01
Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco
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Omar Abu Arqub
2012-01-01
Full Text Available This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution ( is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution ( is obtained and it is proved to converge to the exact solution (. Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.
Numerical Integration and Synchronization for the 3-Dimensional Metriplectic Volterra System
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Gheorghe Ivan
2011-01-01
Full Text Available The main purpose of this paper is to study the metriplectic system associated to 3-dimensional Volterra model. For this system we investigate the stability problem and numerical integration via Kahan's integrator. Finally, the synchronization problem for two coupled metriplectic Volterra systems is discussed.
Institute of Scientific and Technical Information of China (English)
鲁世平
2003-01-01
By employing the theory of differential inequality and some analysis methods, a nonlinear boundary value problem subject to a general kind of second-order Volterra functional differential equation was considered first. Then, by constructing the right-side layer function and the outer solution, a nonlinear boundary value problem subject to a kind of second- order Volterra functional differential equation with a small parameter was studied further. By using the differential mean value theorem and the technique of upper and lower solution, a new result on the existence of the solutions to the boundary value problem is obtained, and a uniformly valid asymptotic expansions of the solution is given as well.
Institute of Scientific and Technical Information of China (English)
LI Shoufu
2005-01-01
A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.
B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations
Institute of Scientific and Technical Information of China (English)
LI; Shoufu(李寿佛)
2003-01-01
B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations (VFDEs) are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differentialequations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs ofother type which appear in practice.
Ullah, Saif; Farooq, Muhammad; Ahmad, Latif; Abdullah, Saleem
2014-01-01
Fuzzy partial integro-differential equations have a major role in the fields of science and engineering. In this paper, we propose the solution of fuzzy partial Volterra integro-differential equation with convolution type kernel using fuzzy Laplace transform method (FLTM) under Hukuhara differentiability. It is shown that FLTM is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated with few examples to show the ability of the proposed method.
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John A. D. Appleby
2008-01-01
Full Text Available This paper considers necessary and sufficient conditions for the solution of a stochastically and deterministically perturbed Volterra equation to converge exponentially to a nonequilibrium and nontrivial limit. Convergence in an almost sure and pth mean sense is obtained.
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Behzad Ghanbari
2014-01-01
Full Text Available We aim to study the convergence of the homotopy analysis method (HAM in short for solving special nonlinear Volterra-Fredholm integrodifferential equations. The sufficient condition for the convergence of the method is briefly addressed. Some illustrative examples are also presented to demonstrate the validity and applicability of the technique. Comparison of the obtained results HAM with exact solution shows that the method is reliable and capable of providing analytic treatment for solving such equations.
Numerical solutions of stochastic Lotka-Volterra equations via operational matrices
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F. Hosseini Shekarabi
2016-03-01
Full Text Available In this paper, an efficient and convenient method for numerical solutions of stochastic Lotka-Volterra dynamical system is proposed. Here, we consider block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the procedure and accuracy of the operational matrices based on block pulse functions.
Difference equations and cluster algebras I: Poisson bracket for integrable difference equations
Inoue, Rei
2010-01-01
We introduce the cluster algebraic formulation of the integrable difference equations, the discrete Lotka-Volterra equation and the discrete Liouville equation, from the view point of the general T-system and Y-system. We also study the Poisson structure for the cluster algebra, and give the associated Poisson bracket for the two difference equations.
Institute of Scientific and Technical Information of China (English)
Rui QI; Cheng-jian ZHANG; Yu-jie ZHANG
2012-01-01
This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. We investigate the dissipativity properties of (k,l)-algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid.The finitedimensional and infinite-dimensional dissipativity results of (k,l)-algebraically stable Runge-Kutta methods are obtained.
Coexistence for systems governed by difference equations of Lotka-Volterra type.
Hofbauer, J; Hutson, V; Jansen, W
1987-01-01
The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.
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Salih Yalcinbas
2016-01-01
Full Text Available In this paper, a new collocation method based on the Fibonacci polynomials is introduced to solve the high-order linear Volterra integro-differential equations under the conditions. Numerical examples are included to demonstrate the applicability and validity of the proposed method and comparisons are made with the existing results. In addition, an error estimation based on the residual functions is presented for this method. The approximate solutions are improved by using this error estimation.
Tricomi, Francesco Giacomo
1957-01-01
This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient. The book is divided into four chapters, with two useful
An integrable Poisson map generated from the eigenvalue problem of the Lotka-Volterra hierarchy
Energy Technology Data Exchange (ETDEWEB)
Wu Yongtang [Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong (China); Wang Hongye [Department of Mathematics, Zhengzhou University, Henan (China); Du Dianlou [Department of Mathematics, Zhengzhou University, Henan (China)]. E-mail: ddl@zzu.edu.cn
2002-05-03
A 3x3 discrete eigenvalue problem associated with the Lotka-Volterra hierarchy is studied and the corresponding nonlinearized one, an integrable Poisson map with a Lie-Poisson structure, is also presented. Moreover, a 2x2 nonlinearized eigenvalue problem, which also begets the Lotka-Volterra hierarchy, is proved to be a reduction of the Poisson map on the leaves of the symplectic foliation. (author)
Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays
Institute of Scientific and Technical Information of China (English)
Chengming HUANG; Stefan VANDEWALLE
2009-01-01
This paper is concerned with the study of the stability of Runge Kutta-Pouzet methods for Volterra integro-differential equations with delays.We are interested in the comparison between the analytical and numerical stability regions.First,we focus on scalar equations with real coefficients.It is proved that all Gauss-Pouzet methods can retain the asymptotic stability of the analytical solution.Then,we consider the multidimensional case.A new stability condition for the stability of the analytical solution is given.Under this condition,the asymptotic stability of Gauss-Pouzet methods is investigated.
The Lotka-Volterra equation over a finite ring Z/p{sup N}Z
Energy Technology Data Exchange (ETDEWEB)
Matsutani, Shigeki E-mail: RXB01142@nifty.ne.jp
2001-12-07
The discrete Lotka-Volterra equation over p-adic space was constructed since p-adic space is a prototype of spaces with non-Archimedean valuations and the space given by taking the ultra-discrete limit studied in soliton theory should be regarded as a space with the non-Archimedean valuations given in my previous paper (Matsutani S 2001 Int. J. Math. Math. Sci.). In this paper, using the natural projection from a p-adic integer to a ring Z/p{sup n}Z, a soliton equation is defined over the ring. Numerical computations show that it behaves regularly. (author)
Institute of Scientific and Technical Information of China (English)
Tie Zhang; Yan-ping Lin; Robert J.Tait
2002-01-01
In this paper, we present a general error analysis framework for the finite volume element (FVE) approximation to the Ritz-Volterra projection, the Sobolev equations and parabolic integro-differential equations. The main idea in our paper is to consider the FVE methods as perturbations of standard finite element methods which enables us to derive the optimal L2 and H1 norm error estimates, and the L∞ and W1∞ norm error estimates by means of the time dependent Green functions. Our disc ussions also include elliptic and parabolic problems as the special cases.
Analog integrated circuits for the Lotka-Volterra competitive neural networks.
Asai, T; Ohtani, M; Yonezu, H
1999-01-01
A subthreshold MOS integrated circuit (IC) is designed and fabricated for implementing a competitive neural network of the Lotka-Volterra (LV) type which is derived from conventional membrane dynamics of neurons and is used for the selection of external inputs. The steady-state solutions to the LV equation can be classified into three types, each of which represents qualitatively different selection behavior. Among the solutions, the winners-share-all (WSA) solution in which a certain number of neurons remain activated in steady states is particularly useful owing to robustness in the selection of inputs from a noisy environment. The measured results of the fabricated LV IC's agree well with the theoretical prediction as long as the influence of device mismatches is small. Furthermore, results of extensive circuit simulations prove that the large-scale LV circuit producing the WSA solution does exhibit a reliable selection compared with winner-take-all circuits, in the possible presence of device mismatches.
Energy Technology Data Exchange (ETDEWEB)
Moriceau, Y. [Commissariat a l' Energie Atomique, Centre d' Etudes de Limeil, 94 - Villeneuve-Saint-Georges (France)
1968-03-01
It is well known, if not well explained, that photo cross-sections curves depend on numerical resolution; as well as many other physical solutions from integral equations of the first kind, they are oscillating. In the first part of this report, a typical example points out how oscillations are growing. In the second part, a new method is explained yielding a smooth resolution. From experimental data on equidistant intervals, we build functions expanded in Tchebycheff polynomials; the solution is of this kind. Then, the third part points out that semi-analytical resolutions of this problem are illusive. (author) [French] C'est un fait reconnu mais mal explique, que les courbes de sections efficaces photonucleaires dependent de la resolution numerique adoptee. Beaucoup d'autres solutions physiques extraites d'une equation integrale de 1ere espece sont dans ce cas; elles sont arbitraires et oscillatoires. Dans la 1ere partie de ce rapport, on montre, dans un cas particulier typique, comment se forment les oscillations. Dans la 2eme partie, on presente une methode originale qui permet d'obtenir une resolution exempte d'oscillations. A partir de donnees experimentales a intervalles equidistants, on construit des fonctions developpees en polynomes de Tchebycheff; la solution est de ce type. Enfin, on montre dans la 3eme partie que les resolutions semi-analytiques de ce probleme sont illusoires. (auteur)
DEFF Research Database (Denmark)
E. Barndorff-Nielsen, Ole; Benth, Fred Espen; Szozda, Benedykt
This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G* of Potthoff-Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed...
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole E.; Benth, Fred Espen; Szozda, Benedykt
This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G∗ of Potthoff--Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed...
Optimal Parametric Iteration Method for Solving Multispecies Lotka-Volterra Equations
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Vasile Marinca
2012-01-01
Full Text Available We apply an analytical method called the Optimal Parametric Iteration Method (OPIM to multispecies Lotka-Volterra equations. By using initial values, accurate explicit analytic solutions have been derived. The method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement has been demonstrated between the obtained solutions and the numerical ones. This new approach, which can be easily applied to other strongly nonlinear problems, is very effective and yields very accurate results.
Solutions of First-Order Volterra Type Linear Integrodifferential Equations by Collocation Method
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Olumuyiwa A. Agbolade
2017-01-01
Full Text Available The numerical solutions of linear integrodifferential equations of Volterra type have been considered. Power series is used as the basis polynomial to approximate the solution of the problem. Furthermore, standard and Chebyshev-Gauss-Lobatto collocation points were, respectively, chosen to collocate the approximate solution. Numerical experiments are performed on some sample problems already solved by homotopy analysis method and finite difference methods. Comparison of the absolute error is obtained from the present method and those from aforementioned methods. It is also observed that the absolute errors obtained are very low establishing convergence and computational efficiency.
Chakrabarti, Anindya S.
2016-01-01
We present a model of technological evolution due to interaction between multiple countries and the resultant effects on the corresponding macro variables. The world consists of a set of economies where some countries are leaders and some are followers in the technology ladder. All of them potentially gain from technological breakthroughs. Applying Lotka-Volterra (LV) equations to model evolution of the technology frontier, we show that the way technology diffuses creates repercussions in the partner economies. This process captures the spill-over effects on major macro variables seen in the current highly globalized world due to trickle-down effects of technology.
Normalized RBF networks: application to a system of integral equations
Energy Technology Data Exchange (ETDEWEB)
Golbabai, A; Seifollahi, S; Javidi, M [Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844 (Iran, Islamic Republic of)], E-mail: golbabai@iust.ac.ir, E-mail: seif@iust.ac.ir, E-mail: mojavidi@yahoo.com
2008-07-15
Linear integral and integro-differential equations of Fredholm and Volterra types have been successfully treated using radial basis function (RBF) networks in previous works. This paper deals with the case of a system of integral equations of Fredholm and Volterra types with a normalized radial basis function (NRBF) network. A novel learning algorithm is developed for the training of NRBF networks in which the BFGS backpropagation (BFGS-BP) least-squares optimization method as a recursive model is used. In the approach presented here, a trial solution is given by an NRBF network of incremental architecture with a set of unknown parameters. Detailed learning algorithms and concrete examples are also included.
Systems of nonlinear Volterra integro-differential equations of arbitrary order
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Kourosh Parand
2018-10-01
Full Text Available In this paper, a new approximate method for solving of systems of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs. Also, we construct the fractional derivative operational matrix of order $\\alpha$ in the Caputo's definition for GFCFs. This method reduced a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.
A periodic phase soliton of the ultradiscrete hungry Lotka-Volterra equation
Energy Technology Data Exchange (ETDEWEB)
Nakamura, Shinya [Major in Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555 (Japan)], E-mail: s-nakamura@moegi.waseda.jp
2009-12-11
We propose a new type of solution to the ultradiscrete hungry Lotka-Volterra (uhLV) equation. For the solution, the periodic phase is introduced into the known soliton and the extended soliton becomes a traveling wave showing a periodic variation. We call this type of wave a 'periodic phase soliton' (PPS). The solution has two forms of expression: one is the 'perturbation form' and the other is the 'ultradiscrete permanent form'. We analyze the interaction among PPSs and solitons. Moreover, we give the outline of proof to show that the solution satisfies the bilinear equation of the uhLV equation.
Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems
Damianou, P. A.; Evripidou, C. A.; Kassotakis, P.; Vanhaecke, P.
2017-03-01
Given a constant skew-symmetric matrix A, it is a difficult open problem whether the associated Lotka-Volterra system is integrable or not. We solve this problem in a special case when A is a Toeplitz matrix where all off-diagonal entries are plus or minus one. In this case, the associated Lotka-Volterra system turns out to be a reduction of Liouville integrable systems, whose integrability was shown by Bogoyavlenskij and Itoh. We prove that the reduced systems are also Liouville integrable and that they are also non-commutative integrable by constructing a set of independent first integrals, having the required involutive properties (with respect to the Poisson bracket). These first integrals fall into two categories. One set consists of polynomial functions that are restriction of the Bogoyavlenskij-Itoh integrals; their involutivity was already pointed out by Bogoyavlenskij. The other set consists of rational functions which are obtained through a Poisson map from the first integrals of some recently discovered superintegrable Lotka-Volterra systems. The fact that these polynomial and rational first integrals, combined, have the required properties for Liouville and non-commutative integrability is quite remarkable; the quite technical proof of functional independence of the first integrals is given in detail.
线性抛物Volterra差分方程的全局吸引性%Global Attractivity of Linear Parabolic Volterra Difference Equations
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黄立
2004-01-01
In this paper, by constructing Liapunov Sequences, we study the golbal attractivity of linear Parabolic volterra difference equations of neutral type and obtain some sufficient conditions for the global attractivity of the zero solution of above equations.
Global attractor of coupled difference equations and applications to Lotka-Volterra systems
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Pao CV
2005-01-01
Full Text Available This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems with time delays. The aim of the paper is to show the existence and uniqueness of a positive solution and to investigate the asymptotic behavior of the positive solution. Sufficient conditions are given to ensure that a unique positive equilibrium solution exists and is a global attractor of the difference system. Applications are given to three basic types of Lotka-Volterra systems with time delays where some easily verifiable conditions on the reaction rate constants are obtained for ensuring the global attraction of a positive equilibrium solution.
Global attractor of coupled difference equations and applications to Lotka-Volterra systems
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C. V. Pao
2005-03-01
Full Text Available This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems with time delays. The aim of the paper is to show the existence and uniqueness of a positive solution and to investigate the asymptotic behavior of the positive solution. Sufficient conditions are given to ensure that a unique positive equilibrium solution exists and is a global attractor of the difference system. Applications are given to three basic types of Lotka-Volterra systems with time delays where some easily verifiable conditions on the reaction rate constants are obtained for ensuring the global attraction of a positive equilibrium solution.
On the polynomial first integrals of the ({ital a},{ital b},{ital c}) Lotka{endash}Volterra system
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Labrunie, S. [Service de physique de l`etat condense, Centre d`etudes de Saclay, 91191 Gif sur Yvette (France)
1996-11-01
Using elementary differential algebraic techniques, we prove that the 3D Lotka{endash}Volterra dynamical system has no other nontrivial polynomial first integrals than the previously known ones. {copyright} {ital 1996 American Institute of Physics.}
Lotka-Volterra system with Volterra multiplier.
Gürlebeck, Klaus; Ji, Xinhua
2011-01-01
With the aid of Volterra multiplier, we study ecological equations for both tree system and cycle system. We obtain a set of sufficient conditions for the ultimate boundedness to nonautonomous n-dimensional Lotka-Volterra tree systems with continuous time delay. The criteria are applicable to cooperative model, competition model, and predator-prey model. As to cycle system, we consider a three-dimensional predator-prey Lotka-Volterra system. In order to get a condition under which the system is globally asymptotic stable, we obtain a Volterra multiplier, so that in a parameter region the system is with the Volterra multiplier it is globally stable. We have also proved that in regions in which the condition is not satisfied, the system is unstable or at least it is not globally stable. Therefore, we say that the three-dimensional cycle system is with global bifurcation.
Kashani, Mahsa H.; Ghorbani, Mohammad Ali; Dinpashoh, Yagob; Shahmorad, Sedaghat
2016-09-01
Rainfall-runoff simulation is an important task in water resources management. In this study, an integrated Volterra model with artificial neural networks (IVANN) was presented to simulate the rainfall-runoff process. The proposed integrated model includes the semi-distributed forms of the Volterra and ANN models which can explore spatial variation in rainfall-runoff process without requiring physical characteristic parameters of the catchments, while taking advantage of the potential of Volterra and ANNs models in nonlinear mapping. The IVANN model was developed using hourly rainfall and runoff data pertaining to thirteen storms to study short-term responses of a forest catchment in northern Iran; and its performance was compared with that of semi-distributed integrated ANN (IANN) model and lumped Volterra model. The Volterra model was applied as a nonlinear model (second-order Volterra (SOV) model) and solved using the ordinary least square (OLS) method. The models performance were evaluated and compared using five performance criteria namely coefficient of efficiency, root mean square error, error of total volume, relative error of peak discharge and error of time for peak to arrive. Results showed that the IVANN model performs well than the other semi-distributed and lumped models to simulate the rainfall-runoff process. Comparing to the integrated models, the lumped SOV model has lower precision to simulate the rainfall-runoff process.
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shadan sadigh behzadi
2012-03-01
Full Text Available In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integro-differential equation by using the following powerful, efficient but simple methods: (i Modified Adomian decomposition method (MADM, (ii Variational iteration method (VIM, (iii Homotopy analysis method (HAM and (iv Modified homotopy perturbation method (MHPM. The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate the accuracy of the presented methods.
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M.H. Tiwana
2017-04-01
Full Text Available This work investigates the fractional non linear reaction diffusion (FNRD system of Lotka-Volterra type. The system of equations together with the boundary conditions are solved by Homotopy perturbation transform method (HPTM. The series solutions are obtained for the two cases (homogeneous and non-homogeneous of FNRD system. The effect of fractional parameter on the mass concentration of two species are shown and discussed with the help of 3D graphs.
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Mohammed M. Matar
2009-12-01
Full Text Available In this article we study the fractional semilinear mixed Volterra-Fredholm integrodifferential equation $$ frac{d^{alpha }x(t}{dt^{alpha }} =Ax(t+fBig(t,x(t, int_{t_0}^tk(t,s,x(sds,int_{t_0}^{T}h(t,s,x(sdsBig , $$ where $tin [t_0,T]$, $t_0geq 0$, $0
Controllability of Urysohn integral inclusions of Volterra type
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Thomas S. Angell
2010-06-01
Full Text Available The aim of this paper is to study the controllability of a system described by an integral inclusion of Urysohn type with delay. In our approach we reduce the controllability problem of the nonlinear system into solvability problem of another integral inclusion. The solvability of this integral inclusion is subsequently established by imposing suitable standard boundedness, convexity and semicontinuity conditions on the set-valued mapping defining the integral inclusion, and by employing Bohnenblust-Karlin extension of Kakutani's fixed point theorem for set-valued mappings.
Frank, T. D.
The Lotka-Volterra-Haken equations have been frequently used in ecology and pattern formation. Recently, the equations have been proposed by several research groups as amplitude equations for task-related patterns of brain activity. In this theoretical study, the focus is on the circular causality aspect of pattern formation systems as formulated within the framework of synergetics. Accordingly, the stable modes of a pattern formation system inhibit the unstable modes, whereas the unstable modes excite the stable modes. Using this circular causality principle it is shown that under certain conditions the Lotka-Volterra-Haken amplitude equations can be derived from a general model of brain activity akin to the Wilson-Cowan model. The model captures the amplitude dynamics for brain activity patterns in experiments involving several consecutively performed multiple-choice tasks. This is explicitly demonstrated for two-choice tasks involving grasping and walking. A comment on the relevance of the theoretical framework for clinical psychology and schizophrenia is given as well.
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Monte, Luigi [ENEA CR Casaccia, Via P. Anguillarese 301 00100 Rome (Italy)], E-mail: luigi.monte@enea.it
2009-06-15
The present work describes a model for predicting the population dynamics of the main components (resources and consumers) of terrestrial ecosystems exposed to ionising radiation. The ecosystem is modelled by the Lotka-Volterra equations with consumer competition. Linear dose-response relationships without threshold are assumed to relate the values of the model parameters to the dose rates. The model accounts for the migration of consumers from areas characterised by different levels of radionuclide contamination. The criteria to select the model parameter values are motivated by accounting for the results of the empirical studies of past decades. Examples of predictions for long-term chronic exposure are reported and discussed.
Monte, Luigi
2009-06-01
The present work describes a model for predicting the population dynamics of the main components (resources and consumers) of terrestrial ecosystems exposed to ionising radiation. The ecosystem is modelled by the Lotka-Volterra equations with consumer competition. Linear dose-response relationships without threshold are assumed to relate the values of the model parameters to the dose rates. The model accounts for the migration of consumers from areas characterised by different levels of radionuclide contamination. The criteria to select the model parameter values are motivated by accounting for the results of the empirical studies of past decades. Examples of predictions for long-term chronic exposure are reported and discussed.
Numerical solution of functional integral equations by using B-splines
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Reza Firouzdor
2014-05-01
Full Text Available This paper describes an approximating solution, based on Lagrange interpolation and spline functions, to treat functional integral equations of Fredholm type and Volterra type. This method can be extended to functional dierential and integro-dierential equations. For showing eciency of the method we give some numerical examples.
Equivalent HPM with ADM and Convergence of the HPM to a Class of Nonlinear Integral Equations
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J. Manafian Heris
2013-03-01
Full Text Available The purpose of this study is to implement homotopy perturbation method, for solving nonlinear Volterra integral equations. In this work, a reliable approach for convergence of the HPM when applied to a class of nonlinear Volterra integral equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The results obtained by using HPM, are compared to those obtained by using Adomian decomposition method alone. The numerical results, demonstrate that HPM technique, gives the approximate solution with faster convergence rate and higher accuracy than using the standard ADM
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Lee, C.E.
1976-08-01
The Volterra method of the multiplicative integral is used to determine the isotopic density, mass, and energy production in linear systems. The solution method, assumptions, and limitations are discussed. The method allows a rapid accurate calculation of the change in isotopic density, mass, and energy production independent of the magnitude of the time steps, production or decay rates, or flux levels.
Energy Technology Data Exchange (ETDEWEB)
Cairo, Laurent [MAPMO/CNRS-Departement de Mathematiques, Universite d' Orleans, 45067 Orleans, Cedex 2 (France); Llibre, Jaume [Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona (Spain)
2007-06-15
We classify all the global phase portraits of the cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2. For such vector fields there are exactly 28 different global phase portraits in the Poincare disc up to a reversal of sense of all orbits.
Lotka-Volterra equations with chemotaxis: walls, barriers and travelling waves.
Pettet, G J; McElwain, D L; Norbury, J
2000-12-01
In this paper we consider a simple two species model for the growth of new blood vessels. The model is based upon the Lotka-Volterra system of predator and prey interaction, where we identify newly developed capillary tips as the predator species and a chemoattractant which directs their motion as the prey. We extend the Lotka-Volterra system to include a one-dimensional spatial dependence, by allowing the predators to migrate in a manner modelled on the phenomenon of chemotaxis. A feature of this model is its potential to support travelling wave solutions. We emphasize that in order to determine the existence of such travelling waves it is essential that the global relationships of a number of phase plane features other than the equilibria be investigated.
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A. Sami Bataineh
2008-01-01
Full Text Available The time evolution of the multispecies Lotka-Volterra system is investigated by the homotopy analysis method (HAM. The continuous solution for the nonlinear system is given, which provides a convenient and straightforward approach to calculate the dynamics of the system. The HAM continuous solution generated by polynomial base functions is of comparable accuracy to the purely numerical fourth-order Runge-Kutta method. The convergence theorem for the three-dimensional case is also given.
Bouse, E
2010-01-01
We consider systems where a cycle born via the Hopf bifurcation blows up to infinity as a parameter ranges over a finite interval. Two examples demonstrating this effect are presented: planar Lotka-Volterra type systems with a competition-cooperation term and quasi-linear higher order equations.
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Itoh, Yoshiaki [Institute of Statistical Mathematics and the Graduate University for Advanced Studies, 4-6-7 Minami-Azabu Minatoku, Tokyo 106-8569 (Japan)], E-mail: itoh@ism.ac.jp
2009-01-16
The combinatorial method is useful to obtain conserved quantities for some nonlinear integrable systems, as an alternative to the Lax representation method. Here we extend the combinatorial method and introduce an elementary geometry to show the vanishing of the Poisson brackets of the Hamiltonian structure for a Lotka-Volterra system of competing species. We associate a set of points on a circle with a set of species of the Lotka-Volterra system, where the dominance relations between points are given by the dominance relations between the species. We associate each term of the conserved quantities with a subset of points on the circle, which simplifies to show the vanishing of the Poisson brackets.
Institute of Scientific and Technical Information of China (English)
Shu-hua Zhang; Tao Lin; Yan-ping Lin; Ming Rao
2001-01-01
In this paper we will show that the Richardson extrapolation can be used to enhance the numerical solution generated by a Petrov-Galerkin finite element method for the initialvalue problem for a nonlinear Volterra integro-differential equation. As by-products, we will also show that these enhanced approximations can be used to form a class of aposteriori estimators for this Petrov-Galerkin finite element method. Numerical examples are supplied to illustrate the theoretical results.
Numerical treatments for solving nonlinear mixed integral equation
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M.A. Abdou
2016-12-01
Full Text Available We consider a mixed type of nonlinear integral equation (MNLIE of the second kind in the space C[0,T]×L2(Ω,T<1. The Volterra integral terms (VITs are considered in time with continuous kernels, while the Fredholm integral term (FIT is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.
