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1.
DKLAG6 is a FORTRAN 77 code widely used to solve delay differential equations (DDEs). Like all the popular Fortran DDE solvers, new users find it formidable and in many respects, it is not easy to use. We have applied our experience writing DDE solvers in Matlab and the capabilities of Fortran 90 to the development of a friendly Fortran DDE solver. How we accomplished this may be of interest to others who would like to modernize legacy code. In the course of developing a completely new user interface, we have added significantly to the capabilities of DKLAG6.  相似文献   

2.
A Galerkin projection scheme to obtain low dimensional approximations of delay differential equations (DDEs) involving state-dependent delays is developed. The current scheme is an extension of a similar, recently proposed scheme for DDEs with constant delays in the publication by P. Wahi, A. Chatterjee 2005. The resulting ordinary differential equations (ODEs) from the Galerkin scheme are easier to integrate using commercial ODE solvers, and are amenable to stability and bifurcation analysis using standard techniques. First, the application of the formulation is demonstrated through a scalar delay differential equation, and the performance of the formulation is assessed. Next, the scheme is applied to a two degrees-of-freedom model describing the coupled axial and torsional vibrations of oil well drill-strings. In both cases, the Galerkin approximations show an excellent agreements with the direct numerical simulations of the original systems.  相似文献   

3.
In this work we present a new method to compute the delays of delay-differential equations (DDEs), such that the DDE has a purely imaginary eigenvalue. For delay-differential equations with multiple delays, the critical curves or critical surfaces in delay space (that is, the set of delays where the DDE has a purely imaginary eigenvalue) are parameterized. We show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value, i.e., commensurate delays.  相似文献   

4.
Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.  相似文献   

5.
We analyze the asymptotic stability of collocation solutions in spaces of globally continuous piecewise polynomials on uniform meshes for linear delay differential equations with vanishing proportional delay qt (0<q<1) (pantograph DDEs). It is shown that if the collocation points are such that the analogous collocation solution for ODEs is A-stable, then this asymptotic behaviour is inherited by the collocation solution for the pantograph DDE.  相似文献   

6.
This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability of general linear methods for systems of neutral DDEs with multiple delays. A type of interpolation procedure is considered for general linear methods. Linear stability properties of general linear methods with this interpolation procedure are investigated. Many extant results are unified.  相似文献   

7.
We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.  相似文献   

8.
The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.  相似文献   

9.
This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We exploit the equivalence of the linearized collocation system and the discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using a variant of the Newton-Picard method [Int. J. Bifurcation Chaos, 7 (1997), pp. 2547–2560]. This method combines a direct method in the low-dimensional subspace of the weakly stable and unstable modes with an iterative solver in the high-dimensional orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm. AMS subject classification (2000) 65J15, 65P30, 65Q05  相似文献   

10.
The stability of Runge-Kutta methods for systems of delay differentialequations (DDEs) with multiple delays is considered. The stabilityregions of explicit and implicit Runge-Kutta methods are discussedwhen they are applied to asymptotically stable linear DDEs withmultiple delays. A simple estimate on the stability regionsof explicit Runge-Kutta methods is presented. It is shown thatthe stable step-size for numerical integration of DDEs withmultiple delays can be easily selected by means of the estimate.  相似文献   

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