Showing papers on "Integrating factor published in 2006"
01 Jan 2006
1,096 citations
TL;DR: In this paper, integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order, and the concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation.
183 citations
TL;DR: This paper examines global optimization of an integral objective function subject to nonlinear ordinary differential equations and develops a fully automated implementation of the theory for deriving a convex relaxation for an integral based on the composition result defined by McCormick.
Abstract: This paper examines global optimization of an integral objective function subject to nonlinear ordinary differential equations. Theory is developed for deriving a convex relaxation for an integral by utilizing the composition result defined by McCormick (Mathematical Programming 10, 147---175, 1976) in conjunction with a technique for constructing convex and concave relaxations for the solution of a system of nonquasimonotone ordinary differential equations defined by Singer and Barton (SIAM Journal on Scientific Computing, Submitted). A fully automated implementation of the theory is briefly discussed, and several literature case study problems are examined illustrating the utility of the branch-and-bound algorithm based on these relaxations.
113 citations
15 Nov 2006
TL;DR: In this article, the authors compare the performance of several fourth-order methods for the Korteweg-de Vries and the focusing and defocusing nonlinear Schr¤ odinger equations in the small dispersion limit.
Abstract: Purely dispersive equations, such as the Korteweg-de Vries and the nonlinear Schrequations in the limit of small dispersion, have solutions to Cauchy problems with smooth initial data which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow- up. Fourth order time-stepping in combination with spectral methods is benecial to numerically resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Korteweg-de Vries and the focusing and defocusing nonlinear Schr¤ odinger equations in the small dispersion limit: an exponential time-differencing fourth-order Runge-Kutta method as proposed by Cox and Matthews in the implementation by Kassam and Trefethen, integrating factors, time-splitting, Fornberg and Driscoll's 'sliders', and an ODE solver in Matlab.
107 citations
TL;DR: A novel method for solving ordinary and partial differential equations, based on grammatical evolution, forms generations of trial solutions expressed in an analytical closed form with a controlled level of accuracy.
Abstract: A novel method for solving ordinary and partial differential equations, based on grammatical evolution is presented. The method forms generations of trial solutions expressed in an analytical closed form. Several examples are worked out and in most cases the exact solution is recovered. When the solution cannot be expressed in a closed analytical form then our method produces an approximation with a controlled level of accuracy. We report results on several problems to illustrate the potential of this approach.
93 citations
Book•
03 Oct 2006
TL;DR: Systems of Ordinary Differential Equations Bifurcations in Non-linear Systems of ODEs and Chaotic Systems of ODDEs are discussed in detail in this article.
Abstract: Systems of Ordinary Differential Equations Bifurcations in Nonlinear Systems of Ordinary Differential Equations Chaotic Systems of Ordinary Differential Equations Principles of the Theory of Dynamical Chaos in Dissipative Systems of Ordinary Differential Equations Dynamical Chaos in Infinitely-Dimensional Systems of Differential Equations Chaos Control in Systems of Differential Equations.
71 citations
24 Mar 2006
63 citations
01 Jan 2006
61 citations
Journal Article•
TL;DR: In this article, the equivalence between RFDEs with pre-assigned moments of impulse effects and generalized ordinary differential equations (GODEs) was proved and the existence, uniqueness and continuous dependence of solutions for this class of GODEs were shown.
Abstract: It is known that retarded functional differential equations (RFDEs) can be regarded as
generalized ordinary differential equations (we write GODEs). See [2, 6, 7]. In this
paper, we prove the equivalence between RFDEs with pre-assigned moments of impulse effects
and a certain class of GODEs introduced in [8] using some ideas of [2, 6, 7]. We state
results on the existence, uniqueness and continuous dependence of solutions for this class
of GODEs and we use them to obtain fine results concerning the corresponding impulsive
RFDEs.
53 citations
01 Jan 2006
TL;DR: In this paper, the authors present a general method to solve locally in time the initial value problem for non-linear Schrodinger equations under some natural hypotheses of decay and regularity of the coefficients.
Abstract: I will review some recent work done in collaboration with C. E. Kenig, G. Ponce and
C. Rolvung on a general method to solve locally in time the initial value problem for non-linear
Schrodinger equations under some natural hypotheses of decay and regularity of the coefficients.
Also some non-trapping conditions of the solutions of the hamiltonian flowassociated to the initial
data is needed. We will not assume ellipticity on the matrix of the leading order coefficients
but just a non-degeneracy condition. The method is based on energy estimates which can be
performed thanks to the construction of an integrating factor. This construction is of independent
interest and relies on the analysis of some new pseudo-differential operators.
46 citations
TL;DR: In this paper, the scaling limits of these random walks to a diffusion process in the sense of distributions are proved and the multi-dimensional random walk models governed by distributed fractional order differential equations are constructed.