Wilson, Alan
2008-08-01
It is shown that Boltzmann's methods from statistical physics can be applied to a much wider range of systems, and in a variety of disciplines, than has been commonly recognized. A similar argument can be applied to the ecological models of Lotka and Volterra. Furthermore, it is shown that the two methodologies can be applied in combination to generate the Boltzmann, Lotka and Volterra (BLV) models. These techniques enable both spatial interaction and spatial structural evolution to be modelled, and it is argued that they potentially provide a much richer modelling methodology than that currently used in the analysis of 'scale-free' networks.
Multidimensional singular integrals and integral equations
Mikhlin, Solomon Grigorievich; Stark, M; Ulam, S
1965-01-01
Multidimensional Singular Integrals and Integral Equations presents the results of the theory of multidimensional singular integrals and of equations containing such integrals. Emphasis is on singular integrals taken over Euclidean space or in the closed manifold of Liapounov and equations containing such integrals. This volume is comprised of eight chapters and begins with an overview of some theorems on linear equations in Banach spaces, followed by a discussion on the simplest properties of multidimensional singular integrals. Subsequent chapters deal with compounding of singular integrals
On generalized Volterra systems
Charalambides, S. A.; Damianou, P. A.; Evripidou, C. A.
2015-01-01
We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The algorithm uses the root system of a complex simple Lie algebra. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a system to a quadratic Lotka-Volterra system. We present in detail all such systems in the cases of A3, A4 and we also give some examples from higher dimensions. We classify all possible Lotka-Volterra systems that arise via this algorithm in the An case.
Romano, Alessandro
This article is a first application of an integrable nonautonomous Lotka-Volterra (LV) model to the study of tourism dynamics. In particular, we analyze the interaction in terms of touristic flows among three Italian regions. Confirming an hypothesis advanced by recent theoretical works, we find that these regions not only compete against each other, but at times they also proceed in mutualism. Moreover, the kind and the intensity of the interaction changes over time, suggesting that dynamic models can play a vital role in the study of touristic flows.
Romano, Alessandro
2016-01-01
This article is a first application of an integrable nonautonomous Lotka–Volterra (LV) model to the study of tourism dynamics. In particular, we analyze the interaction in terms of touristic flows among three Italian regions. Confirming an hypothesis advanced by recent theoretical works, we find that these regions not only compete against each other, but at times they also proceed in mutualism. Moreover, the kind and the intensity of the interaction changes over time, suggesting that dynamic models can play a vital role in the study of touristic flows. PMID:27661615
Integral equations and their applications
Rahman, M
2007-01-01
For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of in
A Two-Species Cooperative Lotka-Volterra System of Degenerate Parabolic Equations
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Jiebao Sun
2011-01-01
parabolic equations. We are interested in the coexistence of the species in a bounded domain. We establish the existence of global generalized solutions of the initial boundary value problem by means of parabolic regularization and also consider the existence of the nontrivial time-periodic solution for this system.
Enclosing Solutions of Integral Equations
DEFF Research Database (Denmark)
Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole
1996-01-01
We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because...
Enclosing Solutions of Integral Equations
DEFF Research Database (Denmark)
Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole
1996-01-01
We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because...
Integrable Equations on Time Scales
Gurses, Metin; Guseinov, Gusein Sh.; Silindir, Burcu
2005-01-01
Integrable systems are usually given in terms of functions of continuous variables (on ${\\mathbb R}$), functions of discrete variables (on ${\\mathbb Z}$) and recently in terms of functions of $q$-variables (on ${\\mathbb K}_{q}$). We formulate the Gel'fand-Dikii (GD) formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations. In particular they yield integrable equations over integers (difference equations) and over $q...
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Roberts Jason A
2006-01-01
Full Text Available We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation , where α, β, σ, τ ≥ 0 are real constants, and W(t is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.
Stochastic integral equations without probability
Mikosch, T; Norvaisa, R
2000-01-01
A pathwise approach to stochastic integral equations is advocated. Linear extended Riemann-Stieltjes integral equations driven by certain stochastic processes are solved. Boundedness of the p-variation for some 0
On filtering over Îto-Volterra observations
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Michael V. Basin
2000-01-01
Full Text Available In this paper, the Kalman-Bucy filter is designed for an Îto-Volterra process over Ito-Volterra observations that cannot be reduced to the case of a differential observation equation. The Kalman-Bucy filter is then designed for an Ito-Volterra process over discontinuous Ito-Volterra observations. Based on the obtained results, the filtering problem over discrete observations with delays is solved. Proofs of the theorems substantiating the filtering algorithms are given.
Block-pulse functions approach to numerical solution of Abel’s integral equation
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Monireh Nosrati Sahlan
2015-12-01
Full Text Available This study aims to present a computational method for solving Abel’s integral equation of the second kind. The introduced method is based on the use of Block-pulse functions (BPFs via collocation method. Abel’s integral equations as singular Volterra integral equations are hard and heavy in computation, but because of the properties of BPFs, as is reported in examples, this method is more efficient and more accurate than some other methods for solving this class of integral equations. On the other hand, the benefit of this method is low cost of computing operations. The applied method transforms the singular integral equation into triangular linear algebraic system that can be solved easily. An error analysis is worked out and applications are demonstrated through illustrative examples.
Geophysical interpretation using integral equations
Eskola, L
1992-01-01
Along with the general development of numerical methods in pure and applied to apply integral equations to geophysical modelling has sciences, the ability improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assu med to have a back...
Integral equation methods for electromagnetics
Volakis, John
2012-01-01
This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo
Tang, Xianhua; Cao, Daomin; Zou, Xingfu
We consider a periodic Lotka-Volterra competition system without instantaneous negative feedbacks (i.e., pure-delay systems) x(t)=x(t)[r(t)-∑j=1na(t)x(t-τ(t))], i=1,2,…,n. We establish some 3/2-type criteria for global attractivity of a positive periodic solution of the system, which generalize the well-known Wright's 3/2 criteria for the autonomous delay logistic equation, and thereby, address the open problem proposed by both Kuang [Y. Kuang, Global stability in delayed nonautonomous Lotka-Volterra type systems without saturated equilibria, Differential Integral Equations 9 (1996) 557-567] and Teng [Z. Teng, Nonautonomous Lotka-Volterra systems with delays, J. Differential Equations 179 (2002) 538-561].
Institute of Scientific and Technical Information of China (English)
夏述
2010-01-01
通过对一类分数阶Volterra-Lotka捕食方程模型的研究,并利用Krasovskii方法构造出Lyapunov函数,证明了分数阶Volterra-Lotka捕食方程在一定条件下的渐近稳定性.例子仿真说明了充分条件的有效性.
Generalized Volterra lattices: Binary Darboux transformations and self-consistent sources
Müller-Hoissen, F.; Chvartatskyi, O.; Toda, K.
2017-03-01
We study two families of matrix versions of generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete variables, which is a realization of a general integrable equation in bidifferential calculus. This allows to derive a binary Darboux transformation and also self-consistent source extensions via general results of bidifferential calculus. Exact solutions are constructed from the simplest seed solutions.
Institute of Scientific and Technical Information of China (English)
让光林; 徐侃
2003-01-01
In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the nonLipschitz conditions.We use a martingale formula in stead of Ito formula,which leads to simplicity the process of proof and extends the result to unbounded coefficients case.
Integration Rules for Scattering Equations
Baadsgaard, Christian; Bourjaily, Jacob L; Damgaard, Poul H
2015-01-01
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any M\\"obius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.
Integration rules for scattering equations
Baadsgaard, Christian; Bjerrum-Bohr, N. E. J.; Bourjaily, Jacob L.; Damgaard, Poul H.
2015-09-01
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints fo any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.
Integral equations on time scales
Georgiev, Svetlin G
2016-01-01
This book offers the reader an overview of recent developments of integral equations on time scales. It also contains elegant analytical and numerical methods. This book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses. The students in mathematical and physical sciences will find many sections of direct relevance. The book contains nine chapters and each chapter is pedagogically organized. This book is specially designed for those who wish to understand integral equations on time scales without having extensive mathematical background.
A Biological Least-Action Principle for the Ecological Model of Volterra-Lotka
Samuelson, Paul A.
1974-01-01
The conservative model of Volterra for more-than-two predator-prey species is shown to be generated as extremals that minimize a definable Lagrange-Hamilton integral involving half the species and their rates of change. This least-action formulation differs from that derived two generations ago by Volterra, since his involves twice the number of phase variables and it employs as variables the cumulative integrals of the numbers of each species that have ever lived. The present result extends the variational, teleological formulations found a decade ago by the author to the more-than-two species case. The present result is anything but surprising, in view of the works by Kerner, Montroll, and others which apply Gibbs' statistical mechanics to the all-but-canonical equations of the standard Volterra model. By a globally linear transformation of coordinates, the Volterra equations are here converted into a completely canonical system isomorphic with the classical mechanics models of Newton, Lagrange, Hamilton, Jacobi, Boltzmann, Gibbs, Poincaré, and G. D. Birkhoff. The conservative nature of the Lotka-Volterra model, whatever its realism, is a crucially necessary condition for the applicability of the variational formalisms, microscopically and macroscopically. PMID:4528377
Lagrangian structures, integrability and chaos for 3D dynamical equations
Bustamante, M D; Bustamante, Miguel D.; Hojman, Sergio A.
2003-01-01
In this paper we consider the general setting for constructing Action Principles for three-dimensional first order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and we show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behavior or homoclinic orbits have not been verified up to now. The Euler-Lagrange equations we get for these systems usually present "time reparameterization" symmetry, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrabi...
Equivalent boundary integral equations for plane elasticity
Institute of Scientific and Technical Information of China (English)
胡海昌; 丁皓江; 何文军
1997-01-01
Indirect and direct boundary integral equations equivalent to the original boundary value problem of differential equation of plane elasticity are established rigorously. The unnecessity or deficiency of some customary boundary integral equations is indicated by examples and numerical comparison.
Hamiltonian Structures and Integrability for a Discrete Coupled KdV-Type Equation Hierarchy
Institute of Scientific and Technical Information of China (English)
ZHAO Hai-Qiong; ZHU Zuo-Nong; ZHANG Jing-Li
2011-01-01
@@ Coupled Korteweg-de Vries(KdV) systems have many important physical applications.By considering a 4 × 4spectral problem,we derive a discrete coupled KdV-type equation hierarchy.Our hierarchy includes the coupled Volterra system proposed by Lou et al.(e-print arXiv:0711.0420) as the first member which is a discrete version of the coupled KdV equation.We also investigate the integrability in the Liouville sense and the multi-Hamiltonian structures for the obtained hierarchy.%Coupled Korteweg-de Vries (KdV) systems have many important physical applications.By considering a 4 × 4 spectral problem, we derive a discrete coupled KdV-type equation hierarchy.Our hierarchy includes the coupled Volterra system proposed by Lou et al.(e-print arXiv: 0711.0420) as the first member which is a discrete version of the coupled KdV equation.We also investigate the integrability in the Liouville sense and the multi-Hamiltonian structures for the obtained hierarchy.
Solving Volterra's Population Model Using New Second Derivative Multistep Methods
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K. Parand
2008-01-01
Full Text Available In this study new second derivative multistep methods (denoted SDMM are used to solve Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential where the integral term represents the effect of toxin. This model is first converted to a nonlinear ordinary differential equation and then the new SDMM, which has good stability and accuracy properties, are applied to solve this equation. We compare this method with the others and show that new SDMM gives excellent results.
Partial differential equations possessing Frobenius integrable decompositions
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Ma, Wen-Xiu [Department of Mathematics, University of South Florida, Tampa, FL 33620-5700 (United States)]. E-mail: mawx@cas.usf.edu; Wu, Hongyou [Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115-2888 (United States)]. E-mail: wu@math.niu.edu; He, Jingsong [Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026 (China)]. E-mail: jshe@ustc.edu.cn
2007-04-16
Frobenius integrable decompositions are introduced for partial differential equations. A procedure is provided for determining a class of partial differential equations of polynomial type, which possess specified Frobenius integrable decompositions. Two concrete examples with logarithmic derivative Baecklund transformations are given, and the presented partial differential equations are transformed into Frobenius integrable ordinary differential equations with cubic nonlinearity. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations.
Numerical Quadrature of Periodic Singular Integral Equations
DEFF Research Database (Denmark)
Krenk, Steen
1978-01-01
This paper presents quadrature formulae for the numerical integration of a singular integral equation with Hilbert kernel. The formulae are based on trigonometric interpolation. By integration a quadrature formula for an integral with a logarithmic singularity is obtained. Finally...... it is demonstrated how a singular integral equation with infinite support can be solved by use of the preceding formulae....
Stochastic integration and differential equations
Protter, Philip E
2003-01-01
It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...
Zhao, Peng; Fan, Engui
2015-04-01
In this paper, a new type of integrable differential-difference hierarchy, namely, the generalized relativistic Lotka-Volterra (GRLV) hierarchy, is introduced. This hierarchy is closely related to Lotka-Volterra lattice and relativistic Lotka-Volterra lattice, which allows us to provide a unified and effective way to obtain some exact solutions for both the Lotka-Volterra hierarchy and the relativistic Lotka-Volterra hierarchy. In particular, we shall construct algebro-geometric quasiperiodic solutions for the LV hierarchy and the RLV hierarchy in a unified manner on the basis of the finite gap integration theory.
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Chang Tan
2013-01-01
Full Text Available By piecewise Euler method, a discrete Lotka-Volterra predator-prey model with impulsive effect at fixed moment is proposed and investigated. By using Floquets theorem, we show that a globally asymptotically stable pest-eradication periodic solution exists when the impulsive period is less than some critical value. Further, we prove that the discrete system is permanence if the impulsive period is larger than some critical value. Finally, some numerical experiments are given.
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Masato Shinjo
2015-12-01
Full Text Available The Hankel determinant appears in representations of solutions to several integrable systems. An asymptotic expansion of the Hankel determinant thus plays a key role in the investigation of asymptotic analysis of such integrable systems. This paper presents an asymptotic expansion formula of a certain Casorati determinant as an extension of the Hankel case. This Casorati determinant is then shown to be associated with the solution to the discrete hungry Lotka–Volterra (dhLV system, which is an integrable variant of the famous prey–predator model in mathematical biology. Finally, the asymptotic behavior of the dhLV system is clarified using the expansion formula for the Casorati determinant.
On third order integrable vector Hamiltonian equations
Meshkov, A. G.; Sokolov, V. V.
2017-03-01
A complete list of third order vector Hamiltonian equations with the Hamiltonian operator Dx having an infinite series of higher conservation laws is presented. A new vector integrable equation on the sphere is found.
Integral equations with contrasting kernels
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Theodore Burton
2008-01-01
Full Text Available In this paper we study integral equations of the form $x(t=a(t-\\int^t_0 C(t,sx(sds$ with sharply contrasting kernels typified by $C^*(t,s=\\ln (e+(t-s$ and $D^*(t,s=[1+(t-s]^{-1}$. The kernel assigns a weight to $x(s$ and these kernels have exactly opposite effects of weighting. Each type is well represented in the literature. Our first project is to show that for $a\\in L^2[0,\\infty$, then solutions are largely indistinguishable regardless of which kernel is used. This is a surprise and it leads us to study the essential differences. In fact, those differences become large as the magnitude of $a(t$ increases. The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient. The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t$ as the sum of a bounded, but badly behaved function, and a large well behaved function.
Hypersingular integral equations and their applications
Lifanov, IK; Vainikko, MGM
2003-01-01
A number of new methods for solving singular and hypersingular integral equations have emerged in recent years. This volume presents some of these new methods along with classical exact, approximate, and numerical methods. The authors explore the analysis of hypersingular integral equations based on the theory of pseudodifferential operators and consider one-, two- and multi-dimensional integral equations. The text also presents the discrete closed vortex frame method and some other numerical methods for solving hypersingular integral equations. The treatment includes applications to problems in areas such as aerodynamics, elasticity, diffraction, and heat and mass transfer.
Modified Heisenberg Ferromagnet Model and Integrable Equation
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
We investigate some integrable modified Heisenberg ferromagnet models by using the prolongation structure theory. Through associating them with the motion of curve in Minkowski space, the corresponding coupled integrable equations are presented.
Individual based modeling and parameter estimation for a Lotka-Volterra system.
Waniewski, J; Jedruch, W
1999-03-15
Stochastic component, inevitable in biological systems, makes problematic the estimation of the model parameters from a single sequence of measurements, despite the complete knowledge of the system. We studied the problem of parameter estimation using individual-based computer simulations of a 'Lotka-Volterra world'. Two kinds (species) of particles--X (preys) and Y (predators)--moved on a sphere according to deterministic rules and at the collision (interaction) of X and Y the particle X was changed to a new particle Y. Birth of preys and death of predators were simulated by addition of X and removal of Y, respectively, according to exponential probability distributions. With this arrangement of the system, the numbers of particles of each kind might be described by the Lotka-Volterra equations. The simulations of the system with low (200-400 particles on average) number of individuals showed unstable oscillations of the population size. In some simulation runs one of the species became extinct. Nevertheless, the oscillations had some generic properties (e.g. mean, in one simulation run, oscillation period, mean ratio of the amplitudes of the consecutive maxima of X and Y numbers, etc.) characteristic for the solutions of the Lotka-Volterra equations. This observation made it possible to estimate the four parameters of the Lotka-Volterra model with high accuracy and good precision. The estimation was performed using the integral form of the Lotka-Volterra equations and two parameter linear regression for each oscillation cycle separately. We conclude that in spite of the irregular time course of the number of individuals in each population due to stochastic intraspecies component, the generic features of the simulated system evolution can provide enough information for quantitative estimation of the system parameters.
Asymptotic integration of differential and difference equations
Bodine, Sigrun
2015-01-01
This book presents the theory of asymptotic integration for both linear differential and difference equations. This type of asymptotic analysis is based on some fundamental principles by Norman Levinson. While he applied them to a special class of differential equations, subsequent work has shown that the same principles lead to asymptotic results for much wider classes of differential and also difference equations. After discussing asymptotic integration in a unified approach, this book studies how the application of these methods provides several new insights and frequent improvements to results found in earlier literature. It then continues with a brief introduction to the relatively new field of asymptotic integration for dynamic equations on time scales. Asymptotic Integration of Differential and Difference Equations is a self-contained and clearly structured presentation of some of the most important results in asymptotic integration and the techniques used in this field. It will appeal to researchers i...
Quasipolynomial generalization of Lotka-Volterra mappings
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Hernandez-Bermejo, Benito; Brenig, Leon [Service de Physique Theorique et Mathematique, Universite Libre de Bruxelles, Campus Plaine - CP 231, Brussels (Belgium)]. E-mails: bhernand@ulb.ac.be; lbrenig@ulb.ac.be
2002-07-05
In recent years, it has been shown that Lotka-Volterra mappings constitute a valuable tool from both the theoretical and the applied points of view, with developments in very diverse fields such as physics, population dynamics, chemistry and economy. The purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism (originally conceived for the analysis of ordinary differential equations) can be extended into the mapping domain. The extension of the formalism into the discrete-time context is remarkable as far as the quasipolynomial methodology had never been shown to be applicable beyond the differential case. It will be demonstrated that Lotka-Volterra mappings play a central role in the quasipolynomial formalism for the discrete-time case. Moreover, the extension of the formalism into the discrete-time domain allows a significant generalization of Lotka-Volterra mappings as well as a whole transfer of algebraic methods into the discrete-time context. The result is a novel and more general conceptual framework for the understanding of Lotka-Volterra mappings as well as a new range of possibilities that become open not only for the theoretical analysis of Lotka-Volterra mappings and their generalizations, but also for the development of new applications. (author)
Multi-Hamiltonian structure of Lotka-Volterra and quantum Volterra models
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Cronstroem, C. [Nordisk Inst. for Teoretisk Fysik (NORDITA), Copenhagen (Denmark); Noga, M. [Department of Theoretical Physics, Comenius University, Mlynska Dolina, Bratislava (Slovakia)
1995-07-10
We consider evolution equations of the Lotka-Volterra type, and elucidate especially their formulation as canonical Hamiltonian systems. The general conditions under which these equations admit several conserved quantities (multi-Hamiltonians) are analysed. A special case, which is related to the Liouville model on a lattice, is considered in detail, both as a classical and as a quantum system. (orig.).
Multi-hamiltonian structure of Lotka-Volterra and quantum Volterra models
Cronström, C; Cronström, C; Noga, M
1994-01-01
We consider evolution equations of the Lotka-Volterra type, and elucidate especially their formulation as canonical Hamiltonian systems. The general conditions under which these equations admit several conserved quantities (multi-Hamiltonians) are analysed. A special case, which is related to the Liouville model on a lattice, is considered in detail, both as aclassical and as aquantal system
Lectures on differential equations for Feynman integrals
Henn, Johannes M
2014-01-01
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to differential equations for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that allows based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the differential equations. Finally, as an application of the differential equations method we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a differential equation.
Conditional generalized analytic Feynman integrals and a generalized integral equation
Seung Jun Chang; Soon Ja Kang; David Skoug
2000-01-01
We use a generalized Brownian motion process to define a generalized Feynman integral and a conditional generalized Feynman integral. We then establish the existence of these integrals for various functionals. Finally we use the conditional generalized Feynman integral to derive a Schrödinger integral equation.
Integration of quantum hydrodynamical equation
Ulyanova, Vera G.; Sanin, Andrey L.
2007-04-01
Quantum hydrodynamics equations describing the dynamics of quantum fluid are a subject of this report (QFD).These equations can be used to decide the wide class of problem. But there are the calculated difficulties for the equations, which take place for nonlinear hyperbolic systems. In this connection, It is necessary to impose the additional restrictions which assure the existence and unique of solutions. As test sample, we use the free wave packet and study its behavior at the different initial and boundary conditions. The calculations of wave packet propagation cause in numerical algorithm the division. In numerical algorithm at the calculations of wave packet propagation, there arises the problem of division by zero. To overcome this problem we have to sew together discrete numerical and analytical continuous solutions on the boundary. We demonstrate here for the free wave packet that the numerical solution corresponds to the analytical solution.
New integrability case for the Riccati equation
Mak, M K
2012-01-01
A new integrability condition of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ is presented. By introducing an auxiliary equation depending on a generating function $f(x)$, the general solution of the Riccati equation can be obtained if the coefficients $a(x)$, $b(x)$, $c(x)$, and the function $f(x)$ satisfy a particular constraint. The validity and reliability of the method are tested by obtaining the general solutions of some Riccati type differential equations. Some applications of the integrability conditions for the case of the damped harmonic oscillator with time dependent frequency, and for solitonic wave, are briefly discussed.
Bahadur Zada, Mian; Sarwar, Muhammad; Radenović, Stojan
2017-01-01
In this article, we apply common fixed point results in incomplete metric spaces to examine the existence of a unique common solution for the following systems of Urysohn integral equations and Volterra-Hammerstein integral equations, respectively: [Formula: see text] where [Formula: see text]; [Formula: see text] and [Formula: see text], [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text], u, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], are real-valued measurable functions both in s and r on [Formula: see text].
On the asymptotic behaviour of solutions of an asymptotically Lotka-Volterra model
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Attila Dénes
2016-09-01
Full Text Available We make more realistic our model [Nonlinear Anal. 73(2010, 650-659] on the coexistence of fishes and plants in Lake Tanganyika. The new model is an asymptotically autonomous system whose limiting equation is a Lotka-Volterra system. We give conditions for the phenomenon that the trajectory of any solution of the original non-autonomous system "rolls up"' onto a cycle of the limiting Lotka-Volterra equation as $t\\to\\infty$, which means that the limit set of the solution of the non-autonomous system coincides with the cycle. A counterexample is constructed showing that the key integral condition on the coefficient function in the original non-autonomous model cannot be dropped. Computer simulations illustrate the results.
Counting master integrals. Integration by parts vs. functional equations
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Kniehl, Bernd A.; Tarasov, Oleg V. [Hamburg Univ. (Germany). II. Inst. fuer Theoretische Physik
2016-01-15
We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.
Renormalization group and linear integral equations
Klein, W.
1983-04-01
We develop a position-space renormalization-group transformation which can be employed to study general linear integral equations. In this Brief Report we employ our method to study one class of such equations pertinent to the equilibrium properties of fluids. The results of applying our method are in excellent agreement with known numerical calculations where they can be compared. We also obtain information about the singular behavior of this type of equation which could not be obtained numerically.
(2+1)-dimensional supersymmetric integrable equations
Yan, Zhao-Wen; Tala; Chen, Fang; Liu, Tao-Ran; Han, Jing-Min
2017-09-01
By means of two different approaches, we construct the (2+1)-dimensional supersymmetric integrable equations based on the super Lie algebra osp(3/2). We relax the constraint condition of homogenous space of super Lie algebra osp(3/2) in the first approach. In another one, the technique of extending the dimension of the systems is used. Furthermore for the (2 + 1)-dimensional supersymmetric integrable equations, we also derive their Bäcklund transformations.
Model uncertainty in economic impacts of climate change: Bernoulli versus Lotka Volterra dynamics.
Cooke, Roger M
2013-01-01
The dynamic economic behavior in most integrated assessment models linking economic growth to climate change involves a differential equation solved by Jacob Bernoulli in 1695. Using the dynamic integrated climate economy (DICE) model and freezing exogenous variables at initial values, this dynamic is shown to produce implausible projections on a 60-year time frame. If world capital started at US$1, after 60 years the world economy would be indistinguishable from one starting with 10 times the current capitalization. Such behavior points to uncertainty at the level of the fundamental dynamics, and suggests that discussions of discounting, utility, damage functions, and ethics should be conducted within a more general modeling vocabulary. Lotka Volterra dynamics is proposed as an alternative with greater prime facie plausibility. With near universality, economists assume that economic growth will go on forever. Lotka Volterra dynamics alert us to the possibility of collapse.
Linear integral equations and renormalization group
Klein, W.; Haymet, A. D. J.
1984-08-01
A formulation of the position-space renormalization-group (RG) technique is used to analyze the singular behavior of solutions to a number of integral equations used in the theory of the liquid state. In particular, we examine the truncated Kirkwood-Salsburg equation, the Ornstein-Zernike equation, and a simple nonlinear equation used in the mean-field theory of liquids. We discuss the differences in applying the position-space RG to lattice systems and to fluids, and the need for an explicit free-energy rescaling assumption in our formulation of the RG for integral equations. Our analysis provides one natural way to define a "fractal" dimension at a phase transition.
[Analysis of seasonal fluctuations in the Lotka-Volterra model].
Lobanov, A I; Sarancha, D A; Starozhilova, T K
2002-01-01
A modification of the Lotka-Volterra model was proposed. The modification takes into account the factor of seasonal fluctuations in a "predator-prey" model. In this modification, interactions between species in summer are described by the Lotka-Volterra equations; in winter, individuals of both species extinct. This generalization makes the classic model unrough, which substantially extends the field of its application. The results of numerical simulation illustrate the statement formulated above.
Explicit Integration of Friedmann's Equation with Nonlinear Equations of State
Chen, Shouxin; Yang, Yisong
2015-01-01
This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshev's result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics ...
Integrability of Lie Systems Through Riccati Equations
Cariñena, José F.; de Lucas, Javier
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides us with a unified geometrical viewpoint that allows us to analyse some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalised to describe integrability conditions for any Lie system. Finally, we show the usefulness of our treatment in order to study the problem of the linearisability of Riccati equations.
Integrability of Lie systems through Riccati equations
Cariñena, José F
2010-01-01
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides us with a unified geometrical viewpoint that allows us to analyse some previous works on the topic and explain new properties. Moreover, this new approach can be straightforwardly generalised to describe integrability conditions for any Lie system. Finally, we show the usefulness of our treatment in order to study the problem of the linearisability of Riccati equations.
A phenomenological Hamiltonian for the Lotka-Volterra problem
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Georgian, T. [Corps of Engineers, Omaha, NE (United States); Findley, G.L. [Northeast Louisiana Univ., Monroe, LA (United States)
1996-12-31
We have presented a Hamiltonian theory of phenomenological chemical kinetics. In the present paper, we extend this treatment to the Lotka-Volterra model of sustained oscillations. Our approach begins with the usual definition of an intrinsic reaction coordinate space (x{sub 1},x{sub 2}) for the Lotka-Volterra problem, which leads to the rate equations x{sub 1}=ax{sub 1}-bx{sub 1}x{sub 2}, x{sub 2}=-cx{sub 2}+bx{sub 1}x{sub 2}, with a,b and c being real constants. We thereafter present a Hamiltonian function H(x,y)[y{sub 1} = x{sub 1} and y{sub 2} = x{sub 2}] and an associated holonomic constraint, which give rise to the above rates as half of Hamilton`s equations. We provide trajectories by numerical integration (4th order Runge-Kutta) and show that H(x,y) is a constant of the motion. Finally, issues involved in developing an analytic solution to this problem are discussed.
Variational integrators for nonvariational partial differential equations
Kraus, Michael; Maj, Omar
2015-08-01
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.
Volterra integrodifferential systems
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K. Balachandran
1995-01-01
Full Text Available Sufficient conditions for the complete controllability of nonlinear perturbations of Volterra integrodifferential systems with implicit derivative are established. The results generalize the results of Dauer and Balachandran [9] and are obtained through the notions of condensing map and measure of noncompactness of a set.
Langevin equation path integral ground state.
Constable, Steve; Schmidt, Matthew; Ing, Christopher; Zeng, Tao; Roy, Pierre-Nicholas
2013-08-15
We propose a Langevin equation path integral ground state (LePIGS) approach for the calculation of ground state (zero temperature) properties of molecular systems. The approach is based on a modification of the finite temperature path integral Langevin equation (PILE) method (J. Chem. Phys. 2010, 133, 124104) to the case of open Feynman paths. Such open paths are necessary for a ground state formulation. We illustrate the applicability of the method using model systems and the weakly bound water-parahydrogen dimer. We show that the method can lead to converged zero point energies and structural properties.
Scattering integral equations for distorted transition operators
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Kowalski, K.L.; Siciliano, E.R.; Thaler, R.M.
1978-11-01
Methods for embedding phenomenological distorted-wave techniques for rearrangement and inelastic scattering within well-defined theories of multiparticle scattering are developed. The essential point of contact between the two approaches is in the definition and choice of distorting potential. It is shown that the concept of a channel coupling scheme allows a comparative freedom of choice for these potentials; if they are connected operators, such as optical potentials, then it is possible to obtain connected-kernel equations for the distorted transition operators. The latter are introduced in the course of exploiting the two-potential formula for the full transition operator and have the property that their matrix elements with respect to distorted waves are the physical scattering amplitudes. It is found that the distorted counterparts of the Kouri, Levin, and Tobocman and the Bencze-Redish integral equations maintain their connected-kernel and minimally coupled properties. These equations can be used to derive other integral equations with the same properties for the distorted-wave operators which consist of the product of the distorted transition operators and the wave operators corresponding to distorted waves. These simplifications are not realized for arbitrary channel coupling schemes. In order to deal with the general situation an alternative approach employing a subtraction technique which involves projections on the bound two-cluster channel states is introduced. When the distorting potentials are essentially the optical potentials in the entrance and exit channels a set of multichannel two-particle Lippmann-Schwinger integral equations for the two-cluster distorted-wave transition operators are obtained. Input into these two-particle integral equations involves the solution of a modified N-particle equation. Approximations to the latter are discussed in the particular cases of the Kouri, Levin, and Tobocman and Bencze-Redish channel coupling schemes.
PREFACE: Symmetries and Integrability of Difference Equations
Doliwa, Adam; Korhonen, Risto; Lafortune, Stéphane
2007-10-01
The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations (DE), like differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, and quantum field theory. It is thus crucial to develop tools to study and solve DEs. While the theory of symmetry and integrability for differential equations is now largely well-established, this is not yet the case for discrete equations. Although over recent years there has been significant progress in the development of a complete analytic theory of difference equations, further tools are still needed to fully understand, for instance, the symmetries, asymptotics and the singularity structure of difference equations. The series of SIDE meetings on Symmetries and Integrability of Difference Equations started in 1994. Its goal is to provide a platform for an international and interdisciplinary communication for researchers working in areas associated with integrable discrete systems, such as classical and quantum physics, computer science and numerical analysis, mathematical biology and economics, discrete geometry and combinatorics, theory of special functions, etc. The previous SIDE meetings took place in Estérel near Montréal, Canada (1994), at the University of
Representation of neural networks as Lotka-Volterra systems
Moreau, Yves; Louiès, Stéphane; Vandewalle, Joos; Brenig, Léon
1999-03-01
We study changes of coordinates that allow the representation of the ordinary differential equations describing continuous-time recurrent neural networks into differential equations describing predator-prey models—also called Lotka-Volterra systems. We transform the equations for the neural network first into quasi-monomial form, where we express the vector field of the dynamical system as a linear combination of products of powers of the variables. In practice, this transformation is possible only if the activation function is the hyperbolic tangent or the logistic sigmoïd. From this quasi-monomial form, we can directly transform the system further into Lotka-Volterra equations. The resulting Lotka-Volterra system is of higher dimension than the original system, but the behavior of its first variables is equivalent to the behavior of the original neural network.
Unconditionally stable integration of Maxwell's equations
Verwer, J.G.; Botchev, M.A.
2008-01-01
Numerical integration of Maxwell''s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit-finite differenc
Delay differential equations with homogeneous integral conditions
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Abdur Raheem
2013-03-01
Full Text Available In this article we prove the existence and uniqueness of a strong solution of a delay differential equation with homogenous integral conditions using the method of semidiscretization in time. As an application, we include an example that illustrates the main result.
Three New Integrable Hierarchies of Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
A general Lie algebra Vs and the corresponding loop algebra ～Vs are constructed, from which the linear isospectral Lax pairs are established, whose compatibility presents the zero curvature equation. As its application, a new Lax integrable hierarchy containing two parameters is worked out. It is not Liouville-integrable, however, its two reduced systems are Liouville-integrable, whose Hamiltonian structures are derived by making use of the quadratic-form identity and the γ formula (i.e. the computational formula on the constant γ appeared in the trace identity and the quadratic-form identity).
The matrix realization of affine Jacobi varieties and the extended Lotka-Volterra lattice
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Inoue, Rei [Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba 3-8-1, Meguro, Tokyo 153-8902 (Japan)
2004-01-30
We study completely integrable Hamiltonian systems whose monodromy matrices are related to the representatives for the set of gauge equivalence classes M{sub F} of polynomial matrices. Let X be the algebraic curve given by the common characteristic equation for M{sub F}. We construct the isomorphism from the set of representatives to an affine part of the Jacobi variety of X. This variety corresponds to the invariant manifold of the system, where the Hamiltonian flow is linearized. As an application, we discuss the algebraic complete integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.
Volterra, Fascism, and France.
Capristo, Annalisa
2015-12-01
My contribution focuses on two aspects strictly related each other. On one hand, the progressive marginalization of Volterra from Italian scientific and political life after the rise of Fascism - because of his public anti-Fascist stance, both as a senator and as a professor - until his definitive exclusion on racial grounds in 1938. On the other hand, the reactions of his French colleagues and friends to this ostracism, and the support he received from them. As it emerges from several sources (Volterra's correspondence, institutional documentation, conference proceedings, etc.), it was mainly thanks to their support that he was able to escape the complete isolation and the "civil death" to which the regime condemned many of its adversaries.
Coupled Nonlinear Schr\\"{o}dinger equation and Toda equation (the Root of Integrability)
Hisakado, Masato
1997-01-01
We consider the relation between the discrete coupled nonlinear Schr\\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\\"{o}dinger equation. In the same way we can show the integrability in coupled case of dark and bright equations. Using this method we obtain several integrable equations.
Institute of Scientific and Technical Information of China (English)
XU Xi-Xiang; ZHANG Yu-Feng
2004-01-01
A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann .constraint of the resulting hierarchy.Finally, an infinite set of conservation laws is given for the resulting hierarchy.
Integrable version of Burgers equation in magnetohydrodynamics.
Olesen, P
2003-07-01
It is pointed out that for the case of (compressible) magnetohydrodynamics (MHD) with the fields v(y)(y,t) and Bx(y,t), one can have equations of the Burgers type which are integrable. We discuss the solutions. It turns out that the propagation of the nonlinear effects is governed by the initial velocity (as in Burgers case) as well as by the initial Alfvén velocity. Many results previously obtained for the Burgers equation can be transferred to the MHD case. We also discuss equipartition v(y)=+/-Bx. It is shown that an initial localized small scale magnetic field will end up in fields moving to the left and the right, thus transporting energy from smaller to larger distances.
Bounded solutions for fuzzy differential and integral equations
Energy Technology Data Exchange (ETDEWEB)
Nieto, Juan J. [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amnieto@usc.es; Rodriguez-Lopez, Rosana [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amrosana@usc.es
2006-03-01
We find sufficient conditions for the boundness of every solution of first-order fuzzy differential equations as well as certain fuzzy integral equations. Our results are based on several theorems concerning crisp differential and integral inequalities.
A New Integral Equation for the Spheroidal equations in case of m equal 1
Tian, Guihua
2012-01-01
The spheroidal wave functions are investigated in the case m=1. The integral equation is obtained for them. For the two kinds of eigenvalues in the differential and corresponding integral equations, the relation between them are given explicitly. Though there are already some integral equations for the spheroidal equations, the relation between their two kinds of eigenvalues is not known till now. This is the great advantage of our integral equation, which will provide useful information through the study of the integral equation. Also an example is given for the special case, which shows another way to study the eigenvalue problem.
A NOVEL BOUNDARY INTEGRAL EQUATION METHOD FOR LINEAR ELASTICITY--NATURAL BOUNDARY INTEGRAL EQUATION
Institute of Scientific and Technical Information of China (English)
Niu Zhongrong; Wang Xiuxi; Zhou Huanlin; Zhang Chenli
2001-01-01
The boundary integral equation (BIE) of displacement derivatives is put at a disadvantage for the difficulty involved in the evaluation of the hypersingular integrals. In this paper, the operators δij and εij are used to act on the derivative BIE. The boundary displacements, tractions and displacement derivatives are transformed into a set of new boundary tensors as boundary variables. A new BIE formulation termed natural boundary integral equation (NBIE) is obtained. The NBIE is applied to solving two-dimensional elasticity problems. In the NBIE only the strongly singular integrals are contained. The Cauchy principal value integrals occurring in the NBIE are evaluated. A combination of the NBIE and displacement BIE can be used to directly calculate the boundary stresses. The numerical results of several examples demonstrate the accuracy of the NBIE.
Recovering an obstacle using integral equations
Rundell, William
2009-05-01
We consider the inverse problem of recovering the shape, location and surface properties of an object where the surrounding medium is both conductive and homogeneous and we measure Cauchy data on an accessible part of the exterior boundary. It is assumed that the physical situation is modelled by harmonic functions and the boundary condition on the obstacle is one of Dirichlet type. The purpose of this paper is to answer some of the questions raised in a recent paper that introduced a nonlinear integral equation approach for the solution of this type of problem.
Integral solutions of fractional evolution equations with nondense domain
Directory of Open Access Journals (Sweden)
Haibo Gu
2017-06-01
Full Text Available In this article, we study the existence of integral solutions for two classes of fractional order evolution equations with nondensely defined linear operators. First, we consider the nonhomogeneous fractional order evolution equation and obtain its integral solution by Laplace transform and probability density function. Subsequently, based on the form of integral solution for nonhomogeneous fractional order evolution equation, we investigate the existence of integral solution for nonlinear fractional order evolution equation by noncompact measure method.
Integral Equation Methods for Electromagnetic and Elastic Waves
Chew, Weng; Hu, Bin
2008-01-01
Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral eq
A Baecklund transformation between two integrable discrete hungry systems
Energy Technology Data Exchange (ETDEWEB)
Fukuda, Akiko, E-mail: j1409704@ed.kagu.tus.ac.j [Department of Mathematical Information Science, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan); Yamamoto, Yusaku [Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501 (Japan); Iwasaki, Masashi [Department of Informatics and Environmental Science, Kyoto Prefectural University, 1-5, Nakaragi-cho, Shimogamo, Sakyo-ku, Kyoto 606-8522 (Japan); Ishiwata, Emiko [Department of Mathematical Information Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan); Nakamura, Yoshimasa [Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501 (Japan)
2011-01-17
The discrete hungry Toda (dhToda) equation and the discrete hungry Lotka-Volterra (dhLV) system are known as integrable discrete hungry systems. In this Letter, through finding the LR transformations associated with the dhToda equation and the dhLV system, we present a Baecklund transformation between these integrable systems.
Evolution of Black-Box Models Based on Volterra Series
Directory of Open Access Journals (Sweden)
Daniel D. Silveira
2015-01-01
Full Text Available This paper presents a historical review of the many behavioral models actually used to model radio frequency power amplifiers and a new classification of these behavioral models. It also discusses the evolution of these models, from a single polynomial to multirate Volterra models, presenting equations and estimation methods. New trends in RF power amplifier behavioral modeling are suggested.
Competitive Lotka-Volterra Population Dynamics with Jumps
Bao, Jianhai; Yin, Geroge; Yuan, Chenggui
2011-01-01
This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) We discuss the uniform boundedness of $p$th moment with $p>0$ and reveal the sample Lyapunov exponents; (c) Using a variation-of-constants formula for a class of SDEs with jumps, we provide explicit solution for 1-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our $n$-dimensional model.
Integrable Heisenberg Ferromagnet Equations with self-consistent potentials
Zhunussova, Zh Kh; Tungushbaeva, D I; Mamyrbekova, G K; Nugmanova, G N; Myrzakulov, R
2013-01-01
In this paper, we consider some integrable Heisenberg Ferromagnet Equations with self-consistent potentials. We study their Lax representations. In particular we give their equivalent counterparts which are nonlinear Schr\\"odinger type equations. We present the integrable reductions of the Heisenberg Ferromagnet Equations with self-consistent potentials. These integrable Heisenberg Ferromagnet Equations with self-consistent potentials describe nonlinear waves in ferromagnets with magnetic fields.
Institute of Scientific and Technical Information of China (English)
邢维; 田华
2014-01-01
We considered the polynomial first integral for reversible Lotka-Volterra model which describes the oscillatory chemical dynamics in a closed isothermal reaction.Making use of integrability theories of semi-quasihomogeneous systems,we proved that the Lotka-Votterra model is a unique polynomial first integral.%考虑一类描述闭等温反应中振荡化学动力学行为的可逆 Lotka-Volterra模型的多项式首次积分，利用半拟齐次系统的可积性理论，找到 Lotka-Volterra模型的一个多项式首次积分，并证明了它是Lotka-Volterra模型唯一的多项式首次积分。
Integrable coupling system of fractional soliton equation hierarchy
Energy Technology Data Exchange (ETDEWEB)
Yu Fajun, E-mail: yfajun@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)
2009-10-05
In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.
Integrability of two coupled Kadomtsev–Petviashvili equations
Indian Academy of Sciences (India)
Abdul-Majid Wazwaz
2011-08-01
The integrability of two coupled KP equations is studied. The simpliﬁed Hereman form of Hirota’s bilinear method is used to examine the integrability of each coupled equation. Multiplesoliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation.
Directory of Open Access Journals (Sweden)
Yange Huang
2014-01-01
Full Text Available We discuss a class of Volterra-Fredholm type difference inequalities with weakly singular. The upper bounds of the embedded unknown functions are estimated explicitly by analysis techniques. An application of the obtained inequalities to the estimation of Volterra-Fredholm type difference equations is given.
THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
This paper applies the singular integral operators,singular quadrature operators and discretization matrices associated withsingular integral equations with Cauchy kernels, which are established in [1],to give a unified framework for various collocation methods of numericalsolutions of singular integral equations with Cauchy kernels. Under theframework, the coincidence of the direct quadrature method and the indirectquadrature method is very simple and obvious.
Solving Abel integral equations of first kind via fractional calculus
Directory of Open Access Journals (Sweden)
Salman Jahanshahi
2015-04-01
Full Text Available We give a new method for numerically solving Abel integral equations of first kind. An estimation for the error is obtained. The method is based on approximations of fractional integrals and Caputo derivatives. Using trapezoidal rule and Computer Algebra System Maple, the exact and approximation values of three Abel integral equations are found, illustrating the effectiveness of the proposed approach.
The Lamb-Bateman integral equation and the fractional derivatives
Babusci, D; Sacchetti, D
2010-01-01
The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be obtained using the formalism of fractional calculus.
Energy Technology Data Exchange (ETDEWEB)
Mancas, Stefan C. [Department of Mathematics, Embry–Riddle Aeronautical University, Daytona Beach, FL 32114-3900 (United States); Rosu, Haret C., E-mail: hcr@ipicyt.edu.mx [IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potosí, SLP (Mexico)
2013-09-02
We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.
Exactly integrable hyperbolic equations of Liouville type
Energy Technology Data Exchange (ETDEWEB)
Zhiber, A V [Institute of Mechanics, Ufa Centre of the Russian Academy of Sciences, Ufa (Russian Federation); Sokolov, Vladimir V [Centre for Non-linear Studies Landau Institute for Theoretical Physics, Moscow (Russian Federation)
2001-02-28
This is a survey of the authors' results concerning non-linear hyperbolic equations of Liouville type. The definition is based on the condition that the chain of Laplace invariants of the linearized equation be two-way finite. New results include a procedure for finding the general solution and a solution of the classification problem for Liouville type equations.
Transformations of Heun's equation and its integral relations
El-Jaick, Léa Jaccoud
2010-01-01
For each variable transformation which preserves the form of Heun's equation we find a transformation which leaves invariant the form of the equation for the kernels of integral relations among solutions of the former equation. This enables us to generate new kernels for the Heun equation, given by single hypergeometric functions (Lambe-Ward-type kernels) and by products of two hypergeometric functions (Erd\\'elyi-type). Such kernels, by a limiting process, afford new kernels for the confluent Heun equation as well.
Partial differential equations of mathematical physics and integral equations
Guenther, Ronald B
1996-01-01
This book was written to help mathematics students and those in the physical sciences learn modern mathematical techniques for setting up and analyzing problems. The mathematics used is rigorous, but not overwhelming, while the authors carefully model physical situations, emphasizing feedback among a beginning model, physical experiments, mathematical predictions, and the subsequent refinement and reevaluation of the physical model itself. Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the t
Poisson brackets of mappings obtained as ( q,- p) reductions of lattice equations
Tran, Dinh T.; van der Kamp, Peter H.; Quispel, G. R. W.
2016-11-01
In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The ( q,- p) reductions are ( p + q)-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka-Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the (3,-2) reductions of the integrable partial difference equations are Liouville integrable in their own right.
Unification of integrable q-difference equations
Directory of Open Access Journals (Sweden)
Burcu Silindir
2015-10-01
Full Text Available This article presents a unifying framework for q-discrete equations. We introduce a generalized q-difference equation in Hirota bilinear form and develop the associated three-q-soliton solutions which are described in polynomials of power functions by utilizing Hirota direct method. Furthermore, we present that the generalized q-difference soliton equation reduces to q-analogues of Toda, KdV and sine-Gordon equations equipped with their three-q-soliton solutions by appropriate
Financial integration in Europe : Evidence from Euler equation tests
Lemmen, J.J.G.; Eijffinger, S.C.W.
1995-01-01
This paper applies Obstfeld's Euler equation tests to assess the degree of financial integration in the European Union. In addition, we design a new Euler equation test which is intimately related to Obstfeld's Euler equation tests. Using data from the latest Penn World Table (Mark 6), we arrive at
Täuber, Uwe C.
2013-03-01
Field theory tools are applied to analytically study fluctuation and correlation effects in spatially extended stochastic predator-prey systems. In the mean-field rate equation approximation, the classic Lotka-Volterra model is characterized by neutral cycles in phase space, describing undamped oscillations for both predator and prey populations. In contrast, Monte Carlo simulations for stochastic two-species predator-prey reaction systems on regular lattices display complex spatio-temporal structures associated with persistent erratic population oscillations. The Doi-Peliti path integral representation of the master equation for stochastic particle interaction models is utilized to arrive at a field theory action for spatial Lotka-Volterra models in the continuum limit. In the species coexistence phase, a perturbation expansion with respect to the nonlinear predation rate is employed to demonstrate that spatial degrees of freedom and stochastic noise induce instabilities toward structure formation, and to compute the fluctuation corrections for the oscillation frequency and diffusion coefficient. The drastic downward renormalization of the frequency and the enhanced diffusivity are in excellent qualitative agreement with Monte Carlo simulation data.
On integrable rational potentials of the Dirac equation
Energy Technology Data Exchange (ETDEWEB)
Stachowiak, Tomasz, E-mail: stachowiak@cft.edu.pl [Center for Theoretical Physics PAS, Al. Lotnikow 32/46, 02-668 Warszawa (Poland); Przybylska, Maria, E-mail: M.Przybylska@proton.if.uz.zgora.pl [Institute of Physics, University of Zielona Góra, Licealna 9, 65-417 Zielona Góra (Poland)
2013-05-03
The one-dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.
Polyconvolution and the Toeplitz plus Hankel integral equation
Directory of Open Access Journals (Sweden)
Nguyen Xuan Thao
2014-04-01
Full Text Available In this article we introduce a polyconvolution which related to the Hartley and Fourier cosine transforms. We prove some properties of this polyconvolution, and then solve a class of Toeplitz plus Hankel integral equations and systems of two Toeplitz plus Hankel integral equations.
Integral Equation Solution for Biopotentials of Single Cells
Klee, Maurice; Plonsey, Robert
1972-01-01
A Fredholm integral equation of the second type is developed for the biopotentials of single cells. Two singularities arise in the numerical solution of this integral equation and methods for handling them are presented. The problem of a spherical cell in an applied uniform field is used to illustrate the technique. PMID:4655666
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Directory of Open Access Journals (Sweden)
Angelo B. Mingarelli
2007-03-01
Full Text Available Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear differential equations on a half-axis. In addition, we extend the methods and present new similar results for integral equations and Volterra-Stieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of Volterra-Stieltjes integral equations is also presented.
Stability for a class of semilinear fractional stochastic integral equations
Fiel, Allan; Jorge A. León; Márquez-Carreras, David
2015-01-01
In this paper we study some stability criteria for some semilinear integral equations with a function as initial condition and with additive noise, which is a Young integral that could be a functional of fractional Brownian motion. Namely, we consider stability in the mean, asymptotic stability, stability, global stability and Mittag-Leffler stability. To do so, we use comparison results for fractional equations and an equation (in terms of Mittag-Leffler functions) whose family of solutions ...
Institute of Scientific and Technical Information of China (English)
Wei-jun Tang; Hong-yuan Fu; Long-jun Shen
2001-01-01
Consider solving the Dirichlet problem of Helmholtz equation on unbounded region R2\\Г with Г a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.
Numerical evaluation of renewal equations: applications to risk theory and financial models
Usábel Rodrigo, Miguel Arturo
1997-01-01
The so-called Renewal Theory is a frequently used methodology in applied mathematics. Renewal Theory is mainly focussed on solving a Volterra integral equation of the second kind known as Renewal Integral EquationAn interesting problem arises when choosing the appropriate numerical tool in order to approximate the solution of the former integral. The decision will be based on the degree of knowledge of function F(x) and some properties of (u). Three methods based in classical methodologies (s...
A New (2+1)-Dimensional Integrable Equation
Institute of Scientific and Technical Information of China (English)
PEN Bo; LIN Ji
2009-01-01
A new nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained from the mKP equation by means of an asymptotically exact reduction method based on Fourier expansion and spatio-temporal rescaling. In order to demonstrate integrability property of the new equation, the corresponding Lax pair is obtained by applying the reduction technique to the Lax pair of the mKP equation.
Nonsingular Integral Equation for Stability of a Bunched Beam
Energy Technology Data Exchange (ETDEWEB)
Warnock, R
2004-02-25
The linearized Vlasov equation for longitudinal motion of a bunched beam leads to a singular integral equation, the singularity being associated with the tune spectrum of the single-particle motion. A discretization for numerical solution of the equation in this form is not well justified. A simple change of the unknown function gives an equation that can more readily be approximated by a matrix equation. In contrast to the usual approach (Oide-Yokoya) the equation for eigen-frequencies does not have a continuum of solutions corresponding to single-particle frequencies, but only a few solutions corresponding to coherent modes.
On integrable rational potentials of the Dirac equation
Stachowiak, Tomasz
2012-01-01
The Dirac equation, when reducible to an ordinary second order linear equation, exhibits a form of quasi-integrability, i.e. exact solutions exist only for a particular subset of energies. The differential Galois theory can be used to identify the integrable cases, recover integrable rational potentials, explicit solutions and strictly rule out the remaining cases as non-integrable. The effectiveness of this approach is demonstrated by providing a new class of potentials for which the equation in question can be transformed to the Whittaker form.