Abstract: In this paper the multi-dimensional random walk models governed by distributed fractional order differential equations and multi-term fractional order differential equations are constructed. The scaling limits of these random walks to a diffusion process in the sense of distributions is proved.
TL;DR: In this paper, a new algorithm for linearization of a third-order ODE is presented, which consists of composition of two operations: reducing order of an ODE and using the Lie linearization test for the obtained second order ODE.
Abstract: A new algorithm for linearization of a third-order ordinary differential equation is presented. The algorithm consists of composition of two operations: reducing order of an ordinary differential equation and using the Lie linearization test for the obtained second-order ordinary differential equation. The application of the algorithm to several ordinary differential equations is given.
TL;DR: The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly.
Abstract: The implementation of the two-dimensional differential transform method (DTM), Adomian's decomposition method (ADM), and the variational iteration method (VIM) in the mathematical applications of partial differential equations is examined in this paper. The VIM has been found to be particularly valuable as a tool for the solution of differential equations in engineering, science, and applied mathematics. The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly. Numerical solutions of two examples are calculated and the results are presented in tables and figures.
TL;DR: In this paper, a numerical method for estimating the blowup time and the blow-up rate of the solution of an ODE was proposed, when the solution diverges at a finite time.
Proceedings Article•
21 Aug 2006TL;DR: A new method for solving unstable differential ordinary equations is proposed using a computational scheme of Evolutionary Computing (Genetic Algorithms) and the Nelder-Mead method.
Abstract: In this paper, a new method for solving unstable differential ordinary equations is proposed using a computational scheme of Evolutionary Computing (Genetic Algorithms) and the Nelder-Mead method.
TL;DR: This paper introduces and analyzes rigorous conditions under which the later series (solutions) can be reduced to a finite sum of standard functions.
Abstract: It is well known that solutions of many ordinary differential equations allow presentation in the power series form. In the paper we introduce and analyze rigorous conditions under which the later series (solutions) can be reduced to a finite sum of standard functions. Some areas of practical applicability of the proposed algorithm are discussed.
TL;DR: In this article, a generalized linearizing transformation (GLT) for second-order nonlinear ODEs is proposed, which is a sub-case of the GLT for non-point transformations.
Abstract: In this letter, we introduce a new generalized linearizing transformation (GLT) for second-order nonlinear ordinary differential equations (SNODEs). The well-known invertible point (IPT) and non-point transformations (NPT) can be derived as sub-cases of the GLT. A wider class of nonlinear ODEs that cannot be linearized through NPT and IPT can be linearized by this GLT. We also illustrate how to construct GLTs and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this GLT for the SNODEs. We demonstrate the theory with two examples which are of contemporary interest.
TL;DR: In this article, the relation between the existence of an inverse integrating factor and the center problem is studied, and the relationship between the conditions for a center using the Poincare formal series and the inverse integration factor formal series for systems with a linear center perturbed by homogeneous polynomials is proved.
Abstract: In this paper we study some aspects of the nondegenerate center problem for analytic and, in particular, for polynomial vector fields. The relation between the existence of an inverse integrating factor and the center problem is studied. The relationship between the conditions for a center using the Poincare formal series and the inverse integrating factor formal series for systems with a linear center perturbed by homogeneous polynomials is proved.
TL;DR: In this paper, a nonlinear dynamical system with dimensions n and with k constraints is embedded into a mathematically equivalent n+k-dimensional nonlinear system, which including k integrating factors.
Abstract: The system we consider consists of two parts: a purely algebraic system describing the manifold of constraints and a differential part describing the dynamics on this manifold. For the constrained dynamical problem in its engineering application, it is utmost important to developing numerical methods that can preserve the constraints. We embed the nonlinear dynamical system with dimensions n and with k constraints into a mathematically equivalent n+k-dimensional nonlinear system, which including k integrating factors. Each subsystem of the k independent sets constitutes a Lie type system of Ẋi = AiXi with Ai ∈ so(ni,1) and n1 + · · ·+ nk = n. Then, we can apply the exponential mapping technique to integrate the augmented systems and use the k freedoms to adjust the k integrating factors such that the k constraints are satisfied. A similar procedure is also applied to the case when one integrates the k augmented systems by the fourth-order Runge-Kutta method. Since all constraints are included in the newly developed integrating schemes, it is guaranteed that all algebraic equations that describe the manifold are satisfied up to an accuracy that is used to integrate these dynamical equations and hence a drift from the solution manifold can be avoided. Several numerical examples, including differential algebraic equations (DAEs), are investigated to confirm that the new numerical methods are effective to integrate the constrained dynamical systems by preserving the constraints. keyword: Nonlinear dynamical system, Preserving constraints, Integrating factors, Cones, Minkowski space, Group preserving scheme
Posted Content•
TL;DR: Genetic algorithms can be used in order to solve optimization problems and such a technique may be used to solve differential equations.
Abstract: Genetic algorithms can be used in order to solve optimization problems. Such a technique may be used in order to solve differential equations.