Volterra Series Based Distortion Effect
DEFF Research Database (Denmark)
Agerkvist, Finn T.
2010-01-01
A large part of the characteristic sound of the electric guitar comes from nonlinearities in the signal path. Such nonlinearities may come from the input- or output-stage of the amplier, which is often equipped with vacuum tubes or a dedicated distortion pedal. In this paper the Volterra series e...
Master equations and the theory of stochastic path integrals
Weber, Markus F
2016-01-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. We discuss analytical and numerical methods for the solution of master equations, keeping our focus on methods that are applicable even when stochastic fluctuations are strong. The reviewed methods include the generating function technique and the Poisson representation, as well as novel ways of mapping the forward and backward master equations onto linear partial differential equations (PDEs). Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE obeyed by the generating function. After outlining these methods, we solve the derived PDEs in terms of two path integrals. The path integrals provide distinct exact representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Furthermore, we review a method for the approxima...
Volterra series truncation and kernel estimation of nonlinear systems in the frequency domain
Zhang, B.; Billings, S. A.
2017-02-01
The Volterra series model is a direct generalisation of the linear convolution integral and is capable of displaying the intrinsic features of a nonlinear system in a simple and easy to apply way. Nonlinear system analysis using Volterra series is normally based on the analysis of its frequency-domain kernels and a truncated description. But the estimation of Volterra kernels and the truncation of Volterra series are coupled with each other. In this paper, a novel complex-valued orthogonal least squares algorithm is developed. The new algorithm provides a powerful tool to determine which terms should be included in the Volterra series expansion and to estimate the kernels and thus solves the two problems all together. The estimated results are compared with those determined using the analytical expressions of the kernels to validate the method. To further evaluate the effectiveness of the method, the physical parameters of the system are also extracted from the measured kernels. Simulation studies demonstrates that the new approach not only can truncate the Volterra series expansion and estimate the kernels of a weakly nonlinear system, but also can indicate the applicability of the Volterra series analysis in a severely nonlinear system case.
Isogeometric Analysis of Boundary Integral Equations
2015-04-21
obtains high-order collocation methods based on superior approximation and numerical integration schemes and well-conditioned systems of linear algebraic ...matrices associated with the operators 12I+K and 1 2I−K ′. This construction results in well-conditioned linear algebraic systems [2], and it is superior ...for regularizing integral operators. As a result one obtains high-order collocation methods based on superior approximation and numerical integration
Multicomponent integrable wave equations: II. Soliton solutions
Energy Technology Data Exchange (ETDEWEB)
Degasperis, A [Dipartimento di Fisica, Universita di Roma ' La Sapienza' , and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Rome (Italy); Lombardo, S [School of Mathematics, University of Manchester, Alan Turing Building, Upper Brook Street, Manchester M13 9EP (United Kingdom)], E-mail: antonio.degasperis@roma1.infn.it, E-mail: sara.lombardo@manchester.ac.uk, E-mail: sara@few.vu.nl
2009-09-25
The Darboux-dressing transformations developed in Degasperis and Lombardo (2007 J. Phys. A: Math. Theor. 40 961-77) are here applied to construct soliton solutions for a class of boomeronic-type equations. The vacuum (i.e. vanishing) solution and the generic plane wave solution are both dressed to yield one-soliton solutions. The formulae are specialized to the particularly interesting case of the resonant interaction of three waves, a well-known model which is of boomeronic type. For this equation a novel solution which describes three locked dark pulses (simulton) is introduced.
The Kadomtsev{endash}Petviashvili equation as a source of integrable model equations
Energy Technology Data Exchange (ETDEWEB)
Maccari, A. [Technical Institute ``G. Cardano,`` Piazza della Resistenza 1, 00015 Monterotondo Rome (Italy)
1996-12-01
A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev{endash}Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev{endash}Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs. {copyright} {ital 1996 American Institute of Physics.}
Integrable hierarchies of Heisenberg ferromagnet equation
Nugmanova, G.; Azimkhanova, A.
2016-08-01
In this paper we consider the coupled Kadomtsev-Petviashvili system. From compatibility conditions we obtain the form of matrix operators. After using a gauge transformation, obtained a new type of Lax representation for the hierarchy of Heisenberg ferromagnet equation, which is equivalent to the gauge coupled Kadomtsev-Petviashvili system.
The Explicit Solutions of Riccati Equation by Integral Series
Yan, Yimin
2010-01-01
The paper aims at exactly solving the linear differential equation and the matrix Riccati equation with variable coefficients. Starting with the simplest structure of them, this article promotes the exponential function by introducing two maps with integral series: $\\mathcal{E}(X)$ and $\\mathcal{F}(X)$, which extend the important properties of exponential function: convergence, reversibility, and the relationship of determinant. Then,the article shows two approaches to the Riccati equation solving: (1)one is the Simplified Way: summed up the Riccati equation into the simplest form $\\frac{\\partial}{\\partial x}W+WPW-Q=0$, and gets the accurate solution;(2) the other is Matrix Way: directly deal with the general Riccati equation, and transform into the matrix linear differential equation. And then the solution could be further expresses with the elementary form with the particular solution. At the end, we can see that Riccati equation solving is somehow a particular case of linear differential equation when view...
Darboux polynomials for Lotka-Volterra systems in three dimensions
Christodoulides, Yiannis T
2008-01-01
We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding cofactors. More precisely, we show that a Darboux polynomial of degree greater than one is reducible. In fact, it is a product of linear Darboux polynomials and first integrals.
Optoacoustic inversion via Volterra kernel reconstruction
Melchert, O; Roth, B
2016-01-01
In this letter we address the numeric inversion of optoacoustic signals to initial stress profiles. Therefore we put under scrutiny the optoacoustic kernel reconstruction problem in the paraxial approximation of the underlying wave-equation. We apply a Fourier-series expansion of the optoacoustic Volterra kernel and obtain the respective expansion coefficients for a given "apparative" setup by performing a gauge procedure using synthetic input data. The resulting effective kernel is subsequently used to solve the optoacoustic source reconstruction problem for general signals. We verify the validity of the proposed inversion protocol for synthetic signals and explore the feasibility of our approach to also account for the diffraction transformation of signals beyond the paraxial approximation.
Ecological communities with Lotka-Volterra dynamics
Bunin, Guy
2017-04-01
Ecological communities in heterogeneous environments assemble through the combined effect of species interaction and migration. Understanding the effect of these processes on the community properties is central to ecology. Here we study these processes for a single community subject to migration from a pool of species, with population dynamics described by the generalized Lotka-Volterra equations. We derive exact results for the phase diagram describing the dynamical behaviors, and for the diversity and species abundance distributions. A phase transition is found from a phase where a unique globally attractive fixed point exists to a phase where multiple dynamical attractors exist, leading to history-dependent community properties. The model is shown to possess a symmetry that also establishes a connection with other well-known models.
Integrability Estimates for Gaussian Rough Differential Equations
Cass, Thomas; Lyons, Terry
2011-01-01
We derive explicit tail-estimates for the Jacobian of the solution flow of stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H>1/4. We remark on the relevance of such estimates to a number of significant open problems.
Lattice defects as Lotka-Volterra societies
Energy Technology Data Exchange (ETDEWEB)
Yost, F.G.
1995-07-01
Since the early part of this century the Lotka-Volterra or predator-prey equations have been known to simulate the stability, instability, and persistent oscillations observed in many biological and ecological societies. These equations have been modified in many ways and have been used to model phenomena as varied as childhood epidemics, enzyme reactions, and conventional warfare. In the work to be described, similarities are drawn between various lattice defects and Lotka-Volterra (LV) societies. Indeed, grain boundaries are known to ``consume`` dislocations, inclusions ``infect`` grain boundaries, and dislocations ``annihilate`` dislocations. Several specific cases of lattice defect interaction kinetics models are drawn from the materials science literature to make these comparisons. Each model will be interpreted as if it were a description of a biological system. Various approaches to the modification of this class of interaction kinetics will be presented and discussed. The earliest example is the Damask-Dienes treatment of vacancy-divacancy annealing kinetics. This historical model will be modified to include the effects of an intermediate species and the results will be compared with the original model. The second example to be examined is the Clark-Alden model for deformation-enhanced grain growth. Dislocation kinetics will be added to this model and results will be discussed considering the original model. The third example to be presented is the Ananthakrishna-Sahoo model of the Portevin-Le Chatelier effect that was offered in 1985 as an extension of the classical Cottrell atmosphere explanation. Their treatment will be modified by inclusion of random interference from a pesky but peripheral species and by allowing a rate constant to be a function of time.
Kernel approximation for solving few-body integral equations
Christie, I.; Eyre, D.
1986-06-01
This paper investigates an approximate method for solving integral equations that arise in few-body problems. The method is to replace the kernel by a degenerate kernel defined on a finite dimensional subspace of piecewise Lagrange polynomials. Numerical accuracy of the method is tested by solving the two-body Lippmann-Schwinger equation with non-separable potentials, and the three-body Amado-Lovelace equation with separable two-body potentials.
Adomian solution of a nonlinear quadratic integral equation
Directory of Open Access Journals (Sweden)
E.A.A. Ziada
2013-04-01
Full Text Available We are concerned here with a nonlinear quadratic integral equation (QIE. The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Two methods are used to solve these type of equations; ADM and repeated trapezoidal method. The obtained results are compared.
DUAL INTEGRAL EQUATIONS INVOLVING LEGENDRE FUNCTIONS IN DISTRIBUTION SPACES
Directory of Open Access Journals (Sweden)
P. K. BANERJI, DESHNA LOONKER
2010-11-01
Full Text Available In this paper we use the Mehler-Fock transformation to obtain thesolution of dual integral equations involving Legendre functions. The solutionso obtained is proved to be distributional because they satisfy properties ofdistribution space.
A geometric approach to integrability conditions for Riccati equations
Directory of Open Access Journals (Sweden)
Arturo Ramos
2007-09-01
Full Text Available Several instances of integrable Riccati equations are analyzed from the geometric perspective of the theory of Lie systems. This provides us a unifying viewpoint for previous approaches.
Coverings and integrability of the Gauss-Mainardi-Codazzi equations
Krasilchchik, I; Krasil'shchik, Joseph; Marvan, Michal
1998-01-01
Using covering theory approach (zero-curvature representations with the gauge group SL2), we insert the spectral parameter into the Gauss-Mainardi-Codazzi equations in Tchebycheff and geodesic coordinates. For each choice, four integrable systems are obtained.
Integral conditions in the theory of the Beltrami equations
Ryazanov, V; Yakubov, E
2010-01-01
It is shown that many recent and new results on the existence of ACL homeomorphic solutions for the degenerate Beltrami equations with integral constraints follow from our extension of the well--known Lehto existence theorem.
Improved non-singular local boundary integral equation method
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
When the source nodes are on the global boundary in the implementation of local boundary integral equation method (LBIEM), singularities in the local boundary integrals need to be treated specially. In the current paper, local integral equations are adopted for the nodes inside the domain and moving least square approximation (MLSA)for the nodes on the global boundary, thus singularities will not occur in the new algorithm. At the same time, approximation errors of boundary integrals are reduced significantly. As applications and numerical tests, Laplace equation and Helmholtz equation problems are considered and excellent numerical results are obtained. Furthermore,when solving the Helmholtz problems, the modified basis functions with wave solutions areadapted to replace the usually-used monomial basis functions. Numerical results show that this treatment is simple and effective and its application is promising in solutions for the wave propagation problem with high wave number.
APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
Directory of Open Access Journals (Sweden)
Vorona Yu.V.
2015-12-01
Full Text Available Boundary Integral Equation Method is used for solving analytically the problems of coupled thermoelastic spherical wave propagation. The resulting mathematical expressions coincide with the solutions obtained in a conventional manner.
A New Algorithm for System of Integral Equations
Directory of Open Access Journals (Sweden)
Abdujabar Rasulov
2014-01-01
Full Text Available We develop a new algorithm to solve the system of integral equations. In this new method no need to use matrix weights. Beacause of it, we reduce computational complexity considerable. Using the new algorithm it is also possible to solve an initial boundary value problem for system of parabolic equations. To verify the efficiency, the results of computational experiments are given.
Nonlinear partial differential equations: Integrability, geometry and related topics
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
Riccati equation-based generalization of Dawson's integral function
Messina, R; Messina, A; Napoli, A
2007-01-01
A new generalization of Dawson's integral function based on the link between a Riccati nonlinear differential equation and a second-order ordinary differential equation is reported. The MacLaurin expansion of this generalized function is built up and to this end an explicit formula for a generic cofactor of a triangular matrix is deduced.
Numerical Inversion of Integral Equations for Medical Imaging and Geophysics
1988-12-13
Equations for Medical Imaging and Geophysics (Unclassified) 12 PERSONAL AUTHOR(S) Frank Stenger 13a. TYPE OF REPORT 13b TIME COVERED 14. DATE OF REPORT...9r~S NUMERICAL INVERSION OF INTEGRAL EQUATIONS FOR MEDICAL IMAGING AND GEOPHYSICS FINAL REPORT AUTHOR OF REPORT: Frank Stenger December 13, 1988
Kleinert, H.; Zatloukal, V.
2015-01-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Positive and Nontrivial Solutions for the Urysohn Integral Equation
Institute of Scientific and Technical Information of China (English)
Daniel FRANCO; Gennaro INFANTE; Donal O'REGAN
2006-01-01
We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.
Exponential Observers for Lotka-Volterra Systems
Directory of Open Access Journals (Sweden)
Dr. V. Sundarapandian
2011-03-01
Full Text Available This paper solves the exponential observer design problem for Lotka-Volterra systems. Explicitly, Sundarapandian’s theorem (2002 for observer design for exponential observer design is used to solve the nonlinear observer design problem for 2-species, 3-species and 4-species Lotka-Volterra systems. Numerical examples are provided to illustrate the effectiveness of the proposed exponential observer design for the Lotka-Volterra systems.
Fringe integral equation method for a truncated grounded dielectric slab
DEFF Research Database (Denmark)
Jørgensen, Erik; Maci, S.; Toccafondi, A.
2001-01-01
The problem of scattering by a semi-infinite grounded dielectric slab illuminated by an arbitrary incident TMz polarized electric field is studied by solving a new set of “fringe” integral equations (F-IEs), whose functional unknowns are physically associated to the wave diffraction processes...... occurring at the truncation. The F-IEs are obtained by subtracting from the surface/surface integral equations pertinent to the truncated slab, an auxiliary set of equations obtained for the canonical problem of an infinite grounded slab illuminated by the same source. The F-IEs are solved by the method...
A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure.A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a B(a)cklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.
Differential equations and integrable models the $SU(3)$ case
Dorey, P; Dorey, Patrick; Tateo, Roberto
2000-01-01
We exhibit a relationship between the massless $a_2^{(2)}$ integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schrödinger equation. This forms part of a more general correspondence involving $A_2$-related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the nonlinear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators $\\phi_{12}$, $\\phi_{21}$ and $\\phi_{15}$. This is checked against previous results obtained using the thermodynamic Bethe ansatz.
NOVEL REGULARIZED BOUNDARY INTEGRAL EQUATIONS FOR POTENTIAL PLANE PROBLEMS
Institute of Scientific and Technical Information of China (English)
ZHANG Yao-ming; L(U) He-xiang; WANG Li-min
2006-01-01
The universal practices have been centralizing on the research of regularization to the direct boundary integal equations (DBIEs). The character is elimination of singularities by using the simple solutions. However, up to now the research of regularization to the first kind integral equations for plane potential problems has never been found in previous literatures. The presentation is mainly devoted to the research on the regularization of the singular boundaryintegral equations with indirect unknowns. A novel view and idea is presented herein, in which the regularized boundary integral equations with indirect unknowns without including the Cauchy principal value (CPV) and Hadamard-finite-part (HFP) integrals are established for the plane potential problems.With some numerical results, it is shown that the better accuracy and higher efficiency,especially on the boundary, can be achieved by the present system.
Rebelo, Raphaël; Winternitz, Pavel
2017-01-01
This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers...
An algorithm of computing inhomogeneous differential equations for definite integrals
Nakayama, Hiromasa; Nishiyama, Kenta
2010-01-01
We give an algorithm to compute inhomogeneous differential equations for definite integrals with parameters. The algorithm is based on the integration algorithm for $D$-modules by Oaku. Main tool in the algorithm is the Gr\\"obner basis method in the ring of differential operators.
LINEAR SINGULAR INTEGRAL EQUATION ON DOMAINS COMPOSED BY BALLS
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
For domains composed by balls in Cn, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.
Nonzero solutions of nonlinear integral equations modeling infectious disease
Energy Technology Data Exchange (ETDEWEB)
Williams, L.R. (Indiana Univ., South Bend); Leggett, R.W.
1982-01-01
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.
Lotka-Volterra systems satisfying a strong Painlevé property
Bountis, Tassos; Vanhaecke, Pol
2016-12-01
We use a strong version of the Painlevé property to discover and characterize a new class of n-dimensional Hamiltonian Lotka-Volterra systems, which turn out to be Liouville integrable as well as superintegrable. These systems are in fact Nambu systems, they posses Lax equations and they can be explicitly integrated in terms of elementary functions. We apply our analysis to systems containing only quadratic nonlinearities of the form aijxixj , i ≠ j, and require that all variables diverge as t-1. We also require that the leading terms depend on n - 2 free parameters. We thus discover a cocycle relation among the coefficients aij of the equations of motion and by integrating the cocycle equations we show that they are equivalent to the above strong version of the Painlevé property. We also show that these systems remain explicitly solvable even if a linear term bixi is added to the i-th equation, even though this violates the Painlevé property, as logarithmic singularities are introduced in the Laurent solutions, at the first terms following the leading order pole.
Energy Technology Data Exchange (ETDEWEB)
Zenchuk, A I, E-mail: zenchuk@itp.ac.r [Institute of Problems of Chemical Physics, RAS Acad. Semenov av., 1 Chernogolovka, Moscow region 142432 (Russian Federation)
2010-06-18
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical C- and S-integrable partial differential equations (PDEs). Generalizations of (1+1)-dimensional C-integrable and (2+1)-dimensional S-integrable N-wave equations are derived among examples. Examples of multidimensional second-order PDEs are represented as well.
Energy Technology Data Exchange (ETDEWEB)
Bustamante, Miguel D [Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Chile); Hojman, Sergio A [Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Chile)
2003-01-10
In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behaviour has not been verified up to now. The Euler-Lagrange equations we get for these systems usually present 'time reparametrization' invariance, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion.
Distribution theory for Schrödinger’s integral equation
Energy Technology Data Exchange (ETDEWEB)
Lange, Rutger-Jan, E-mail: rutger-jan.lange@cantab.net [VU University Amsterdam, 1081 HV Amsterdam (Netherlands)
2015-12-15
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics.
Distribution theory for Schrödinger's integral equation
Lange, Rutger-Jan
2015-12-01
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger's equation. This paper, in contrast, investigates the integral form of Schrödinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's [J. Math. Anal. Appl. 201(1), 297-323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger's differential equation. Third, we derive boundary conditions for "super-singular" potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger's integral equation is a viable tool for studying singular interactions in quantum mechanics.
First integrals and stability of second-order differential equations
Institute of Scientific and Technical Information of China (English)
Xu Xue-Jun; Mei Feng-Xiang
2006-01-01
The stability of second-order differential equations is studied by using their integrals. A system of second-order differential equations can be considered as a mechanical system with holonomic constraints. A conserved quantity of the mechanical system or a part of the system is obtained by using the Noether theory. It is possible that the conserved quantity becomes a Liapunov function and the stability of the system is proved by the Liapunov theorem.
Coherence and Chaos in Integrable PDEs (Partial Differential Equations)
1991-03-01
01 Aug 88 to 30 Sep 9n 4. AMSUB"=S. PUNOUUS NU"蕁 COHERENCE AND CHAOS IN INTEGRABLE PDEs ( PARTIAL DIFFERENTIAL EQUATIONS ) AFOSR-83-0195 _61102F... Differential Equations , Parts 1 and 2; Lectures in Appl. Math. 23, edited by Basil Nicolaenko, Darrel Holm, and and J. Mac Hyman (American Mathematical...Coherent Structures, edited by David Campbell, Alan C. Newell, R. Schrieffer, and Harvey Segur, Physica 18D (1986). 4. Nonlinear Systems of Partial
Calculation of multi frequency of Helmholtz boundary integral equation
Institute of Scientific and Technical Information of China (English)
ZHAO Zhigao; HUANG Qibai
2005-01-01
The method using series expansion is presented, and the wavenumber is separated from fundamental solution of Helmholtz boundary element equation, then the system matrices dependent of wavenumber are the matrices series associated with wavenumber, and the astringency of the method is proved. The numerical results show that combined with the CHIEFmethod, the SECHIEF (Series Expansion Combined Helmholtz Integral Equation Formulation) method can not only provide uniqueness of solution and reduce the computational time but also give accurate results under the coarse elements.
Fractional Calculus: Integral and Differential Equations of Fractional Order
Gorenflo, Rudolf
2008-01-01
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.
Yang-Baxter Maps, Discrete Integrable Equations and Quantum Groups
Bazhanov, Vladimir V
2015-01-01
For every quantized Lie algebra there exists a map from the tensor square of the algebra to itself, which by construction satisfies the set-theoretic Yang-Baxter equation. This map allows one to define an integrable discrete quantum evolution system on quadrilateral lattices, where local degrees of freedom (dynamical variables) take values in a tensor power of the quantized Lie algebra. The corresponding equations of motion admit the zero curvature representation. The commuting Integrals of Motion are defined in the standard way via the Quantum Inverse Problem Method, utilizing Baxter's famous commuting transfer matrix approach. All elements of the above construction have a meaningful quasi-classical limit. As a result one obtains an integrable discrete Hamiltonian evolution system, where the local equation of motion are determined by a classical Yang-Baxter map and the action functional is determined by the quasi-classical asymptotics of the universal R-matrix of the underlying quantum algebra. In this paper...
Field Method for Integrating the First Order Differential Equation
Institute of Scientific and Technical Information of China (English)
JIA Li-qun; ZHENG Shi-wang; ZHANG Yao-yu
2007-01-01
An important modern method in analytical mechanics for finding the integral, which is called the field-method, is used to research the solution of a differential equation of the first order. First, by introducing an intermediate variable, a more complicated differential equation of the first order can be expressed by two simple differential equations of the first order, then the field-method in analytical mechanics is introduced for solving the two differential equations of the first order. The conclusion shows that the field-method in analytical mechanics can be fully used to find the solutions of a differential equation of the first order, thus a new method for finding the solutions of the first order is provided.
Excited state nonlinear integral equations for an integrable anisotropic spin-1 chain
Energy Technology Data Exchange (ETDEWEB)
Suzuki, J [Department of Physics, Faculty of Science, Shizuoka University, Ohya 836, Shizuoka (Japan)
2004-12-17
We propose a set of nonlinear integral equations to describe the excited states of an integrable the spin-1 chain with anisotropy. The scaling dimensions, evaluated numerically in previous studies, are recovered analytically by using the equations. This result may be relevant to the study of the supersymmetric sine-Gordon model.
A spectral boundary integral equation method for the 2-D Helmholtz equation
Hu, Fang Q.
1994-01-01
In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the nonsmoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the nonsmoothness of the integral kernels in the spectral implementation. The present method is robust for a general boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. A numerical example of wave scattering is given in which the exponential accuracy of the present numerical method is demonstrated.
Master equations and the theory of stochastic path integrals.
Weber, Markus F; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon
Master equations and the theory of stochastic path integrals
Weber, Markus F.; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from
Plasmonic properties of metal nanoislands: surface integral equations approach
Scherbak, S. A.; Lipovskii, A. A.
2016-08-01
The surface integral equations method is used to analyse the surface plasmon resonance position in a metal island film formed by non-interacting axisymmetrical prolate/oblate hemispheroids placed on a dielectric substrate. The approach is verified via the comparison of results obtained for a hemisphere on a substrate with the ones obtained using the multipole expansion method. The preference of the integral equations method is in obtaining a simple final analytical expression for a particle polarizability in which any dielectric function of a metal can be substituted. Such simple formulae for the hemispherical particle on the substrate and calculated dependences of the hemispheroid resonant wavelength on its aspect ratio are presented.
The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation
Dong, Huanhe; Zhang, Yong; Zhang, Xiaoen
2016-07-01
A discrete matrix spectral problem is presented and the hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are established. As to the discrete integrable system, nonlinearization of the spatial parts of the Lax pairs and the adjoint Lax pairs generate a new integrable symplectic map. Based on the theory, a new integrable symplectic map and a family of finite-dimension completely integrable systems are given. Especially, two explicit equations are obtained under the Bargmann constraint. Finally, the symmetry of the discrete equation is provided according to the recursion operator and the seed symmetry. Although the solutions of the discrete equations have been gained by many methods, there are few articles that solving the discrete equation via the symmetry. So the solution of the discrete lattice equation is obtained through the symmetry theory.
Lotka-Volterra representation of general nonlinear systems.
Hernández-Bermejo, B; Fairén, V
1997-02-01
In this article we elaborate on the structure of the generalized Lotka-Volterra (GLV) form for nonlinear differential equations. We discuss here the algebraic properties of the GLV family, such as the invariance under quasimonomial transformations and the underlying structure of classes of equivalence. Each class possesses a unique representative under the classical quadratic Lotka-Volterra form. We show how other standard modeling forms of biological interest, such as S-systems or mass-action systems, are naturally embedded into the GLV form, which thus provides a formal framework for their comparison and for the establishment of transformation rules. We also focus on the issue of recasting of general nonlinear systems into the GLV format. We present a procedure for doing so and point at possible sources of ambiguity that could make the resulting Lotka-Volterra system dependent on the path followed. We then provide some general theorems that define the operational and algorithmic framework in which this is not the case.
Seismic traveltime inversion based on tomographic equation without integral terms
Huang, Guangnan; Zhou, Bing; Li, Hongxing; Nobes, David C.
2017-07-01
The Jacobian matrix in the seismic traveltime tomographic equations usually contains several integral terms. These integral expressions not only greatly increase the computational complexity of seismic traveltime tomography, but also increase difficulty for programming these expressions. Therefore, if these integral expressions of the Jacobian matrix can be eliminated, the program of seismic traveltime tomography can be greatly simplified. In order to solve the computational complexity of the traditional seismic traveltime tomography, we found an anisotropic seismic traveltime tomographic equation which does not contain integral expressions. Then, it is degenerated into an isotropic seismic traveltime tomographic equation. In order to verify the effectiveness of this seismic traveltime tomographic equation based on the node network, a program has been coded to execute seismic traveltime inversion. For a crosswell checkerboard velocity model, the same results are obtained by this proposed tomographic method and the traditional method (with integral terms). Besides, two undulating topography velocity models are used as testing models. Numerical simulation results show that this proposed tomographic method can achieve good tomograms. Finally, this proposed tomographic method is used to investigate near surface velocity distribution near a power plant. Tomogram indicates that contaminated liquid diffuses and aggregates along strata at a certain depth. And velocity is lower near pollutant source than that away from it.
The Pentabox Master Integrals with the Simplified Differential Equations approach
Papadopoulos, Costas G; Wever, Christopher
2015-01-01
We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many $2\\to 3$ amplitudes of interest at the LHC, as for instance three jet production, $\\gamma, V, H +2$ jets etc., based on the Simplified Differential Equations approach.
On the numeric integration of dynamic attitude equations
Crouch, P. E.; Yan, Y.; Grossman, Robert
1992-01-01
We describe new types of numerical integration algorithms developed by the authors. The main aim of the algorithms is to numerically integrate differential equations which evolve on geometric objects, such as the rotation group. The algorithms provide iterates which lie on the prescribed geometric object, either exactly, or to some prescribed accuracy, independent of the order of the algorithm. This paper describes applications of these algorithms to the evolution of the attitude of a rigid body.
Yersultanova, Z. S.; Zhassybayeva, M.; Yesmakhanova, K.; Nugmanova, G.; Myrzakulov, R.
2016-10-01
Integrable Heisenberg ferromagnetic equations are an important subclass of integrable systems. The M-XCIX equation is one of a generalizations of the Heisenberg ferromagnetic equation and are integrable. In this paper, the Darboux transformation of the M-XCIX equation is constructed. Using the DT, a 1-soliton solution of the M-XCIX equation is presented.
Bustamante, M D
2003-01-01
In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behaviour has not been verified up to now. The Euler-Lagrange equations we get for these systems usually present 'time reparametrization' invariance, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, w...
Painlevé equations, elliptic integrals and elementary functions
Żołądek, Henryk; Filipuk, Galina
2015-02-01
The six Painlevé equations can be written in the Hamiltonian form, with time dependent Hamilton functions. We present a rather new approach to this result, leading to rational Hamilton functions. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems with two degrees of freedom. We realize the Bäcklund transformations of the Painlevé equations as symplectic birational transformations in C4 and we interpret the cases with classical solutions as the cases of partial integrability of the extended Hamiltonian systems. We prove that the extended Hamiltonian systems do not have any additional algebraic first integral besides the known special cases of the third and fifth Painlevé equations. We also show that the original Painlevé equations admit the first integrals expressed in terms of the elementary functions only in the special cases mentioned above. In the proofs we use equations in variations with respect to a parameter and Liouville's theory of elementary functions.
An integrable semi-discretization of the Boussinesq equation
Energy Technology Data Exchange (ETDEWEB)
Zhang, Yingnan, E-mail: ynzhang@njnu.edu.cn [Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu (China); Tian, Lixin, E-mail: tianlixin@njnu.edu.cn [Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu (China); Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu (China)
2016-10-23
Highlights: • A new integrable semi-discretization of the Boussinesq equation is present. • A Bäcklund transformation and a Lax pair for the differential-difference system is derived by using Hirota's bilinear method. • The soliton solutions of 'good' Boussinesq equation and numerical algorithms are investigated. - Abstract: In this paper, we present an integrable semi-discretization of the Boussinesq equation. Different from other discrete analogues, we discretize the ‘time’ variable and get an integrable differential-difference system. Under a standard limitation, the differential-difference system converges to the continuous Boussinesq equation such that the discrete system can be used to design numerical algorithms. Using Hirota's bilinear method, we find a Bäcklund transformation and a Lax pair of the differential-difference system. For the case of ‘good’ Boussinesq equation, we investigate the soliton solutions of its discrete analogue and design numerical algorithms. We find an effective way to reduce the phase shift caused by the discretization. The numerical results coincide with our analysis.
An integrable semi-discretization of the Boussinesq equation
Zhang, Yingnan; Tian, Lixin
2016-10-01
In this paper, we present an integrable semi-discretization of the Boussinesq equation. Different from other discrete analogues, we discretize the 'time' variable and get an integrable differential-difference system. Under a standard limitation, the differential-difference system converges to the continuous Boussinesq equation such that the discrete system can be used to design numerical algorithms. Using Hirota's bilinear method, we find a Bäcklund transformation and a Lax pair of the differential-difference system. For the case of 'good' Boussinesq equation, we investigate the soliton solutions of its discrete analogue and design numerical algorithms. We find an effective way to reduce the phase shift caused by the discretization. The numerical results coincide with our analysis.
ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
MEI Shu-li; LU Qi-shao; ZHANG Sen-wen; JIN Li
2005-01-01
The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations (ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.
Integrated vehicle dynamics control using State Dependent Riccati Equations
Bonsen, B.; Mansvelders, R.; Vermeer, E.
2010-01-01
In this paper we discuss a State Dependent Riccati Equations (SDRE) solution for Integrated Vehicle Dynamics Control (IVDC). The SDRE approach is a nonlinear variant of the well known Linear Quadratic Regulator (LQR) and implements a quadratic cost function optimization. A modified version of this
Classical integrable systems and Knizhnik-Zamolodchikov-Bernard equations
Aminov, G.; Levin, A.; Olshanetsky, M.; Zotov, A.
2015-05-01
The results obtained in the works supported in part by the Russian Foundation for Basic Research (project 12-02-00594) are briefly reviewed. We mainly focus on interrelations between classical integrable systems, Painlevé-Schlesinger equations and related algebraic structures such as classical and quantum R-matrices. The constructions are explained in terms of simplest examples.
Integrated vehicle dynamics control using State Dependent Riccati Equations
Bonsen, B.; Mansvelders, R.; Vermeer, E.
2010-01-01
In this paper we discuss a State Dependent Riccati Equations (SDRE) solution for Integrated Vehicle Dynamics Control (IVDC). The SDRE approach is a nonlinear variant of the well known Linear Quadratic Regulator (LQR) and implements a quadratic cost function optimization. A modified version of this m
Higher-Order Integral Equation Methods in Computational Electromagnetics
DEFF Research Database (Denmark)
Jørgensen, Erik; Meincke, Peter
Higher-order integral equation methods have been investigated. The study has focused on improving the accuracy and efficiency of the Method of Moments (MoM) applied to electromagnetic problems. A new set of hierarchical Legendre basis functions of arbitrary order is developed. The new basis...
Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation
Kazakov, A. Ya.; Slavyanov, S. Yu.
2008-05-01
Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
The Lyapunov stabilization of satellite equations of motion using integrals
Nacozy, P. E.
1973-01-01
A method is introduced that weakens the Lyapunov or in track instability of satellite equations of motion. The method utilizes a linearized energy integral of satellite motion as a constraint on solutions obtained by numerical integration. The procedure prevents local numerical error from altering the frequency associated with the fast angular variable and thereby reduces the Lyapunov instability and the global numerical error. Applications of the method to satellite motion show accuracy improvements of two to three orders of magnitude in position and velocity after 50 revolutions. A modification of the method is presented that allows the use of slowly varying integrals of motion.
Boundary regularized integral equation formulation of the Helmholtz equation in acoustics.
Sun, Qiang; Klaseboer, Evert; Khoo, Boo-Cheong; Chan, Derek Y C
2015-01-01
A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals.
Boundary regularized integral equation formulation of the Helmholtz equation in acoustics
Sun, Qiang; Klaseboer, Evert; Khoo, Boo-Cheong; Chan, Derek Y. C.
2015-01-01
A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals. PMID:26064591
Kwong-Wong-type integral equation on time scales
Directory of Open Access Journals (Sweden)
Baoguo Jia
2011-09-01
Full Text Available Consider the second-order nonlinear dynamic equation $$ [r(tx^Delta(ho(t]^Delta+p(tf(x(t=0, $$ where $p(t$ is the backward jump operator. We obtain a Kwong-Wong-type integral equation, that is: If $x(t$ is a nonoscillatory solution of the above equation on $[T_0,infty$, then the integral equation $$ frac{r^sigma(tx^Delta(t}{f(x^sigma(t} =P^sigma(t+int^infty_{sigma(t}frac{r^sigma(s [int^1_0f'(x_h(sdh][x^Delta(s]^2}{f(x(s f(x^sigma(s}Delta s $$ is satisfied for $tgeq T_0$, where $P^sigma(t=int^infty_{sigma(t}p(sDelta s$, and $x_h(s=x(s+hmu(sx^Delta(s$. As an application, we show that the superlinear dynamic equation $$ [r(tx^{Delta}(ho(t]^Delta+p(tf(x(t=0, $$ is oscillatory, under certain conditions.
A Multiple Iterated Integral Inequality and Applications
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Zongyi Hou
2014-01-01
Full Text Available We establish new multiple iterated Volterra-Fredholm type integral inequalities, where the composite function w(u(s of the unknown function u with nonlinear function w in integral functions in [Ma, QH, Pečarić, J: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal. 69 (2008 393–407] is changed into the composite functions w1(u(s,w2(u(s,…, wn (u(s of the unknown function u with different nonlinear functions w1,w2,…,wn, respectively. By adopting novel analysis techniques, the upper bounds of the embedded unknown functions are estimated explicitly. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations.
Rational first integrals of geodesic equations and generalised hidden symmetries
Aoki, Arata; Tomoda, Kentaro
2016-01-01
We discuss novel generalisations of Killing tensors, which are introduced by considering rational first integrals of geodesic equations. We introduce the notion of inconstructible generalised Killing tensors, which cannot be constructed from ordinary Killing tensors. Moreover, we introduce inconstructible rational first integrals, which are constructed from inconstructible generalised Killing tensors, and provide a method for checking the inconstructibility of a rational first integral. Using the method, we show that the rational first integral of the Collinson-O'Donnell solution is not inconstructible. We also provide several examples of metrics admitting an inconstructible rational first integral in two and four dimensions, by using the Maciejewski-Przybylska system. Furthermore, we attempt to generalise other hidden symmetries such as Killing-Yano tensors.
Existence of positive almost periodic solutions for delay Lotka-Volterra cooperative systems
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Kaihong Zhao
2013-07-01
Full Text Available In this article, we study a Lotka-Volterra cooperative system of equations with time-varying delays and distributed delays. By using Mawhin's continuation theorem of coincidence degree theory, we obtain sufficient conditions for the existence of positive almost periodic solutions. Also we present an example to illustrate our results.
Sustained dynamic transience in a Lotka–Volterra competition model system for grassland species
Geijzendorffer, I.R.; Werf, van der W.; Bianchi, F.J.J.A.; Schulte, R.P.O.
2011-01-01
Theoretical approaches, such as the Lotka–Volterra framework, enable predictions about long term species coexistence based on stability criteria, but generally assume temporal constancy of system equations and parameters. In real world systems, temporal variability may interfere with the attainment
Sustained dynamic transience in a Lotka–Volterra competition model system for grassland species
Geijzendorffer, I.R.; Werf, van der W.; Bianchi, F.J.J.A.; Schulte, R.P.O.
2011-01-01
Theoretical approaches, such as the Lotka–Volterra framework, enable predictions about long term species coexistence based on stability criteria, but generally assume temporal constancy of system equations and parameters. In real world systems, temporal variability may interfere with the attainment
Singularity Preserving Numerical Methods for Boundary Integral Equations
Kaneko, Hideaki (Principal Investigator)
1996-01-01
In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.
Ultradiscrete soliton equations derived from ultradiscrete permanent formulae
Energy Technology Data Exchange (ETDEWEB)
Nakamura, Shinya, E-mail: s-nakamura@moegi.waseda.jp [Major in Pure and Applied Mathematics, Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555 (Japan)
2011-07-22
We propose formulae of ultradiscrete permanent. Utilizing the formulae, ultradiscrete soliton equations and their multi-soliton solutions are obtained by a simple process. Changing variables and parameters of the formulae, we can derive the ultradiscrete Toda, KdV and hungry Lotka-Volterra equations. An extended version of the ultradiscrete hungry Lotka-Volterra equation is also proposed.
Institute of Scientific and Technical Information of China (English)
Qingfeng ZHU; Yufeng SHI
2012-01-01
Backward doubly stochastic differential equations driven by Brownian motions and Poisson process (BDSDEP) with non-Lipschitz coefficients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.
Extinction dynamics of Lotka-Volterra ecosystems on evolving networks.
Coppex, F; Droz, M; Lipowski, A
2004-06-01
We study a model of a multispecies ecosystem described by Lotka-Volterra-like equations. Interactions among species form a network whose evolution is determined by the dynamics of the model. Numerical simulations show power-law distribution of intervals between extinctions, but only for ecosystems with sufficient variability of species and with networks of connectivity above certain threshold that is very close to the percolation threshold of the network. The effect of slow environmental changes on extinction dynamics, degree distribution of the network of interspecies interactions, and some emergent properties of our model are also examined.
Directory of Open Access Journals (Sweden)
Adriana C. Briozzo
2006-02-01
Full Text Available We prove the existence and uniqueness, local in time, of a solution for a one-phase Stefan problem of a non-classical heat equation for a semi-infinite material with temperature boundary condition at the fixed face. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations.
The Jungle Universe: coupled cosmological models in a Lotka-Volterra framework
Perez, Jérôme; Füzfa, André; Carletti, Timoteo; Mélot, Laurence; Guedezounme, Lazare
2014-06-01
In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann-Lemaître universes is a special case of generalized Lotka-Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka-Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann-Lemaître cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserves the structure of the dynamical equations. Using the standard tools of Lotka-Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids.
Directory of Open Access Journals (Sweden)
Xiao-Ping Chen
2016-01-01
Full Text Available The n-species Lotka-Volterra system with discrete delays is considered. The local asymptotic stability of positive equilibrium is investigated based on a contour integral method. The main purpose of this paper is to propose a new and general algorithm to study the local asymptotic stability of the positive equilibrium for the n-dimensional Lotka-Volterra system. Some numerical experiments are carried out to show the effectiveness of the proposed method.
Stability Criteria for Volterra Integrodynamic System
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Nusrat Yasmin
2015-01-01
Full Text Available We study conditions under which the solutions of linear Volterra integrodynamic system of the form yΔt=Atyt+∫t0tKt,sysΔs are stable on certain time scales. We construct a number of Lyapunov functionals on time scales from which we obtain necessary and sufficient conditions for stability of Volterra integrodynamic system and also we prove several results concerning qualitative behavior of this system.
Boundary Integral Equations and A Posteriori Error Estimates
Institute of Scientific and Technical Information of China (English)
YU Dehao; ZHAO Longhua
2005-01-01
Adaptive methods have been rapidly developed and applied in many fields of scientific and engineering computing. Reliable and efficient a posteriori error estimates play key roles for both adaptive finite element and boundary element methods. The aim of this paper is to develop a posteriori error estimates for boundary element methods. The standard a posteriori error estimates for boundary element methods are obtained from the classical boundary integral equations. This paper presents hyper-singular a posteriori error estimates based on the hyper-singular integral equations. Three kinds of residuals are used as the estimates for boundary element errors. The theoretical analysis and numerical examples show that the hyper-singular residuals are good a posteriori error indicators in many adaptive boundary element computations.
Towards the automatized evaluation of Feynman integrals with differential equations
Energy Technology Data Exchange (ETDEWEB)
Meyer, Christoph; Uwer, Peter [HU Berlin, Berlin (Germany)
2016-07-01
In the past years the method of differential equations has proven itself to be a powerful tool for the computation of multi-loop Feynman integrals. This method relies on the choice of a basis of master integrals in which the dependence on the dimensional regulator factorizes. We present an algorithm which automatizes the transformation to such a basis, starting from a given reduction basis. The algorithm only requires some mild assumptions about the basis. It applies to problems with multiple scales of which we will present some examples.
Introduction to stochastic analysis integrals and differential equations
Mackevicius, Vigirdas
2013-01-01
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion pro
A New time Integration Scheme for Cahn-hilliard Equations
Schaefer, R.
2015-06-01
In this paper we present a new integration scheme that can be applied to solving difficult non-stationary non-linear problems. It is obtained by a successive linearization of the Crank- Nicolson scheme, that is unconditionally stable, but requires solving non-linear equation at each time step. We applied our linearized scheme for the time integration of the challenging Cahn-Hilliard equation, modeling the phase separation in fluids. At each time step the resulting variational equation is solved using higher-order isogeometric finite element method, with B- spline basis functions. The method was implemented in the PETIGA framework interfaced via the PETSc toolkit. The GMRES iterative solver was utilized for the solution of a resulting linear system at every time step. We also apply a simple adaptivity rule, which increases the time step size when the number of GMRES iterations is lower than 30. We compared our method with a non-linear, two stage predictor-multicorrector scheme, utilizing a sophisticated step length adaptivity. We controlled the stability of our simulations by monitoring the Ginzburg-Landau free energy functional. The proposed integration scheme outperforms the two-stage competitor in terms of the execution time, at the same time having a similar evolution of the free energy functional.
The Volterra series as special case of artificial neural network model
Napiorkowski, J.; O Kane, J. P.
2003-04-01
The geophysical processes contributing to the hydrological cycle are described by theoretically sound non-linear partial differential equations of mass and energy transfer. The hydrodynamic equations describing hydrological processes were developed in non-linear form in the nineteenth century. In the case of surface runoff from a natural catchment or flow in an open channel, an accurate application of the hydraulic approach requires a detailed topographical survey and determination of roughness parameters. In order to avoid these difficulties, alternative approaches e.g. via conceptual models and black box models were developed in the second half of the last century. The conceptual model approach is to simulate the nature of the catchment response or the channel response by relatively simple non-linear model built up from simple non-linear elements, e.g. cascade of non-linear reservoirs. Each non-linear reservoir is responsible for part of the attenuation of the system response. This lumped dynamic model can be represented by a set of ordinary differential equations: begin{gathered} dot S_1 (t) = - f[S_1 (t)] + x(t) dot S_2 (t) = - f[S_2 (t)] + f[S_1 (t)] ... dot S_n (t) = - f[S_n (t)] + f[Sn - 1 (t)] y(t) = f[S_n (t)] % MathType!End!2!1! (1) where x is the input signal(rainfall or flow at the upstream end of the channel), Si is the storage in the i-th reservoir, f(.) represents the outflow-storage relation and y is the output signal (surface runoff or flow at the downstream end of the channel). Non-linear black box analysis is concerned with representing a system by a functional Volterra series in the form of a sum of convolution integrals: begin{gathered} y(t) = intlimits_0^t {h_1 (τ )x(t - τ )dτ + intlimits_0^t {intlimits_0^t {h_2 (τ _1 ,τ _2 )x(t - τ _1 )x(t - τ _2 )dτ _1 dτ _2 } } } quad quad + intlimits_0^t {intlimits_0^t {intlimits_0^t {h_3 (τ _1 ,τ _2 ,τ _3 )x(t - τ _1 )x(t - τ _2 )x(t - τ _3 )dτ _1 dτ _2 dτ _3 } + ...} } % MathType!End!2
Directory of Open Access Journals (Sweden)
Gülden Gün
2013-01-01
Full Text Available We analyze Noether and -symmetries of the path equation describing the minimum drag work. First, the partial Lagrangian for the governing equation is constructed, and then the determining equations are obtained based on the partial Lagrangian approach. For specific altitude functions, Noether symmetry classification is carried out and the first integrals, conservation laws and group invariant solutions are obtained and classified. Then, secondly, by using the mathematical relationship with Lie point symmetries we investigate -symmetry properties and the corresponding reduction forms, integrating factors, and first integrals for specific altitude functions of the governing equation. Furthermore, we apply the Jacobi last multiplier method as a different approach to determine the new forms of -symmetries. Finally, we compare the results obtained from different classifications.
INSTANTANEOUS AND INTEGRAL POWER EQUATIONS OF NONSINUSOIDAL 3-PHASE PROCESSES
Directory of Open Access Journals (Sweden)
Iu.A. Sirotin
2016-03-01
Full Text Available Purpose. To identify the mathematical relationship between the instantaneous powers (classical and vectorial and integral powers in non-sinusoidal mode and to get complex form of instantaneous powers in 3-phase 4-wire power supply in terms of the spectral approach. Methodology. We have applied the vector approach with one voice allows you to analyze the energy characteristics of 3-phase power supply circuits (for 4-wire and 3-wire circuits in sinusoidal and non–sinusoidal mode, both the time domain and frequency domain. We have used 3-dimensional representation of the energy waveforms with the complex multi-dimensional Fourier series. Results. For 4-wire network with a non-sinusoidal (regardless of their symmetry processes, we have developed the mathematical model one-dimensional representations of the complex form for the active (scalar instantaneous power (IP and 3-dimensional form (inactive vectorial IP. It is possible to obtain two dual integral power equations for complex scalar and vector integrated power of non-sinusoidal modes. The power equations generalize generalizes the equations of sinusoidal modes for 4-wire network. Originality. In addition to the classification of energy local regimes in the time domain for the first time we spent the classification of non-sinusoidal modes in the spectral region and showed the value and importance of the classification of regimes based on the instantaneous powers. Practical value. The practical value the obtained equations is the possibility of their use for improving the quality of electricity supply and the quality electricity consumption.
On solitary patterns in Lotka-Volterra chains
Zilburg, Alon; Rosenau, Philip
2016-03-01
We present and study a class of Lotka-Volterra chains with symmetric 2N-neighbors interactions. To identify the types of solitary waves which may propagate along the chain, we study their quasi-continuum approximations which, depending on the coupling between neighbors, reduce into a large variety of partial differential equations. Notable among the emerging equations is a bi-cubic equation {u}t={[{{bu}}2+2κ {{uu}}{xx}+{({u}{xx})}2]}x which we study in some detail. It begets remarkably stable topological and non-topological solitary compactons that interact almost elastically. They are used to identify discretons, their solitary discrete antecedents on the lattice, which decay at a doubly exponential rate. Many of the discrete modes are robust while others either decompose or evolve into breathers.
Short Polymer Modeling using Self-Consistent Integral Equation Method
Kim, Yeongyoon; Park, So Jung; Kim, Jaeup
2014-03-01
Self-consistent field theory (SCFT) is an excellent mean field theoretical tool for predicting the morphologies of polymer based materials. In the standard SCFT, the polymer is modeled as a Gaussian chain which is suitable for a polymer of high molecular weight, but not necessarily for a polymer of low molecular weight. In order to overcome this limitation, Matsen and coworkers have recently developed SCFT of discrete polymer chains in which one polymer is modeled as finite number of beads joined by freely jointed bonds of fixed length. In their model, the diffusion equation of the canonical SCFT is replaced by an iterative integral equation, and the full spectral method is used for the production of the phase diagram of short block copolymers. In this study, for the finite length chain problem, we apply pseudospectral method which is the most efficient numerical scheme to solve the iterative integral equation. We use this new numerical method to investigate two different types of polymer bonds: spring-beads model and freely-jointed chain model. By comparing these results with those of the Gaussian chain model, the influences on the morphologies of diblock copolymer melts due to the chain length and the type of bonds are examined. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (no. 2012R1A1A2043633).
Directory of Open Access Journals (Sweden)
L. O. Fichte
2006-01-01
Full Text Available Boundary Integral Equation formulations can be used to describe electromagnetic shielding problems. Yet, this approach frequently leads to integrals which contain a singularity and an oscillating part. Those integrals are difficult to handle when integrated naivly using standard integration techniques, and in some cases even a very high number of integration nodes will not lead to precise results. We present a method for the numerical quadrature of an integral with a logarithmic singularity and a cosine oscillator: a modified Filon-Lobatto quadrature for the oscillating parts and an integral transformation based on the error function for the singularity. Since this integral can be solved analytically, we are in a position to verify the results of our investigations, with a focus on precision and computation time.
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
Generalized Lotka—Volterra systems connected with simple Lie algebras
Charalambides, Stelios A.; Damianou, Pantelis A.; Evripidou, Charalambos A.
2015-06-01
We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type An for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type An, we produce new integrable Hamiltonian systems.
Galerkin boundary integral equation method for spontaneous rupture propagation problems
Goto, H.; Bielak, J.
2007-12-01
We develop a Galerkin finite element boundary integral equation method (GaBIEM) for spontaneous rupture propagation problems for a planar fault embedded in a homogeneous full 2D space. A simple 2D anti plane rupture propagation problem, with a slip-weakening friction law, is simulated by the GaBIEM. This method allows one to separate explicitly the kernel into singular static and time-dependent parts, and a nonsingular dynamic component. The simulated results throw light into the performance of the GaBIEM and highlight differences with respect to that of the traditional, collocation, boundary integral equation method (BIEM). The rate of convergence of the GaBIEM, as measured from a root mean square (RMS) analysis of the difference of approximate solutions corresponding to increasingly finer element sizes is of a higher order than that of the BIEM. There is no restriction on the CFL stability number since an implicit, unconditionally stable method is used for the time integration. The error of the approximation increases with the time step, as expected, and it can remain below that of the BIEM.
A nonlinear wave equation with a nonlinear integral equation involving the boundary value
Directory of Open Access Journals (Sweden)
Thanh Long Nguyen
2004-09-01
Full Text Available We consider the initial-boundary value problem for the nonlinear wave equation $$displaylines{ u_{tt}-u_{xx}+f(u,u_{t}=0,quad xin Omega =(0,1,; 0
Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion
Dan Li,; Jing’an Cui; Guohua Song
2014-01-01
This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associate...
Cross-diffusional effect in a telegraph reaction diffusion Lotka-Volterra two competitive system
Energy Technology Data Exchange (ETDEWEB)
Abdusalam, H.A E-mail: hosny@operamail.com; Fahmy, E.S
2003-10-01
It is known now that, telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion in several branches of sciences. Telegraph reaction diffusion Lotka-Volterra two competitive system is considered. We observed that this system can give rise to diffusive instability only in the presence of cross-diffusion. Local and global stability analysis in the cross-diffusional effect are studied by considering suitable Lyapunov functional.
Hayes, E. F.; Kouri, D. J.
1971-01-01
Coupled integral equations are derived for the full scattering amplitudes for both reactive and nonreactive channels. The equations do not involve any partial wave expansion and are obtained using channel operators for reactive and nonreactive collisions. These coupled integral equations are similar in nature to equations derived for purely nonreactive collisions of structureless particles. Using numerical quadrature techniques, these equations may be reduced to simultaneous algebraic equations which may then be solved.
Exponential integrators for the incompressible Navier-Stokes equations.