TL;DR: In this article, the authors considered rational ODEs with real coefficients and showed that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method.
Abstract: In this work, we consider rational ordinary differential equations dy/dx=Q(x, y)/P(x, y), with
Q(x, y) and P(x, y) coprime polynomials with real coefficients. We give a method to construct equations of this type
for which a first integral can be expressed from two independent solutions of a second–order
homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function.
We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families.
We give an analogous method for constructing rational equations but by means of a linear differential equation of first order
TL;DR: In this article, an existence, uniqueness, and continuous dependence theorem for increasing time is formulated and proved for solutions of an ordinary differential equation the right-hand side of which is the gradient of the sum of a concave and a smooth function.
Abstract: The concept of gradient of smooth functions is generalized for their sums with concave functions. An existence, uniqueness, and continuous dependence theorem for increasing time is formulated and proved for solutions of an ordinary differential equation the right-hand side of which is the gradient of the sum of a concave and a smooth function. With the use of this result a physically natural motion of particles, well defined even at discontinuities of the velocity field, is constructed in the variational problem of the minimal mechanical action in a space of arbitrary dimension. For such a motion of particles in the plane all typical cases of the birth and the interaction of point clusters of positive mass are described.
TL;DR: The concept is to simply build a divergence-free extended ODE whose solution is just a lift-up of the original ODE, and to constitute an efficient integrator that preserves the phase-space volume on the extended system.
Abstract: For an arbitrary ordinary differential equation (ODE), a scheme for constructing an extended ODE endowed with a time-invariant function is here proposed. This scheme enables us to examine the accuracy of the numerical integration of an ODE that may itself have had no invariant. These quantities are constructed by referring to the Nos\'e-Hoover molecular dynamics equation and its related conserved quantity. By applying this procedure to several molecular dynamics equations, the conventional conserved quantity individually defined in each dynamics can be reproduced in a uniform, generalized way; our concept allows a transparent outlook underlying these quantities and ideas. Developing the technique, for a certain class of ODEs we construct a numerical integrator that is not only explicit and symmetric, but preserves a unit Jacobian for a suitably defined extended ODE, which also provides an invariant. Our concept is thus to simply build a divergence-free extended ODE whose solution is just a lift-up of the original ODE, and to constitute an efficient integrator that preserves the phase-space volume on the extended system. We present precise discussions about the general mathematical properties of the integrator and provide specific conditions that should be incorporated for practical applications.
Journal Article•
TL;DR: In this article, the relation between solutions, their 1st, 2nd derivatives, differential polynomial of a class of second order linear differential equations with function of small growth is investigated.
Abstract: This paper investigates the relation between solutions, their 1th, 2th derivatives, differential polynomial of a class of second order linear differential equations with function of small growth.
01 Jan 2006
TL;DR: In this paper, exterior differential systems are used to solve the inverse problem in the calculus of variations for a given set of second-order ordinary differential equations and whether the Lagrangian exists.
Abstract: The inverse problem in the calculus of variations for a given set of second-order ordinary differential equations consists of deciding whether their solutions are those of Euler–Lagrange equations and whether the Lagrangian, if it exists, is unique. This paper discusses the exterior differential systems approach to this problem. In particular, it proposes an algorithmic procedure towards the construction of a certain differential ideal. The emphasis is not so much on obtaining a complete set of integrability conditions for the problem, but rather on producing a minimal set to expedite the differential ideal process.
11 Dec 2006
TL;DR: The thermodynamic formulation of economics is based on the laws of calculus as discussed by the authors, and it is possible to invest little in one way and gain a lot on the way back, and to do this periodically.
Abstract: The thermodynamic formulation of economics is based on the laws of calculus. Differential forms in two dimensions are generally not exact forms (δQ), the integral from (A) to (B) is not always the same as the integral from (B) to (A). It is possible to invest little in one way and gain a lot on the way back, and to do this periodically. This is the mechanism of energy production in heat pumps, of economic production in companies and of growth in economies. Not exact forms may be turned into exact forms (dS) by an integrating factor T, dS = δQ/T. The new function (S) is called entropy and is related to the probability (P) as S = ln P. In economics the function (S) is called production function. The factor (T) is a market index or the standard of living, GNP/capita, of countries. The dynamics of economic growth is based on the Carnot process, which is driven by external resources. Economic growth and capital generation – like heat pumps and electric generators – depend on natural resources like oil. GNP and oil consumption run parallel for all countries. Markets and motors, economic and thermodynamics processes are all based on the same laws of calculus and statistics.
TL;DR: In this article, the authors studied the blow up phenomena for scalar delay differential equations and made connection with the blow-up of ordinary differential equations that are related to the delay differential equation.
Abstract: In this work we study the blow up phenomena for some scalar delay differential equations. In particular, we make connection with the blow up of ordinary differential equations that are related to the delay differential equations.