Energy Technology Data Exchange (ETDEWEB)
Newman, Christopher K.
2004-07-01
We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H
Integral equation study of soft-repulsive dimeric fluids
Munaò, Gianmarco; Saija, Franz
2017-03-01
We study fluid structure and water-like anomalies of a system constituted by dimeric particles interacting via a purely repulsive core-softened potential by means of integral equation theories. In our model, dimers interact through a repulsive pair potential of inverse-power form with a softened repulsion strength. By employing the Ornstein–Zernike approach and the reference interaction site model (RISM) theory, we study the behavior of water-like anomalies upon progressively increasing the elongation λ of the dimers from the monomeric case (λ =0 ) to the tangent configuration (λ =1 ). For each value of the elongation we consider two different values of the interaction potential, corresponding to one and two length scales, with the aim to provide a comprehensive description of the possible fluid scenarios of this model. Our theoretical results are systematically compared with already existing or newly generated Monte Carlo data: we find that theories and simulations agree in providing the picture of a fluid exhibiting density and structural anomalies for low values of λ and for both the two values of the interaction potential. Integral equation theories give accurate predictions for pressure and radial distribution functions, whereas the temperatures where anomalies occur are underestimated. Upon increasing the elongation, the RISM theory still predicts the existence of anomalies; the latter are no longer observed in simulations, since their development is likely precluded by the onset of crystallization. We discuss our results in terms of the reliability of integral equation theories in predicting the existence of water-like anomalies in core-softened fluids.
Construction of N-soliton solutions for a new integrable equation by Darboux transformation
Bai, Shuting; Zhaqilao
2016-10-01
In this paper, the relationship between a new integrable equation and well-known KdV equation is established by a set of transformation. With the help of Darboux transformation of KdV equation and the set of transformation, parametric representations of multi-soliton solutions for the new integrable equation are obtained, and their figures are plotted.
Numerical Analysis for Functional Differential and Integral Equations
Institute of Scientific and Technical Information of China (English)
Hermann BRUNNER; Tao TANG; Stefan VANDEWALLE
2009-01-01
@@ From December 3-6,2007,the Department of Mathematics at Hong Kong Baptist University hosted the International Workshop on Numerical Analysis and Computational Methods for Functional Differential and Integral Equations. This workshop,organized by Hermann Brunner of Memorial University of Newfoundland (Canada) & Hong Kong Baptist University,Leevan Ling and Tao Tang of Hong Kong Baptist University,and Chengjian Zhang of Huazhong University of Science and Technology (China) brought together some 40 members of research groups in Hong Kong,Taiwan and the mainland of China,Belgium,Canada,Japan,and Portugal.
Exact controllability of generalized Hammerstein type integral equation and applications
Directory of Open Access Journals (Sweden)
Dimplekumar N. Chalishajar
2006-11-01
Full Text Available In this article, we study the exact controllability of an abstract model described by the controlled generalized Hammerstein type integral equation $$ x(t = int_0^t h(t,su(sds+ int_0^t k(t,s,xf(s,x(sds, quad 0 leq t leq T less than infty, $$ where, the state $x(t$ lies in a Hilbert space $H$ and control $u(t$ lies another Hilbert space $V$ for each time $t in I=[0,T]$, $T$ greater than 0. We establish the controllability result under suitable assumptions on $h, k$ and $f$ using the monotone operator theory.
Integrable Cosmological Models From Higher Dimensional Einstein Equations
Sano, M; Sano, Masakazu; Suzuki, Hisao
2007-01-01
We consider the cosmological models for the higher dimensional spacetime which includes the curvatures of our space as well as the curvatures of the internal space. We find that the condition for the integrability of the cosmological equations is that the total space-time dimensions are D=10 or D=11 which is exactly the conditions for superstrings or M-theory. We obtain analytic solutions with generic initial conditions in the four dimensional Einstein frame and study the accelerating universe when both our space and the internal space have negative curvatures.
The moment-method form of Pocklington's integral equation above ground
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
Pocklington's integral equation is presented for analysis of current distributions on wire antenna above ground. Sommerfeld type integrals, the kernel functions of the integral equation, can be approximately expressed as the elementary functions using the Fresnel plane-wave reflection coefficients method; and the Pocklington's integral equation will be rearranged into a linear equation with solution easily obtained by using the method of moments, when the sinusoidal sub-domain expansion is chosen to express the current distributions.
Explicit solution of Calderon preconditioned time domain integral equations
Ulku, Huseyin Arda
2013-07-01
An explicit marching on-in-time (MOT) scheme for solving Calderon-preconditioned time domain integral equations is proposed. The scheme uses Rao-Wilton-Glisson and Buffa-Christiansen functions to discretize the domain and range of the integral operators and a PE(CE)m type linear multistep to march on in time. Unlike its implicit counterpart, the proposed explicit solver requires the solution of an MOT system with a Gram matrix that is sparse and well-conditioned independent of the time step size. Numerical results demonstrate that the explicit solver maintains its accuracy and stability even when the time step size is chosen as large as that typically used by an implicit solver. © 2013 IEEE.
A note on dual integral equations involving inverse associated Weber-Orr transforms
Directory of Open Access Journals (Sweden)
Nanigopal Mandal
1996-01-01
Full Text Available We consider dual integral equations involving inverse associated Weber-Orr transforms. Elementary methods have been used to reduce dual integral equations to a Fredholm integral equation of second kind. Some known results are obtained as special case.
Analytic treatment of nonlinear evolution equations using ﬁrst integral method
Indian Academy of Sciences (India)
Ahmet Bekir; Ömer Ünsal
2012-07-01
In this paper, we show the applicability of the ﬁrst integral method to combined KdV-mKdV equation, Pochhammer–Chree equation and coupled nonlinear evolution equations. The power of this manageable method is conﬁrmed by applying it for three selected nonlinear evolution equations. This approach can also be applied to other nonlinear differential equations.
Some Exact Results for the Schroedinger Wave Equation with a Time Dependent Potential
Campbell, Joel
2009-01-01
The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.
Bagci, A
2016-01-01
The author in his previous works were presented a numerical integration method, namely, global-adaptive with the Gauss-Kronrod numerical integration extension in order to accurate calculation of molecular auxiliary functions integrals involve power functions with non-integer exponents. They are constitute elements of molecular integrals arising in Dirac equation when Slater-type orbitals with non-integer principal quantum numbers are used. Binomial series representation of power functions method, so far, is used for analytical evaluation of the molecular auxiliary function integrals however, intervals of integration cover areas beyond the condition of convergence. In the present study, analytical evaluation of these integrals is re-examined. They are expressed via prolate spheroidal coordinates. An alternative analytical approximation are derived. Slowly convergent binomial series representation formulae for power functions is investigated through nonlinear sequence transformations for the acceleration of con...
Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations
Directory of Open Access Journals (Sweden)
Han Guo
2012-01-01
Full Text Available Hierarchical (H- matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE- based computational electromagnetics, H-matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solve H-matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure of H-matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving H-matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.
PERTURBATION FINITE VOLUME METHOD FOR CONVECTIVE-DIFFUSION INTEGRAL EQUATION
Institute of Scientific and Technical Information of China (English)
GAO Zhi; YANG Guowei
2004-01-01
A perturbation finite volume (PFV) method for the convective-diffusion integral equation is developed in this paper. The PFV scheme is an upwind and mixed scheme using any higher-order interpolation and second-order integration approximations, with the least nodes similar to the standard three-point schemes, that is, the number of the nodes needed is equal to unity plus the face-number of the control volume. For instance, in the two-dimensional (2-D) case, only four nodes for the triangle grids and five nodes for the Cartesian grids are utilized, respectively. The PFV scheme is applied on a number of 1-D linear and nonlinear problems, 2-D and 3-D flow model equations. Comparing with other standard three-point schemes, the PFV scheme has much smaller numerical diffusion than the first-order upwind scheme (UDS). Its numerical accuracies are also higher than the second-order central scheme (CDS), the power-law scheme (PLS) and QUICK scheme.
Verhulst-Lotka-Volterra (VLV) model of ideological struggles
Ausloos, Marcel R; Dimitrova, Zlatinka I
2011-01-01
Let the population of e.g. a country where some opinion struggle occurs be varying in time, according to Verhulst equation. Consider next some competition between opinions such as the dynamics be described by Lotka and Volterra equations. Two kinds of influences can be used, in such a model, for describing the dynamics of an agent opinion conversion: this can occur (i) either by means of mass communication tools, under some external field influence, or (ii) by means of direct interactions between agents. It results, among other features, that change(s) in environmental conditions can prevent the extinction of populations of followers of some ideology due to different kinds of resurrection effects. The tension arising in the country population is proposed to be measured by an appropriately defined scale index.
Bargmann Symmetry Constraint for a Family of Liouville Integrable Differential-Difference Equations
Institute of Scientific and Technical Information of China (English)
徐西祥
2012-01-01
A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectie map and a completely integrable tinite-dimensionai Hamiltonian system.
On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation
Directory of Open Access Journals (Sweden)
Hameed Husam Hameed
2015-01-01
Full Text Available We develop the Newton-Kantorovich method to solve the system of 2×2 nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.
Auluck, S K H
2010-01-01
Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients of such expansions are integrals over products of Bessel functions in problems involving cylindrical geometry. In this paper, certain identities involving products of two and three general solutions of Bessel's equation have been derived. Some of these identities have been useful in the study of Turner relaxation of annular magnetized plasma [S.K.H. Auluck, Phys. Plasmas, 16, 122504, 2009], where quadratic integral quantities such as helicity and total energy were expressed as algebraic functions of the arbitrary constants of the general solution of Bessel's equation, allowing their determination by a minimization procedure. Identities involving products of three solutions enable expanding a product of two solutions in a Fourier-Bessel series of single Bessel functions fac...
A bin integral method for solving the kinetic collection equation
Wang, Lian-Ping; Xue, Yan; Grabowski, Wojciech W.
2007-09-01
A new numerical method for solving the kinetic collection equation (KCE) is proposed, and its accuracy and convergence are investigated. The method, herein referred to as the bin integral method with Gauss quadrature (BIMGQ), makes use of two binwise moments, namely, the number and mass concentration in each bin. These two degrees of freedom define an extended linear representation of the number density distribution for each bin following Enukashvily (1980). Unlike previous moment-based methods in which the gain and loss integrals are evaluated for a target bin, the concept of source-bin pair interactions is used to transfer bin moments from source bins to target bins. Collection kernels are treated by bilinear interpolations. All binwise interaction integrals are then handled exactly by Gauss quadrature of various orders. In essence the method combines favorable features in previous spectral moment-based and bin-based pair-interaction (or flux) methods to greatly enhance the logic, consistency, and simplicity in the numerical method and its implementation. Quantitative measures are developed to rigorously examine the accuracy and convergence properties of BIMGQ for both the Golovin kernel and hydrodynamic kernels. It is shown that BIMGQ has a superior accuracy for the Golovin kernel and a monotonic convergence behavior for hydrodynamic kernels. Direct comparisons are also made with the method of Berry and Reinhardt (1974), the linear flux method of Bott (1998), and the linear discrete method of Simmel et al. (2002).
Kouri, Donald J; Vijay, Amrendra
2003-04-01
The most robust treatment of the inverse acoustic scattering problem is based on the reversion of the Born-Neumann series solution of the Lippmann-Schwinger equation. An important issue for this approach to inversion is the radius of convergence of the Born-Neumann series for Fredholm integral kernels, and especially for acoustic scattering for which the interaction depends on the square of the frequency. By contrast, it is well known that the Born-Neumann series for the Volterra integral equations in quantum scattering are absolutely convergent, independent of the strength of the coupling characterizing the interaction. The transformation of the Lippmann-Schwinger equation from a Fredholm to a Volterra structure by renormalization has been considered previously for quantum scattering calculations and electromagnetic scattering. In this paper, we employ the renormalization technique to obtain a Volterra equation framework for the inverse acoustic scattering series, proving that this series also converges absolutely in the entire complex plane of coupling constant and frequency values. The present results are for acoustic scattering in one dimension, but the method is general. The approach is illustrated by applications to two simple one-dimensional models for acoustic scattering.
The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1-dimensional hyperbolic nonlinear Schrodinger (HNLS equation, the generalized nonlinear Schrodinger (GNLS equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.
SINGULAR INTEGRAL EQUATIONS ALONG AN OPEN ARC WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER
Institute of Scientific and Technical Information of China (English)
Zhong Shouguo
2005-01-01
In this paper, the difficulties on calculation in solving singular integral equations are overcome when the restriction of curve of integration to be a closed contour is cancelled. When the curve is an open arc and the solutions for singular integral equations possess singularities of higher order, the solution and the solvable condition for characteristic equations as well as the generalized Noether theorem for complete equations are given.
Laamiri, Imen; Khouaja, Anis; Messaoud, Hassani
2015-03-01
In this paper we provide a convergence analysis of the alternating RGLS (Recursive Generalized Least Square) algorithm used for the identification of the reduced complexity Volterra model describing stochastic non-linear systems. The reduced Volterra model used is the 3rd order SVD-PARAFC-Volterra model provided using the Singular Value Decomposition (SVD) and the Parallel Factor (PARAFAC) tensor decomposition of the quadratic and the cubic kernels respectively of the classical Volterra model. The Alternating RGLS (ARGLS) algorithm consists on the execution of the classical RGLS algorithm in alternating way. The ARGLS convergence was proved using the Ordinary Differential Equation (ODE) method. It is noted that the algorithm convergence canno׳t be ensured when the disturbance acting on the system to be identified has specific features. The ARGLS algorithm is tested in simulations on a numerical example by satisfying the determined convergence conditions. To raise the elegies of the proposed algorithm, we proceed to its comparison with the classical Alternating Recursive Least Squares (ARLS) presented in the literature. The comparison has been built on a non-linear satellite channel and a benchmark system CSTR (Continuous Stirred Tank Reactor). Moreover the efficiency of the proposed identification approach is proved on an experimental Communicating Two Tank system (CTTS).
Steen-Ermakov-Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation
Prykarpatskyy, Yarema
2017-01-01
The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide.
Comparison of four stable numerical methods for Abel's integral equation
Murio, Diego A.; Mejia, Carlos E.
1991-01-01
The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abel-type integral equations. If the experimental information is obtained from measured data, on a discrete set of points, special methods are needed in order to restore continuity with respect to the data. A new combined Regularized-Adjoint-Conjugate Gradient algorithm, together with two different implementations of the Mollification Method (one based on a data filtering technique and the other on the mollification of the kernal function) and a regularization by truncation method (initially proposed for 2-D ray sample schemes and more recently extended to 3-D cone-beam image reconstruction) are extensively tested and compared for accuracy and numerical stability as functions of the level of noise in the data.
An integral equation model for warm and hot dense mixtures
Starrett, C E; Daligault, J; Hamel, S
2014-01-01
In Starrett and Saumon [Phys. Rev. E 87, 013104 (2013)] a model for the calculation of electronic and ionic structures of warm and hot dense matter was described and validated. In that model the electronic structure of one "atom" in a plasma is determined using a density functional theory based average-atom (AA) model, and the ionic structure is determined by coupling the AA model to integral equations governing the fluid structure. That model was for plasmas with one nuclear species only. Here we extend it to treat plasmas with many nuclear species, i.e. mixtures, and apply it to a carbon-hydrogen mixture relevant to inertial confinement fusion experiments. Comparison of the predicted electronic and ionic structures with orbital-free and Kohn-Sham molecular dynamics simulations reveals excellent agreement wherever chemical bonding is not significant.
Fluctuations in a ferrofluid monolayer: an integral equation study.
Luo, Liang; Klapp, Sabine H L
2009-07-21
Using integral equation theory in the reference hypernetted chain (RHNC) approximation we investigate the structure and phase behavior of a monolayer of dipolar spheres. The dipole orientations of the particles fluctuate within the plane. The resulting angle dependence of the correlation functions is treated via an expansion in two-dimensional rotational invariants. For homogeneous, isotropic states the RHNC correlation functions turn out to be in good agreement with Monte Carlo simulation data. We then use the RHNC theory combined with a stability (fluctuation) analysis to identify precursors of the low-temperature behavior. As expected, the fluctuations point to pair and cluster formation in the range of low and moderate densities. At high densities, there is no clear indication for a ferroelectric transition, contrary to what is found in three-dimensional dipolar fluids. The stability analysis rather indicates an alignment of chains supplemented by local crystal-like order.
Kleinert, H; Zatloukal, V
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
The first integral method to study the (2+1)-dimensional Jaulent–Miodek equations
Indian Academy of Sciences (India)
M Matinfar; M Eslami; S Roshandel
2015-10-01
In this paper, we have presented the applicability of the first integral method for constructing exact solutions of (2+1)-dimensional Jaulent–Miodek equations. The first integral method is a powerful and effective method for solving nonlinear partial differential equations which can be applied to nonintegrable as well as integrable equations. The present paper confirms the significant features of the method employed and exact kink and soliton solutions are constructed through the established first integrals.
Lechleiter, Armin; Nguyen, Dinh Liem
2010-01-01
Scattering of acoustic waves from an inhomogeneous medium can be described by the Lippmann-Schwinger integral equation. For scattering problems in free space, Vainikko proposed a fast spectral solution method that exploits the convolution structure of this equation's integral operator by using the fast Fourier transform. In a planar 3--dimensional waveguide, the integral operator of the Lippmann-Schwinger integral equation fails to be a convolution. In this paper, we show that the separable s...
Viscous erosion with a generalized traction integral equation
Mitchell, William H
2016-01-01
A double-layer integral equation for the surface tractions on a body moving in a viscous fluid is derived, allowing for the incorporation of a background flow and/or the presence of a plane wall. The Lorentz reciprocal theorem is used to link the surface tractions on the body to integrals involving the background velocity and stress fields on an imaginary bounding sphere (or hemisphere for wall-bounded flows). The derivation requires the velocity and stress fields associated with numerous fundamental singularity solutions which we provide for free-space and wall-bounded domains. Two sample applications of the method are discussed: we study the tractions on an ellipsoid moving near a plane wall, which provides a more detailed understanding of the well-studied glancing and reversing trajectories, and we explore a new problem, erosion of bodies by a viscous flow, in which the surface is ablated at a rate proportional to the local viscous shear stress. Simulations and analytical estimates suggest that a spherical...
Modern integral equation techniques for quantum reactive scattering theory
Energy Technology Data Exchange (ETDEWEB)
Auerbach, Scott Michael [Univ. of California, Berkeley, CA (United States)
1993-11-01
Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D+H_{2} → H_{2}/DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H+H_{2} state resolved integral cross sections σ{sub v'j',vj}(E) for the transitions (v = 0,j = 0) to (v'} = 1,j' = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expansion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence.
Two Kinds of Square-Conservative Integrators for Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
CHEN Jing-Bo; LIU Hong
2008-01-01
@@ Based on the Lie-group and Gauss-Legendre methods, two kinds of square-conservative integrators for squareconservative nonlinear evolution equations are presented. Lie-group based square-conservative integrators are linearly implicit, while Gauss-Legendre based square-conservative integrators are nonlinearly implicit and iterarive schemes are needed to solve the corresponding integrators. These two kinds of integrators provide natural candidates for simulating square-conservative nonlinear evolution equations in the sense that these integrators not only preserve the square-conservative properties of the continuous equations but also are nonlinearly stable.Numerical experiments are performed to test the presented integrators.
Stability and Hopf bifurcation in a symmetric Lotka-Volterra predator-prey system with delays
Directory of Open Access Journals (Sweden)
Jing Xia
2013-01-01
Full Text Available This article concerns a symmetrical Lotka-Volterra predator-prey system with delays. By analyzing the associated characteristic equation of the original system at the positive equilibrium and choosing the delay as the bifurcation parameter, the local stability and Hopf bifurcation of the system are investigated. Using the normal form theory, we also establish the direction and stability of the Hopf bifurcation. Numerical simulations suggest an existence of Hopf bifurcation near a critical value of time delay.
A maximum principle for diffusive Lotka-Volterra systems of two competing species
Chen, Chiun-Chuan; Hung, Li-Chang
2016-10-01
Using an elementary approach, we establish a new maximum principle for the diffusive Lotka-Volterra system of two competing species, which involves pointwise estimates of an elliptic equation consisting of the second derivative of one function, the first derivative of another function, and a quadratic nonlinearity. This maximum principle gives a priori estimates for the total mass of the two species. Moreover, applying it to the system of three competing species leads to a nonexistence theorem of traveling wave solutions.
The First Integral Method to Study a Class of Reaction-Diffusion Equations
Institute of Scientific and Technical Information of China (English)
KE Yun-Quan; YU Jun
2005-01-01
In this letter, a class of reaction-diffusion equations, which arise in chemical reaction or ecology and other fields of physics, are investigated. A more general analytical solution of the equation is obtained by using the first integral method.
The periodic competing Lotka-Volterra model with impulsive effect.
Liu, Bing; Chen, Lansun
2004-06-01
In this paper, the dynamic behaviour of a classical periodic Lotka-Volterra competing system with impulsive effect is investigated. By applying the Floquet theory of linear periodic impulsive equations, some conditions are obtained for the linear stability of the trivial and semi-trivial periodic solutions. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We use standard bifurcation theory to show the existence of nontrivial periodic solutions which arise near the semi-trivial periodic solution. As an application, a fish harvest problem is considered. We explain how two competing species, one of which in a periodic environment without impulsive effect would be doomed to extinction, can coexist with suitably periodic impulsive harvesting.
AN OPERATORIAL APPROACH TO SINGULAR INTEGRAL EQUATIONS OF A MODIFIED TYPE
Institute of Scientific and Technical Information of China (English)
He Fuli; Du Jinyuan
2008-01-01
In this article, the authors discuss a kind of modified singular integral equa-tions on a disjoint union of closed contours or a disjoint union of open arcs. The authors introduce some singular integral operators associated with this kind of singular integral equations, and obtain some useful properties for them. An operatorial approach is also given together with some illustrated examples.
ON DIRECT METHOD OF SOLUTION FOR A CLASS OF SINGULAR INTEGRAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.
Adaptive integral method with fast Gaussian gridding for solving combined field integral equations
Bakır, O.; Baǧ; Cı, H.; Michielssen, E.
Fast Gaussian gridding (FGG), a recently proposed nonuniform fast Fourier transform algorithm, is used to reduce the memory requirements of the adaptive integral method (AIM) for accelerating the method of moments-based solution of combined field integral equations pertinent to the analysis of scattering from three-dimensional perfect electrically conducting surfaces. Numerical results that demonstrate the efficiency and accuracy of the AIM-FGG hybrid in comparison to an AIM-accelerated solver, which uses moment matching to project surface sources onto an auxiliary grid, are presented.
A new integrable discrete generalized nonlinear Schrodinger equation and its reductions
Li, Hongmin; Li, Yuqi; Chen, Yong
2013-01-01
A new integrable discrete system is constructed and studied, based on the algebraization of the difference operator. The model is named the discrete generalized nonlinear Schrodinger (GNLS) equation for which can be reduced to classical discrete nonlinear Schrodinger (NLS) equation. To show the complete integrability of the discrete GNLS equation, the recursion operator, symmetries and conservation quantities are obtained. Furthermore, all of reductions for the discrete GNLS equation are give...
Energy Technology Data Exchange (ETDEWEB)
Stalin, S. [Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620024, Tamil Nadu (India); Senthilvelan, M., E-mail: velan@cnld.bdu.ac.in [Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620024, Tamil Nadu (India)
2011-10-17
In this Letter, we formulate an exterior differential system for the newly discovered cubically nonlinear integrable Camassa-Holm type equation. From the exterior differential system we establish the integrability of this equation. We then study Cartan prolongation structure of this equation. We also discuss the method of identifying conservation laws and Baecklund transformation for this equation from the identified exterior differential system. -- Highlights: → An exterior differential system for a cubic nonlinear integrable equation is given. → The conservation laws from the exterior differential system is derived. → The Baecklund transformation from the Cartan-Ehresmann connection is obtained.
Nasser, Mohamed M. S.; Murid, Ali H. M.; Sangawi, Ali W. K.
2013-01-01
This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a boundary integral equation with the adjoint generalized Neumann kernel for the derivative of the boundary...
Integral equations in the study of polar and ionic interaction site fluids.
Howard, Jesse J; Pettitt, B Montgomery
2011-10-01
In this review article we consider some of the current integral equation approaches and application to model polar liquid mixtures. We consider the use of multidimensional integral equations and in particular progress on the theory and applications of three dimensional integral equations. The IEs we consider may be derived from equilibrium statistical mechanical expressions incorporating a classical Hamiltonian description of the system. We give example including salt solutions, inhomogeneous solutions and systems including proteins and nucleic acids.
Integral Equations in the Study of Polar and Ionic Interaction Site Fluids
Howard, Jesse J.; Pettitt, B. Montgomery
2011-10-01
We consider some of the current integral equation approaches and application to model polar liquid mixtures. We show the use of multidimensional integral equations and in particular progress on the theory and applications of three dimensional integral equations. The IEs we consider may be derived from equilibrium statistical mechanical expressions incorporating a classical Hamiltonian description of the system. We give example including salt solutions, inhomogeneous solutions and systems including proteins and nucleic acids.
Integrable systems of partial differential equations determined by structure equations and Lax pair
Energy Technology Data Exchange (ETDEWEB)
Bracken, Paul, E-mail: bracken@panam.ed [Department of Mathematics, University of Texas, Edinburg, TX 78541-2999 (United States)
2010-01-11
It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.
Surface integrals approach to solution of some free boundary problems
Directory of Open Access Journals (Sweden)
Igor Malyshev
1988-01-01
Full Text Available Inverse problems in which it is required to determine the coefficients of an equation belong to the important class of ill-posed problems. Among these, of increasing significance, are problems with free boundaries. They can be found in a wide range of disciplines including medicine, materials engineering, control theory, etc. We apply the integral equations techniques, typical for parabolic inverse problems, to the solution of a generalized Stefan problem. The regularization of the corresponding system of nonlinear integral Volterra equations, as well as local existence, uniqueness, continuation of its solution, and several numerical experiments are discussed.
On chaos in Lotka-Volterra systems: an analytical approach
Kozlov, Vladimir; Vakulenko, Sergey
2013-08-01
In this paper, we study Lotka-Volterra systems with N species and n resources. We show that the long time dynamics of these systems may be complicated. Depending on parameter choice, they can generate all types of hyperbolic dynamics, in particular, chaotic ones. Moreover, Lotka-Volterra systems can generate Lorenz dynamics. We state the conditions on the strong persistence of Lotka-Volterra systems when the number of resources is less than the number of species.
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
Radhakrishnan, K.
1984-01-01
The efficiency of several algorithms used for numerical integration of stiff ordinary differential equations was compared. The methods examined included two general purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes were applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code available for the integration of combustion kinetic rate equations. It is shown that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient then evaluating the temperature by integrating its time-derivative.
Gesztesy, Fritz; Makarov, Konstantin A.
2003-01-01
We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green's functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants. We...
Integral and integrable algorithms for a nonlinear shallow-water wave equation
Camassa, Roberto; Huang, Jingfang; Lee, Long
2006-08-01
An asymptotic higher-order model of wave dynamics in shallow water is examined in a combined analytical and numerical study, with the aim of establishing robust and efficient numerical solution methods. Based on the Hamiltonian structure of the nonlinear equation, an algorithm corresponding to a completely integrable particle lattice is implemented first. Each "particle" in the particle method travels along a characteristic curve. The resulting system of nonlinear ordinary differential equations can have solutions that blow-up in finite time. We isolate the conditions for global existence and prove l1-norm convergence of the method in the limit of zero spatial step size and infinite particles. The numerical results show that this method captures the essence of the solution without using an overly large number of particles. A fast summation algorithm is introduced to evaluate the integrals of the particle method so that the computational cost is reduced from O( N2) to O( N), where N is the number of particles. The method possesses some analogies with point vortex methods for 2D Euler equations. In particular, near singular solutions exist and singularities are prevented from occurring in finite time by mechanisms akin to those in the evolution of vortex patches. The second method is based on integro-differential formulations of the equation. Two different algorithms are proposed, based on different ways of extracting the time derivative of the dependent variable by an appropriately defined inverse operator. The integro-differential formulations reduce the order of spatial derivatives, thereby relaxing the stability constraint and allowing large time steps in an explicit numerical scheme. In addition to the Cauchy problem on the infinite line, we include results on the study of the nonlinear equation posed in the quarter (space-time) plane. We discuss the minimum number of boundary conditions required for solution uniqueness and illustrate this with numerical
THE PARALLEL RECURSIVE AP ADAPTIVE ALGORITHM BASED ON VOLTERRA SERIES
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
Aiming at the nonlinear system identification problem, a parallel recursive affine projection (AP) adaptive algorithm for the nonlinear system based on Volterra series is presented in this paper. The algorithm identifies in parallel the Volterra kernel of each order, recursively estimate the inverse of the autocorrelation matrix for the Volterra input of each order, and remarkably improve the convergence speed of the identification process compared with the NLMS and conventional AP adaptive algorithm based on Volterra series. Simulation results indicate that the proposed method in this paper is efficient.
Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.
Fokas, A S
2006-05-19
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
ON ASYMTOTIC APPROXIMATIONS OF FIRST INTEGRALS FOR DIFFERENTIAL AND DIFFERENCE EQUATIONS
Directory of Open Access Journals (Sweden)
W.T. van Horssen
2007-04-01
Full Text Available In this paper the concept of integrating factors for differential equations and the concept of invariance factors for difference equations to obtain first integrals or invariants will be presented. It will be shown that all integrating factors have to satisfya system of partial differential equations, and that all invariance factors have to satisfy a functional equation. In the period 1997-2001 a perturbation method based on integrating vectors was developed to approximate first integrals for systems of ordinary differential equations. This perturbation method will be reviewed shortly. Also in the paper the first results in the development of a perturbation method for difference equations based on invariance factors will be presented.
Evolutionary stability in Lotka-Volterra systems.
Cressman, Ross; Garay, József
2003-05-21
The Lotka-Volterra model of population ecology, which assumes all individuals in each species behave identically, is combined with the behavioral evolution model of evolutionary game theory. In the resultant monomorphic situation, conditions for the stability of the resident Lotka-Volterra system, when perturbed by a mutant phenotype in each species, are analysed. We develop an evolutionary ecology stability concept, called a monomorphic evolutionarily stable ecological equilibrium, which contains as a special case the original definition by Maynard Smith of an evolutionarily stable strategy for a single species. Heuristically, the concept asserts that the resident ecological system must be stable as well as the phenotypic evolution on the "stationary density surface". The conditions are also shown to be central to analyse stability issues in the polymorphic model that allows arbitrarily many phenotypes in each species, especially when the number of species is small. The mathematical techniques are from the theory of dynamical systems, including linearization, centre manifolds and Molchanov's Theorem.
Exactly conservation integrators
Energy Technology Data Exchange (ETDEWEB)
Shadwick, B.A.; Bowman, J.C.; Morrison, P.J. [Univ. of Texas, Austin, TX (United States)
1999-03-01
Traditional explicit numerical discretizations of conservative systems generically predict artificial secular drifts of any nonlinear invariants. In this work the authors present a general approach for developing explicit nontraditional algorithms that conserve such invariants exactly. They illustrate the method by applying it to the three-wave truncation of the Euler equations, the Lotka-Volterra predator-prey model, and the Kepler problem. The ideas are discussed in the context of symplectic (phase-space-conserving) integration methods as well as nonsymplectic conservative methods. They comment on the application of the method to general conservative systems.
Exactly conservative integrators
Energy Technology Data Exchange (ETDEWEB)
Shadwick, B.A.; Bowman, J.C.; Morrison, P.J.
1995-07-19
Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves invariants. We illustrate the general method by applying it to the Three-Wave truncation of the Euler equations, the Volterra-Lotka predator-prey model, and the Kepler problem. We discuss our method in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.
Exactly conservative integrators
Shadwick, B A; Morrison, P J; Bowman, John C
1995-01-01
Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants. We illustrate the general method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator--prey model, and the Kepler problem. This method is discussed in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.
Exactly conservative integrators
Energy Technology Data Exchange (ETDEWEB)
Shadwick, B.A.; Bowman, J.C.; Morrison, P.J.
1995-07-19
Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves invariants. We illustrate the general method by applying it to the Three-Wave truncation of the Euler equations, the Volterra-Lotka predator-prey model, and the Kepler problem. We discuss our method in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.
Integration of equations of parabolic type by the method of nets
Saul'Yev, V K; Stark, M; Ulam, S
1964-01-01
International Series of Monographs in Pure and Applied Mathematics, Volume 54: Integration of Equations of Parabolic Type by the Method of Nets deals with solving parabolic partial differential equations using the method of nets. The first part of this volume focuses on the construction of net equations, with emphasis on the stability and accuracy of the approximating net equations. The method of nets or method of finite differences (used to define the corresponding numerical method in ordinary differential equations) is one of many different approximate methods of integration of partial diff
Potential method of integration for solving the equations of mechanical systems
Institute of Scientific and Technical Information of China (English)
Wu Hui-Bin
2006-01-01
This paper is intended to apply a potential method of integration to solving the equations of holonomic and nonholonomic systems. For a holonomic system, the differential equations of motion can be written as a system of differential equations of first order and its fundamental partial differential equation is solved by using the potential method of integration. For a nonholonomic system, the equations of the corresponding holonomic system are solved by using the method and then the restriction of the nonholonomic constraints on the initial conditions of motion is added.
Time integration of the shallow water equations in spherical geometry
D. Lanser; J.G. Blom (Joke); J.G. Verwer (Jan)
2000-01-01
textabstractThe shallow water equations in spherical geometry provide a prototype for developing and testing numerical algorithms for atmospheric circulation models. In a previous paper we have studied a spatial discretization of these equations based on an Osher-type finite-volume method on stereog
Time integration of the shallow water equations in spherical geometry
Lanser, D.; Blom, J.G.; Verwer, J.G.
2000-01-01
The shallow water equations in spherical geometry provide a prototype for developing and testing numerical algorithms for atmospheric circulation models. In a previous paper we have studied a spatial discretization of these equations based on an Osher-type finite-volume method on stereographic and l
On adjoint symmetry equations, integrating factors and solutions of nonlinear ODEs
Energy Technology Data Exchange (ETDEWEB)
Guha, Partha [Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig (Germany); Choudhury, A Ghose [Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Calcutta-700 009 (India); Khanra, Barun [Sailendra Sircar Vidyalaya, 62A Shyampukur Street, Calcutta-700 004 (India)], E-mail: partha.guha@mis.mpg.de, E-mail: a_ghosechoudhury@rediffmail.com, E-mail: barunkhanra@rediffmail.com
2009-03-20
We consider the role of the adjoint equation in determining explicit integrating factors and first integrals of nonlinear ODEs. In Chandrasekar et al (2006 J. Math. Phys. 47 023508), the authors have used an extended version of the Prelle-Singer method for a class of nonlinear ODEs of the oscillator type. In particular, we show that their method actually involves finding a solution of the adjoint symmetry equation. Next, we consider a coupled second-order nonlinear ODE system and derive the corresponding coupled adjoint equations. We illustrate how the coupled adjoint equations can be solved to arrive at a first integral.
Complex Modified Korteweg--DeVries equation, a non-integrable evolution equation
Energy Technology Data Exchange (ETDEWEB)
Karney, C.F.F.; Sen, A.; Chu, F.Y.F.
1978-06-01
The two-dimensional steady-state propagation of electrostatic waves is governed by delta v/delta tau + delta/sup 3/v/delta xi/sup 3/ + delta((absolute value of v)/sup 2/v)/delta xi = 0, the Complex Modified Korteweg-DeVries equation. The properties of this equation are studied.
Toeplitz matrix and product Nystrom methods for solving the singular integral equation
Directory of Open Access Journals (Sweden)
M. A. Abdou
2002-05-01
Full Text Available The Toeplitz matrix and the product Nystrom methods are applied to an integral equation of the second kind. We consider two cases: logarithmic kernel and Hilbert kernel. The two methods are applied to two integral equations with known exact solutions. The error in each case is calculated.
First integrals for time-dependent higher-order Riccati equations by nonholonomic transformation
Guha, Partha; Ghose Choudhury, A.; Khanra, Barun
2011-08-01
We exploit the notion of nonholonomic transformations to deduce a time-dependent first integral for a (generalized) second-order nonautonomous Riccati differential equation. It is further shown that the method can also be used to compute the first integrals of a particular class of third-order time-dependent ordinary differential equations and is therefore quite robust.
Lie systems and integrability conditions of differential equations and some of its applications
Cariñena, J F
2009-01-01
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the literature will be analysed from this new perspective, and some applications in physics will be given.
Lie Algebraic Structures and Integrability of Long-Short Wave Equation in (2+1) Dimensions
Institute of Scientific and Technical Information of China (English)
ZHAO Xue-Qing; L(U)Jing-Fa
2004-01-01
The hidden symmetry and integrability of the long-short wave equation in (2+1) dimensions are considered using the prolongation approach. The internal algebraic structures and their linear spectra are derived in detail which show that the equation is integrable.
Integrability of Nonlinear Equations of Motion on Two-Dimensional World Sheet Space-Time
Institute of Scientific and Technical Information of China (English)
YAN Jun
2005-01-01
The integrability character of nonlinear equations of motion of two-dimensional gravity with dynamical torsion and bosonic string coupling is studied in this paper. The space-like and time-like first integrals of equations of motion are also found.
TIME-DOMAIN VOLUME INTEGRAL EQUATION FOR TRANSIENT SCATTERING FROM INHOMOGENEOUS OBJECTS-2D TM CASE
Institute of Scientific and Technical Information of China (English)
Wang Jianguo; Fan Ruyu
2001-01-01
This letter proposes a time-domain volume integral equation based method for analyzing the transient scattering from a 2D inhomogeneous cylinder by involking the volume equivalence principle for the transverse magnetic case. This integral equation is solved by using an MOT scheme. Numerical results obtained using this method agree very well with those obtained using the FDTD method.
TIME-DOMAIN VOLUME INTEGRAL EQUATION FOR TRANSIENT SCATTERING FROM INHOMOGENEOUS OBJECTS-2D TE CASE
Institute of Scientific and Technical Information of China (English)
Wang Jianguo; Fan Ruyu
2001-01-01
This letter proposes a time-domain volume integral equation based method for analyzing the transient scattering from a 2D inhomogeneous cylinder by involking the volume equivalence principle for the transverse electric case. This integral equation is solved by using an MOT scheme. Numerical results obtained using this method agree very well with those obtained using the FDTD method.
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2014-01-01
Full Text Available The Fourier law of one-dimensional heat conduction equation in fractal media is investigated in this paper. An approximate solution to one-dimensional local fractional Volterra integral equation of the second kind, which is derived from the transformation of Fourier flux equation in discontinuous media, is considered. The Picard successive approximation method is applied to solve the temperature field based on the given Mittag-Leffler-type Fourier flux distribution in fractal media. The nondifferential approximate solutions are given to show the efficiency of the present method.
Dynamics of a Lotka-Volterra type model with applications to marine phage population dynamics
Energy Technology Data Exchange (ETDEWEB)
Gavin, C [School of Mathematical Sciences University College Cork, Cork (Ireland); Pokrovskii, A [School of Mathematical Sciences University College Cork, Cork (Ireland); Prentice, M [Department of Microbiology University College Cork, Cork (Ireland); Sobolev, V [Department of Differential Equations and Control Theory Samara State University, Akademika Pavlova Street, 1, 443011 (Russian Federation)
2006-12-01
The famous Lotka-Volterra equations play a fundamental role in the mathematical modeling of various ecological and chemical systems. A new modification of these equations has been recently suggested to model the structure of marine phage populations, which are the most abundant biological entities in the biosphere. The purpose of the paper is: (i) to make some methodical remarks concerning this modification; (ii) to discuss new types of canards which arise naturally in this context; (iii) to present results of some numerical experiments.
Shiryaeva, E V
2014-01-01
In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of the hodograph method based on the conservation laws for two hyperbolic quasilinear equations of the first order is described. Using these results we propose a method which allows to reduce the Cauchy problem for the two quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is actually some similar method of characteristics for a system of two hyperbolic quasilinear equations. The method can be used effectively in all cases, when the linear hyperbolic equation in partial derivatives of the second order with variable coefficients, resulting from the application of the hodograph method, has an explicit expression for the Riemann-Green function. One of the method's features is the possibility to construct a multi-valued solutions. In this paper we present examples of method application for solving the classical shallow water equations.
Dimitrova, Zlatinka I
2013-01-01
We present a brief overview of integrability of nonlinear ordinary and partial differential equations with a focus on the Painleve property: an ODE of second order has the Painleve property if the only movable singularities connected to this equation are single poles. The importance of this property can be seen from the Ablowitz-Ramani-Segur conhecture that states that a nonlinear PDE is solvable by inverse scattering transformation only if each nonlinear ODE obtained by exact reduction of this PDE has the Painleve property. The Painleve property motivated motivated much research on obtaining exact solutions on nonlinear PDEs and leaded in particular to the method of simplest equation. A version of this method called modified method of simplest equation is discussed below.
Integration Processes of Delay Differential Equation Based on Modified Laguerre Functions
Directory of Open Access Journals (Sweden)
Yeguo Sun
2012-01-01
Full Text Available We propose long-time convergent numerical integration processes for delay differential equations. We first construct an integration process based on modified Laguerre functions. Then we establish its global convergence in certain weighted Sobolev space. The proposed numerical integration processes can also be used for systems of delay differential equations. We also developed a technique for refinement of modified Laguerre-Radau interpolations. Lastly, numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.
Dimitriu, G.; Satco, B.
2016-10-01
Motivated by the fact that bounded variation (often discontinuous) functions frequently appear when studying integral equations that describe physical phenomena, we focus on the existence of bounded variation solutions for Urysohn integral measure driven equations. Due to numerous applications of Urysohn integral equations in various domains, problems of this kind have been extensively studied in literature, under more restrictive assumptions. Our approach concerns the framework of Kurzweil-Stieltjes integration, which allows the occurrence of high oscillatory features on the right hand side of the equation. A discussion about interesting consequences of our main result (given by particular cases of the measure driving the equation) is presented. Finally, we show the generality of our results by investigating two examples of impulsive type problems (from both theoretical and numerical perspective) and giving an application in electronics industry concerning polarization properties of ferroelectric materials.
Hou, Gene; Koganti, Gopichand
1993-01-01
Controls-structure integrated design is a complicated multidisciplinary design optimization problem which involves the state equations pertaining to open-loop eigenvalues and control laws. In order to alleviate the intensity of the computation, this study uses the adjoint variable method to derive sensitivity equations for the eigenvalue, Liapunov, and Riccati equations. These individual sensitivity equations are then combined together to form the multidisciplinary sensitivity equations for the control structure integrated design problems. A set of linear sensitivity equations, proportional in number to the number of performance functions involved in the optimization process, are solved. This proposed approach may provide a great saving in computer resources. The validity of the newly developed sensitivity equations is verified by numerical examples.
Directory of Open Access Journals (Sweden)
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
Existence of solutions to nonlinear Hammerstein integral equations and applications
Li, Fuyi; Li, Yuhua; Liang, Zhanping
2006-11-01
In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L2(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.
Integrable achiral D5-brane reflections and asymptotic Bethe equations
Correa, Diego H; Young, Charles A S
2011-01-01
We study the reflection of magnons from a D5-brane in the framework of the AdS/CFT correspondence. We consider two possible orientations of the D5-brane with respect to the reference vacuum state, namely vacuum states aligned along "vertical" and "horizontal" directions. We show that the reflections are of the achiral type. We also show that the reflection matrices satisfy the boundary Yang-Baxter equations for both orientations. In the horizontal case the reflection matrix can be interpreted in terms of a bulk S-matrix, S(p, -p), and factorizability of boundary scattering therefore follows from that of bulk scattering. Finally, we solve the nested coordinate Bethe ansatz for the system in the vertical case to find the Bethe equations. In the horizontal case, the Bethe equations are of the same form as those for the closed string.
An Integral Equation Method for Electromagnetic Scattering by a Periodic Chiral Structure
Institute of Scientific and Technical Information of China (English)
张德悦; 马富明
2005-01-01
In this paper, we consider the electromagnetic scattering by a periodic chiral structure. The media is homogeneous and the structure is periodic in one direction and invariant in another direction. The electromagnetic fields inside the chiral medium are governed by Maxwell equations together with the Drude-BornFedorov equations. We simplify the problem to a two-dimensional scattering problem and discuss the existence and the uniqueness of solutions by an integral equation approach. We show that for all but possibly a discrete set of wave numbers, the integral equation has a unique solution.
Integrability and structural stability of solutions to the Ginzburg-Landau equation
Keefe, Laurence R.
1986-01-01
The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).
Directory of Open Access Journals (Sweden)
Lijun Zhang
2014-01-01
Full Text Available An integral-differential model equation arising from neuronal networks with very general kernel functions is considered in this paper. The kernel functions we study here include pure excitations, lateral inhibition, lateral excitations, and more general synaptic couplings (e.g., oscillating kernel functions. The main goal of this paper is to prove the existence and uniqueness of the traveling wave front solutions. The main idea we apply here is to reduce the nonlinear integral-differential equation into a solvable differential equation and test whether the solution we get is really a wave front solution of the model equation.
The determination of an unknown boundary condition in a fractional diffusion equation
Rundell, William
2013-07-01
In this article we consider an inverse boundary problem, in which the unknown boundary function ∂u/∂v = f(u) is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. © 2013 Copyright Taylor and Francis Group, LLC.
Permanence of Stochastic Lotka-Volterra Systems
Liu, Meng; Fan, Meng
2017-04-01
This paper proposes a new definition of permanence for stochastic population models, which overcomes some limitations and deficiency of the existing ones. Then, we explore the permanence of two-dimensional stochastic Lotka-Volterra systems in a general setting, which models several different interactions between two species such as cooperation, competition, and predation. Sharp sufficient criteria are established with the help of the Lyapunov direct method and some new techniques. This study reveals that the stochastic noises play an essential role in the permanence and characterize the systems being permanent or not.
New integrable lattice hierarchies
Energy Technology Data Exchange (ETDEWEB)
Pickering, Andrew [Area de Matematica Aplicada, ESCET, Universidad Rey Juan Carlos, c/ Tulipan s/n, 28933 Mostoles, Madrid (Spain); Zhu Zuonong [Departamento de Matematicas, Universidad de Salamanca, Plaza de la Merced 1, 37008 Salamanca (Spain) and Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030 (China)]. E-mail: znzhu2@yahoo.com.cn
2006-01-23
In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula.
Seyrich, Jonathan
2013-01-01
In this work, we present the hitherto most efficient and accurate method for the numerical integration of post-Newtonian equations of motion. We first transform the Poisson system as given by the post-Newtonian approximation to canonically symplectic form. Then we apply Gauss Runge-Kutta schemes to numerically integrate the resulting equations. This yields a convenient method for the structure preserving long-time integration of post-Newtonian equations of motion. In extensive numerical experiments, this approach turns out to be faster and more accurate i) than previously proposed structure preserving splitting schemes and ii) than standard explicit Runge-Kutta methods.
Directory of Open Access Journals (Sweden)
Lakshmi Narayan Mishra
2016-04-01
Full Text Available In the present manuscript, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contains various integral and functional equations that considered in nonlinear analysis and its applications. By utilizing the techniques of noncompactness measures, we operate the fixed point theorems such as Darbo's theorem in Banach algebra concerning the estimate on the solutions. The results obtained in this paper extend and improve essentially some known results in the recent literature. We also provide an example of nonlinear functional-integral equation to show the ability of our main result.
Energy Technology Data Exchange (ETDEWEB)
Stoenescu, M.L.
1977-06-01
The terms in Boltzmann kinetic equation corresponding to elastic short range collisions, inelastic excitational collisions, coulomb interactions and electric field acceleration are evaluated numerically for a standard distribution function minimizing the computational volume by expressing the terms as linear combinations with recalculable coefficients, of the distribution function and its derivatives. The present forms are suitable for spatial distribution calculations.
Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition
Directory of Open Access Journals (Sweden)
Malinowski Marek T.
2015-01-01
Full Text Available We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors. The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.
Lotka-Volterra system in a random environment
Dimentberg, Mikhail F.
2002-03-01
Classical Lotka-Volterra (LV) model for oscillatory behavior of population sizes of two interacting species (predator-prey or parasite-host pairs) is conservative. This may imply unrealistically high sensitivity of the system's behavior to environmental variations. Thus, a generalized LV model is considered with the equation for preys' reproduction containing the following additional terms: quadratic ``damping'' term that accounts for interspecies competition, and term with white-noise random variations of the preys' reproduction factor that simulates the environmental variations. An exact solution is obtained for the corresponding Fokker-Planck-Kolmogorov equation for stationary probability densities (PDF's) of the population sizes. It shows that both population sizes are independent γ-distributed stationary random processes. Increasing level of the environmental variations does not lead to extinction of the populations. However it may lead to an intermittent behavior, whereby one or both population sizes experience very rare and violent short pulses or outbreaks while remaining on a very low level most of the time. This intermittency is described analytically by direct use of the solutions for the PDF's as well as by applying theory of excursions of random functions and by predicting PDF of peaks in the predators' population size.
Lotka-Volterra system in a random environment.
Dimentberg, Mikhail F
2002-03-01
Classical Lotka-Volterra (LV) model for oscillatory behavior of population sizes of two interacting species (predator-prey or parasite-host pairs) is conservative. This may imply unrealistically high sensitivity of the system's behavior to environmental variations. Thus, a generalized LV model is considered with the equation for preys' reproduction containing the following additional terms: quadratic "damping" term that accounts for interspecies competition, and term with white-noise random variations of the preys' reproduction factor that simulates the environmental variations. An exact solution is obtained for the corresponding Fokker-Planck-Kolmogorov equation for stationary probability densities (PDF's) of the population sizes. It shows that both population sizes are independent gamma-distributed stationary random processes. Increasing level of the environmental variations does not lead to extinction of the populations. However it may lead to an intermittent behavior, whereby one or both population sizes experience very rare and violent short pulses or outbreaks while remaining on a very low level most of the time. This intermittency is described analytically by direct use of the solutions for the PDF's as well as by applying theory of excursions of random functions and by predicting PDF of peaks in the predators' population size.
Integrable quartic potentials and coupled KdV equations
Baker, S; Fordy, A P; Enolskii, V Z; Fordy, A P
1995-01-01
We show a surprising connection between known integrable Hamiltonian systems with quartic potential and the stationary flows of some coupled KdV systems related to fourth order Lax operators. In particular, we present a connection between the Hirota-Satsuma coupled KdV system and (a generalisation of) the 1:6:1 integrable case quartic potential. A generalisation of the 1:6:8 case is similarly related to a different (but gauge related) fourth order Lax operator. We exploit this connection to derive a Lax representation for each of these integrable systems. In this context a canonical transformation is derived through a gauge transformation.
AN INTEGRAL EQUATION DESCRIBING RIDING WAVES IN SHALLOW WATER OF FINITE DEPTH
Institute of Scientific and Technical Information of China (English)
An Shu-ping; Le Jia-chun; Dai Shi-qiang
2003-01-01
An integral equation describing riding waves, i.e., small-scale perturbation waves superposed on unperturbed surface waves, in shallow water of finite depth was studied via explicit Hamiltonian formulation, and the water was regarded as ideal incompressible fluid of uniform density. The kinetic energy, density of the perturbed fluid motion was formulated with Hamiltonian canonical variables[1], elevation of the free surface and the velocity potential at the free surface. Then the variables were expanded to the first order at the free surface of unperturbed waves. An integal equation for velocity potential of perturbed waves on the unperturbed free surface was derived by conformal mapping and the Fourier transformation. The integral equation could replace the Hamiltonian canonical equations which are difficult to solve. An explicit expression of Lagrangian density function could be obtained by solving the integral equation. The method used in this paper provides a new path to study the Hamiltonian formulation of riding waves and wave interaction problems.
Deformations of surfaces associated with integrable Gauss-Mainardi-Codazzi equations
Ceyhan, Ö.; Fokas, A. S.; Gürses, M.
2000-04-01
Using the formulation of the immersion of a two-dimensional surface into the three-dimensional Euclidean space proposed recently, a mapping from each symmetry of integrable equations to surfaces in R3 can be established. We show that among these surfaces the sphere plays a unique role. Indeed, under the rigid SU(2) rotations all integrable equations are mapped to a sphere. Furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine-Gordon equation are homeomorphic to a sphere. We also find some new Weingarten surfaces arising from the deformations of the modified Kurteweg-de Vries and of the nonlinear Schrödinger equations. Surfaces can also be associated with the motion of curves. We study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity.
Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations
Eden, Burkhard; Smirnov, Vladimir A.
2016-10-01
We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.
Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations
Eden, Burkhard
2016-01-01
We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.
Neglected transport equations: extended Rankine-Hugoniot conditions and J -integrals for fracture
Davey, K.; Darvizeh, R.
2016-09-01
Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine-Hugoniot conditions for fracture are established along with extended forms of J -integrals.
Fokas, A. S.; De Lillo, S.
2014-03-01
So-called inverse scattering provides a powerful method for analyzing the initial value problem for a large class of nonlinear evolution partial differential equations which are called integrable. In the late 1990s, the first author, motivated by inverse scattering, introduced a new method for analyzing boundary value problems. This method provides a unified treatment for linear, linearizable and integrable nonlinear partial differential equations. Here, this method, which is often referred to as the unified transform, is illustrated for the following concrete cases: the heat equation on the half-line; the nonlinear Schrödinger equation on the half-line; Burger's equation on the half-line; and Burger's equation on a moving boundary.
Institute of Scientific and Technical Information of China (English)
J. CABALLERO; B. L(ó)PEZ; K. SADARANGANI
2007-01-01
We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0,1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x(t) = a(t) + (Tx)(t)∫σ(t)o u(t,s,x(s),x(λs))ds 0λ1 which can be considered in connection with the following Cauchy problem x'(t) = u(t,s,x(t),x(λt)), t∈[0,1], 0 λ 1 x(0) = uo.
Imbert, Cyril
2009-01-01
The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an interface (Imbert, 2008) on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic i...
Symmetries, Integrals and Solutions of Ordinary Differential Equations of Maximal Symmetry
Indian Academy of Sciences (India)
P G L Leach; R R Warne; N Caister; V Naicker; N Euler
2010-02-01
Second-and third-order scalar ordinary differential equations of maximal symmetry in the traditional sense of point, respectively contact, symmetry are examined for the mappings they produce in solutions and fundamental first integrals. The properties of the `exceptional symmetries’, i.e. those not considered to be generic to scalar equations of maximal symmetry, can be recast into a form which is applicable to all such equations of maximal symmetry. Some properties of these symmetries are demonstrated.
LINEAR STIELTJES EQUATION WITH GENERALIZED RIEMANN INTEGRAL AND EXISTENCE OF REGULATED SOLUTIONS
Institute of Scientific and Technical Information of China (English)
L. BARBANTI
2001-01-01
In this work we establish an existence theorem of regulated solutions for a class of Stieltjes equations which involve generalized Riemann kind of integrals. The general method applied consists in considering the continuous-time Stieltjes equation as limit of discrete processes. This approach will prove fruitful in the study of the controllability of Stieltjes systems, because it will be possible to get properties on the continuous time equation by transferring properties of the discrete ones.
Relativistic Dirac equation for particles with arbitrary half-integral spin
Guseinov, I I
2008-01-01
The sets of 2(2s+1)-component matrices through the four-component Dirac matrices are suggested, where s=3/2, 5/2,.... Using these matrices sets the Dirac relativistic equation for a description of arbitrary half-integral spin particles is constructed. The new Dirac equation of motion leads to an equation of continuity with a positive-definite probability density.
A calderón multiplicative preconditioner for the combined field integral equation
Bagci, Hakan
2009-10-01
A Calderón multiplicative preconditioner (CMP) for the combined field integral equation (CFIE) is developed. Just like with previously proposed Caldern-preconditioned CFIEs, a localization procedure is employed to ensure that the equation is resonance-free. The iterative solution of the linear system of equations obtained via the CMP-based discretization of the CFIE converges rapidly regardless of the discretization density and the frequency of excitation. © 2009 IEEE.
Directory of Open Access Journals (Sweden)
B. C. Dhage
2004-09-01
Full Text Available In this paper a random version of a fixed-point theorem of Schaefer is obtained and it is further applied to a certain nonlinear functional random integral equation for proving the existence result under Caratheodory conditions.
Properties of the Lennard-Jones dimeric fluid in two dimensions: an integral equation study.
Urbic, Tomaz; Dias, Cristiano L
2014-03-07
The thermodynamic and structural properties of the planar soft-sites dumbbell fluid are examined by Monte Carlo simulations and integral equation theory. The dimers are built of two Lennard-Jones segments. Site-site integral equation theory in two dimensions is used to calculate the site-site radial distribution functions for a range of elongations and densities and the results are compared with Monte Carlo simulations. The critical parameters for selected types of dimers were also estimated. We analyze the influence of the bond length on critical point as well as tested correctness of site-site integral equation theory with different closures. The integral equations can be used to predict the phase diagram of dimers whose molecular parameters are known.
Brahim Tellab; Kamel Haouam
2016-01-01
In this paper, we investigate the existence and uniqueness of solutions for second order nonlinear fractional differential equation with integral boundary conditions. Our result is an application of the Banach contraction principle and the Krasnoselskii fixed point theorem.
On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations
ElHassan ElJaoui; Said Melliani
2016-01-01
We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.
On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations
Directory of Open Access Journals (Sweden)
ElHassan ElJaoui
2016-01-01
Full Text Available We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.
MULTIPLE POSITIVE SOLUTIONS TO A SYSTEM OF NONLINEAR HAMMERSTEIN TYPE INTEGRAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Wang Feng; Zhang Fang; Liu Chunhan
2009-01-01
In this paper, we use cone theory and a new method of computation of fixed point index to study a system of nonlinear Hammerstein type integral equations, and the existence of multiple positive solutions to the system is discussed.
Eigenvalue Problem for Nonlinear Fractional Differential Equations with Integral Boundary Conditions
Directory of Open Access Journals (Sweden)
Guotao Wang
2014-01-01
Full Text Available By employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions.
Institute of Scientific and Technical Information of China (English)
Fu Jing-Li; Xu Shu-Shan; Weng Yu-Quan
2008-01-01
A field method for integrating the equations of motion for mechanico-electrical coupling dynamical systems is studied. Two examples in mechanico-electrical engineering are given to illustrate this method.
Radiative transfer in plane-parallel media and Cauchy integral equations III. The finite case
Rutily, Bernard; Chevallier, Loïc
2006-01-01
We come back to the Cauchy integral equations occurring in radiative transfer problems posed in finite, plane-parallel media with light scattering taken as monochromatic and isotropic. Their solution is calculated following the classical scheme where a Cauchy integral equation is reduced to a couple of Fredholm integral equations. It is expressed in terms of two auxiliary functions $\\zeta_+$ and $\\zeta_-$ we introduce in this paper. These functions show remarkable analytical properties in the complex plane. They satisfy a simple algebraic relation which generalizes the factorization relation of semi-infinite media. They are regular in the domain of the Fredholm integral equations they satisfy, and thus can be computed accurately. As an illustration, the X- and Y-functions are calculated in the whole complex plane, together with the extension in this plane of the so-called Sobouti's functions.
Coombes, S.; Venkov, N.A.; Shiau, L.; Bojak, I.; Liley, D.T.; Laing, C.R.
2007-01-01
Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal dela
Energy Technology Data Exchange (ETDEWEB)
Myrzakulov, R.; Mamyrbekova, G.K.; Nugmanova, G.N.; Yesmakhanova, K.R. [Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana 010008 (Kazakhstan); Lakshmanan, M., E-mail: lakshman@cnld.bdu.ac.in [Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024 (India)
2014-06-13
Motion of curves and surfaces in R{sup 3} lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that a more general situation in which the curves evolve in the presence of additional self-consistent vector potentials can lead to interesting generalized spin systems with self-consistent potentials or soliton equations with self-consistent potentials. We obtain the general form of the evolution equations of underlying curves and report specific examples of generalized spin chains and soliton equations. These include principal chiral model and various Myrzakulov spin equations in (1+1) dimensions and their geometrically equivalent generalized nonlinear Schrödinger (NLS) family of equations, including Hirota–Maxwell–Bloch equations, all in the presence of self-consistent potential fields. The associated gauge equivalent Lax pairs are also presented to confirm their integrability. - Highlights: • Geometry of continuum spin chain with self-consistent potentials explored. • Mapping on moving space curves in R{sup 3} in the presence of potential fields carried out. • Equivalent generalized nonlinear Schrödinger (NLS) family of equations identified. • Integrability of identified nonlinear systems proved by deducing appropriate Lax pairs.
Similarity Reduction and Integrability for the Nonlinear Wave Equations from EPM Model
Institute of Scientific and Technical Information of China (English)
YAN ZhenYa
2001-01-01
Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou. As a result, the nonlinear wave equation is not integrable.``
On the stability of numerical integration routines for ordinary differential equations.
Glover, K.; Willems, J. C.
1973-01-01
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
RBSDE's with jumps and the related obstacle problems for integral-partial differential equations
Institute of Scientific and Technical Information of China (English)
FAN; Yulian
2006-01-01
The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.
New solutions for two integrable cases of a generalized fifth-order nonlinear equation
Wazwaz, Abdul-Majid
2015-05-01
Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.
On quadrature formulas for singular integral equations of the first and the second kind
DEFF Research Database (Denmark)
Krenk, Steen
1975-01-01
It is shown that by proper choice of the collocation points singular integral equations of the first and the second kind can be integrated by use of the usual Gauss-Jacobi quadrature formula. Detailed formulas are given for various values of the index.......It is shown that by proper choice of the collocation points singular integral equations of the first and the second kind can be integrated by use of the usual Gauss-Jacobi quadrature formula. Detailed formulas are given for various values of the index....
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including classical Drinfel'd-Sokolov-Wilson system (DSWE, (2 + 1-dimensional Davey-Stewartson system, and generalized Hirota-Satsuma coupled KdV system. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. This method can also be applied to nonintegrable equations as well as integrable ones.
A review of some basic aspects related to integration of airplane’s equations of motion
Directory of Open Access Journals (Sweden)
Dan TURCANU
2017-09-01
Full Text Available Numerical integration of the airplane’s equations of motion has long been considered among the most fundamental calculations in airplane’s analysis. Numerical algorithms have been implemented and experimentally validated. However, the need for superior speed and accuracy is still very topical, as, nowadays, various optimization algorithms rely heavily on data generated from the integration of the equations of motion and having access to larger amounts of data can increase the quality of the optimization. Now, for a number of decades, engineers have relied heavily on commercial codes based on automatically selected integration steps. However, optimally chosen constant integration steps can save time and allows for larger numbers of integrations to be performed. Yet, the basic papers that presented the fundamentals of numerical integration, as applied to airplane’s equations of motion are nowadays not easy to locate. Consequently, this paper presents a review of basic aspects related to the integration of airplane’s equation of motion. The discussion covers fundamentals of longitudinal and lateral-directional motion as well as the implementation of some numerical integration methods. The relation between numerical integration steps, accuracy, computational resource usage, numerical stability and their relation with the parameters describing the dynamic response of the airplane is considered and suggestions are presented for a faster yet accurate numerical integration.
Multivariate Moran process with Lotka-Volterra phenomenology.
Noble, Andrew E; Hastings, Alan; Fagan, William F
2011-11-25
For a population with any given number of types, we construct a new multivariate Moran process with frequency-dependent selection and establish, analytically, a correspondence to equilibrium Lotka-Volterra phenomenology. This correspondence, on the one hand, allows us to infer the phenomenology of our Moran process based on much simpler Lokta-Volterra phenomenology and, on the other, allows us to study Lotka-Volterra dynamics within the finite populations of a Moran process. Applications to community ecology, population genetics, and evolutionary game theory are discussed.
Fuchsia and master integrals for splitting functions from differential equations in QCD
Gituliar, O
2016-01-01
We report on the recent progress in reducing differential equations for Feynman master integrals to canonical form with the help of a method proposed by Roman Lee. For the first time, we present Fuchsia --- our open-source implementation of the Lee algorithm written in Python using mathematical routines of a free computer algebra system SageMath. We demonstrate Fuchsia by reducing differential equations for NLO contributions to splitting functions in QCD, which contain both loops and legs integrals.
Bednarcyk, Brett A.; Aboudi, Jacob; Arnold, Steven M.
2008-02-01
The radial return method is a well-known algorithm for integrating the classical plasticity equations. Mendelson presented an alternative method for integrating these equations in terms of the so-called plastic strain—total strain plasticity relations. In the present communication, it is shown that, although the two methods appear to be unrelated, they are actually equivalent. A table is provided demonstrating the step by step correspondence of the radial return and Mendelson algorithms in the case of isotropic hardening.
EQUIVALENT BOUNDARY INTEGRAL EQUATIONS WITH INDIRECT VARIABLES FOR PLANE ELASTICITY PROBLEMS
Institute of Scientific and Technical Information of China (English)
张耀明; 温卫东; 张作泉; 孙焕纯; 吕和祥
2003-01-01
The exact form of the exterior problem for plane elasticity problems was produced and fully proved by the variational principle. Based on this, the equivalent boundary integral equations (EBIE) with direct variables, which are equivalent to the original boundary value problem, were deduced rigorously. The conventionally prevailing boundary integral equation with direct variables was discussed thoroughly by some examples and it is shown that the previous results are not EBIE.
Directory of Open Access Journals (Sweden)
Mohammad Almousa
2013-01-01
Full Text Available The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.
Directory of Open Access Journals (Sweden)
Sohrab Bazm
2016-02-01
Full Text Available In this study, the Bernoulli polynomials are used to obtain an approximate solution of a class of nonlinear two-dimensional integral equations. To this aim, the operational matrices of integration and the product for Bernoulli polynomials are derived and utilized to reduce the considered problem to a system of nonlinear algebraic equations. Some examples are presented to illustrate the efficiency and accuracy of the method.
Zink, M.
The integral equation method in the form of the electric field integral equation for wire grid models provides the current distribution on the surface of structures under study. Characteristic parameters such as the input impedance and the radiation diagram are obtained in this fashion. These parameters are determined for a dipole in free space, a monopole over a circular ground plane, and a torus antenna. Good results are obtained for the far field and the variables related to it.
Institute of Scientific and Technical Information of China (English)
QIAO Yong-Fen; ZHAO Shu-Hong
2006-01-01
The conservation theorems of the generalized Lagrangian equations for nonconservative mechanical system are studied by using method of integrating factors. Firstly, the differential equations of motion of system are given, and the definition of integrating factors is given. Next, the necessary conditions for the existence of the conserved quantity are studied in detail. Finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the result.
Continuous limits for an integrable coupling system of Toda equation hierarchy
Energy Technology Data Exchange (ETDEWEB)
Li Li [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China); Yu Fajun, E-mail: yfajun@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)
2009-09-21
In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra sl(4). By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit.
High order explicit symplectic integrators for the Discrete Non Linear Schr\\"odinger equation
Boreux, Jehan; Hubaux, Charles
2010-01-01
We propose a family of reliable symplectic integrators adapted to the Discrete Non-Linear Schr\\"odinger equation; based on an idea of Yoshida (H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A, 150, 5,6,7, (1990), pp. 262.) we can construct high order numerical schemes, that result to be explicit methods and thus very fast. The performances of the integrators are discussed, studied as functions of the integration time step and compared with some non symplectic methods.
Mixed, Nonsplit, Extended Stability, Stiff Integration of Reaction Diffusion Equations
Alzahrani, Hasnaa H.
2016-07-26
A tailored integration scheme is developed to treat stiff reaction-diffusion prob- lems. The construction adapts a stiff solver, namely VODE, to treat reaction im- plicitly together with explicit treatment of diffusion. The second-order Runge-Kutta- Chebyshev (RKC) scheme is adjusted to integrate diffusion. Spatial operator is de- scretised by second-order finite differences on a uniform grid. The overall solution is advanced over S fractional stiff integrations, where S corresponds to the number of RKC stages. The behavior of the scheme is analyzed by applying it to three simple problems. The results show that it achieves second-order accuracy, thus, preserving the formal accuracy of the original RKC. The presented development sets the stage for future extensions, particularly, to multidimensional reacting flows with detailed chemistry.
Liouvillian integrability of gravitating static isothermal fluid spheres
Iacono, Roberto; Llibre, Jaume
2014-10-01
We examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, autonomous second order ordinary differential equation (ODE) for m/R (m is the mass inside the radius R) that has been solved analytically only for n = -1 and n = -3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = -5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here, we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent autonomous Lotka-Volterra quadratic polynomial differential system in {R}^2, and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka-Volterra system possesses Liouvillian first integrals for n = -1, -3, -5, which descend from the existence of invariant algebraic curves of degree one, and for n = -6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka-Volterra system, and consequently of the second order ODE for the mass function must be non-Liouvillian. This makes the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric quite unlikely.
Liouvillian integrability of gravitating static isothermal fluid spheres
Energy Technology Data Exchange (ETDEWEB)
Iacono, Roberto, E-mail: roberto.iacono@enea.it [ENEA-C. R. Casaccia, Via Anguillarese 301, 00123 Roma (Italy); Llibre, Jaume, E-mail: jllibre@mat.uab.cat [Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia (Spain)
2014-10-01
We examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, autonomous second order ordinary differential equation (ODE) for m/R (m is the mass inside the radius R) that has been solved analytically only for n = -1 and n = -3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = -5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here, we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent autonomous Lotka–Volterra quadratic polynomial differential system in R² and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka–Volterra system possesses Liouvillian first integrals for n = -1, -3, -5, which descend from the existence of invariant algebraic curves of degree one, and for n = -6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka–Volterra system, and consequently of the second order ODE for the mass function must be non-Liouvillian. This makes the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric quite unlikely.
Integral geometry and inverse problems for hyperbolic equations
Romanov, V G
1974-01-01
There are currently many practical situations in which one wishes to determine the coefficients in an ordinary or partial differential equation from known functionals of its solution. These are often called "inverse problems of mathematical physics" and may be contrasted with problems in which an equation is given and one looks for its solution under initial and boundary conditions. Although inverse problems are often ill-posed in the classical sense, their practical importance is such that they may be considered among the pressing problems of current mathematical re search. A. N. Tihonov showed [82], [83] that there is a broad class of inverse problems for which a particular non-classical definition of well-posed ness is appropriate. This new definition requires that a solution be unique in a class of solutions belonging to a given subset M of a function space. The existence of a solution in this set is assumed a priori for some set of data. The classical requirement of continuous dependence of the solutio...
Beyond Honour Codes: Bringing Students into the Academic Integrity Equation
Richards, Deborah; Saddiqui, Sonia; McGuigan, Nicholas; Homewood, Judi
2016-01-01
Honour codes represent a successful and unique, student-led, "bottom-up" approach to the promotion of academic integrity (AI). With increased flexibility, globalisation and distance or blended education options, most institutions operate in very different climates and cultures from the US institutions that have a long-established culture…
SOLVING INTEGRAL EQUATIONS WITH LOGARITHMIC KERNEL BY USING PERIODIC QUASI-WAVELET
Institute of Scientific and Technical Information of China (English)
Han-lin Chen; Si-long Peng
2000-01-01
In solving integral equations with logarithmic kernel which arises from the boundary integral equation reformulation of some boundary value problems for the two dimensional Helmholtz equation, we combine the Galerkin method with Beylkin's ([2]) approach, series of dense and nonsymmetric matrices may appear if we use traditional method. By appealing the so-called periodic quasi-wavelet (PQW in abbr.) ([5]), some of these matrices become diagonal, therefore we can find a algorithm with only O(K(m)2) arithmetic operations where m is the highest level. The Galerkin approximation has a polynomial rate of convergence.
Finding Linear Dependencies in Integration-By-Parts Equations: A Monte Carlo Approach
Kant, Philipp
2013-01-01
The reduction of a large number of scalar integrals to a small set of master integrals via Laporta's algorithm is common practice in multi-loop calculations. It is also a major bottleneck in terms of running time and memory consumption. It involves solving a large set of linear equations where many of the equations are linearly dependent. We propose a simple algorithm that eliminates all linearly dependent equations from a given system, reducing the time and space requirements of a subsequent run of Laporta's algorithm.
Directory of Open Access Journals (Sweden)
Mohsen Alipour
2014-01-01
Full Text Available We introduce a new combination of Bernstein polynomials (BPs and Block-Pulse functions (BPFs on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM. This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.
Do All Integrable Evolution Equations Have the Painlevé Property?
Directory of Open Access Journals (Sweden)
K.M. Tamizhmani
2007-06-01
Full Text Available We examine whether the Painlevé property is necessary for the integrability of partial differential equations (PDEs. We show that in analogy to what happens in the case of ordinary differential equations (ODEs there exists a class of PDEs, integrable through linearisation, which do not possess the Painlevé property. The same question is addressed in a discrete setting where we show that there exist linearisable lattice equations which do not possess the singularity confinement property (again in analogy to the one-dimensional case.
Application of the Jacobi method and integrating factors to a class of Painleve-Gambier equations
Energy Technology Data Exchange (ETDEWEB)
Yasar, Emrullah [Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa (Turkey); Reis, Murat, E-mail: eyasar@uludag.edu.t, E-mail: reis@uludag.edu.t [Department of Mechanical Engineering, Faculty of Engineering and Architecture, Uludag University, 16059 Bursa (Turkey)
2010-07-23
In this work, we consider the motion of chain ball drawing with constant force in the frictionless surface which is a class of the Painleve-Gambier equations. We apply Jacobi's method which enables us to obtain Lagrangians of any second-order differential equation. It is comprised that the Lagrangian obtained by Musielak's method is the particular case of the many Lagrangians that can be obtained by Jacobi's method. In addition, we obtain integrating factors and first integrals for the equation in question by Ibragimov's variational derivative approach.
On the integrability and quasi-periodic wave solutions of the Boussinesq equation in shallow water
Ma, Pan-Li; Tian, Shou-Fu; Tu, Jian-Min; Xu, Mei-Juan
2015-05-01
In this paper, the complete integrability of the Boussinesq equation in shallow water is systematically investigated. By using generalized Bell's polynomials, its bilinear formalism, bilinear Bäcklund transformations, Lax pairs of the Boussinesq equation are constructed, respectively. By virtue of its Lax equations, we find its infinite conservation laws. All conserved densities and fluxes are obtained by lucid recursion formulas. Furthermore, based on multidimensional Riemann theta functions, we construct periodic wave solutions of the Boussinesq equation. Finally, the relations between the periodic wave solutions and soliton solutions are strictly constructed. The asymptotic behaviors of the periodic waves are also analyzed by a limiting procedure.
RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Reinhard Hochmuth
2002-01-01
This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.
On discrete fractional integral operators and related Diophantine equations
Kim, Jongchon
2011-01-01
In this paper we show that an arithmetic property of a hypersurface in $\\zn{k+1}$ (a map $\\gamma:\\zn{k} \\to \\mathbb{Z}$) gives $l^p \\to l^q$ bounds of a Radon-type discrete fractional integral operator along the hypersurface. As a corollary, we prove $l^p \\to l^q$ bounds of a Radon-type discrete fractional integral operator along paraboloids in $\\zn{3}$ and some other related operators. As a by-product of this approach, we show that the statement $r_{s,k}(N) = O(N^\\epsilon)$ for any $\\epsilon>0$ is false if $s>k$, where $r_{s,k}(N)$ denotes the number of representations of a positive integer $N$ as a sum of s positive $k$-th powers.
Directory of Open Access Journals (Sweden)
Arthemy V. Kiselev
2006-02-01
Full Text Available We construct new integrable coupled systems of N = 1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian structures. A fermionic extension of the Burgers equation is related with the Burgers flows on associative algebras. A Gardner's deformation is found for the bosonic super-field dispersionless Boussinesq equation, and unusual properties of a recursion operator for its Hamiltonian symmetries are described. Also, we construct a three-parametric supersymmetric system that incorporates the Boussinesq equation with dispersion and dissipation but never retracts to it for any values of the parameters.
Bifurcation Analysis of a Lotka-Volterra Mutualistic System with Multiple Delays
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Xin-You Meng
2014-01-01
Full Text Available A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998. Numerical simulations are given to support the theoretical results.
Multistep epsilon-algorithm, Shanks' transformation, and Lotka-Volterra system by Hirota's method
Brezinski, Claude; Hu, Xing-Biao; Redivo-Zaglia, Michela; Sun, Jian-Qing
2010-01-01
In this paper, we give a multistep extension of the epsilon-algorithm of Wynn, and we show that it implements a multistep extension of the Shanks' sequence transformation which is defined by ratios of determinants. Reciprocally, the quantities defined in this transformation can be recursively computed by the multistep epsilon-algorithm. The multistep epsilon-algorithm and the multistep Shanks' transformation are related to an extended discrete Lotka-Volterra system. These results are obtained by using the Hirota's bilinear method, a procedure quite useful in the solution of nonlinear partial differential and difference equations.
On the maximal cut of Feynman integrals and the solution of their differential equations
Primo, Amedeo
2016-01-01
The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in $\\epsilon = (4-d)/2$, where $d$ are the space-time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exist no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.
Energy Technology Data Exchange (ETDEWEB)
Podesta, John J. [Space Science Institute, Boulder, Colorado 80301 (United States)
2012-08-15
The electric field generated by a time varying point charge in a three-dimensional, unbounded, spatially homogeneous plasma with a uniform background magnetic field and a uniform (static) flow velocity is studied in the electrostatic approximation which is often valid in the near field. For plasmas characterized by Maxwell distribution functions with isotropic temperatures, the linearized Vlasov-Poisson equations may be formulated in terms of an equivalent integral equation in the time domain. The kernel of the integral equation has a relatively simple mathematical form consisting of elementary functions such as exponential and trigonometric functions (sines and cosines), and contains no infinite sums of Bessel functions. Consequently, the integral equation is amenable to numerical solutions and may be useful for the study of the impulse response of magnetized plasmas and, more generally, the response to arbitrary waveforms.
Thirty years of studies of integrable reductions of Einstein's field equations
Alekseev, G A
2010-01-01
More than thirty years passed since the first discoveries of various aspects of integrability of the symmetry reduced vacuum Einstein equations and electrovacuum Einstein - Maxwell equations were made and gave rise to constructions of powerful solution generating methods for these equations. In the subsequent papers, the inverse scattering approach and soliton generating techniques, B\\"acklund and symmetry transformations, formulations of auxiliary Riemann-Hilbert or homogeneous Hilbert problems and various linear integral equation methods have been developed in detail and found different interesting applications. Recently many efforts of different authors were aimed at finding of generalizations of these solution generating methods to various (symmetry reduced) gravity, string gravity and supergravity models in four and higher dimensions. However, in some cases it occurred that even after the integrability of a system was evidenced, some difficulties arise which do not allow the authors to develop some effec...
On the maximal cut of Feynman integrals and the solution of their differential equations
Directory of Open Access Journals (Sweden)
Amedeo Primo
2017-03-01
Full Text Available The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in ϵ=(4−d/2, where d are the space–time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.