Methods For Functional Differential Equations Research Articles

Cattmul-Rom spline approach and the order of convergence of Green’s functional method for functional differential equations

The purpose of this work is to investigate the convergence properties of Green’s function method applied to boundary value problems for functional differential equations. Recently, involving Picard and Mann iterations, a Green’s function technique was developed (in Int. J. Computer Math. 95, no. 10 (2018) 1937-1949) for third order functional differential equations, but without specifying the order of convergence of the proposed method. In order to improve this aspect, here we establish the maximal order of convergence of Green’s function method applied to two-point boundary value problems associated to second and third order functional differential equations. In this context, by using suitable quadrature rule and appropriate spline interpolation procedure, the Picard iterations are approximated by a sequence of cubic splines on uniform mesh. Some numerical experiments are presented in order to test the theoretical results and to illustrate the accuracy of the method.

Application of numerical method of functional differential equations in fair value of financial accounting

Abstract In order to improve the information quality of financial accounting reports, this paper puts forward a functional differential equation, establishes the determination model of fair value by mathematical analysis method and studies the correlation of financial accounting information of fair value through this model. The results show that according to the fair value determination model, in the free accounting environment, fair value financial accounting information has an impact.

Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting

Abstract Financial accounting, the use of historical cost of assets, is an important basic principle of historical cost, which is to become the dominant mode of accounting measurement. Background analyses, as well as the historical cost basis and fair value, result from the development of the theory of historical cost and fair value. Historical cost and fair value measurement model has its own advantages and problems. Based on this background, the paper applies B-theoretical numerical methods to differential equations pan function analysis for calculation of fair value accounting and conducts theoretical analysis of their stability and convergence. Finally, numerical examples with different methods of calculating an approximate solution are provided and a comparison of the various methods is done based on the results obtained. The results show fair value accounting better meets the needs of the target –decision-making availability, compared to historical cost or fair value, more in line with the requirements of Accounting Information Quality.

Application of B-theory for Numerical Method of Functional Differential Equations in the Analysis of Fair Value in Financial Accounting

Application of B-theory for Numerical Method of Functional Differential Equations in the Analysis of Fair Value in Financial Accounting

The numerical solution of functional differential equations, a survey

Significant progress has been made in the last few years in the numerical solution of functional-differential equations. It is the purpose of this paper to review the results in this area. Special attention is given to implementation of one-step methods and predictor-corrector methods for functional-differential equations, including equations of neutral type, and to the stability theory of numerical methods for these equations. MR1146391

Pattern formation in a symmetrical network with delay

In this paper, we investigate the codimension one bifurcation involved in a symmetrical ring network with eight coupled cells. Combining the related theoretical foundations, including the symmetric bifurcation theory, representation theory of Lie groups, center manifold theorem and normal form method for functional differential equations, we address the pattern formation induced from the primary codimension one bifurcation. Numerical simulations are given to illustrate the theoretical predictions.

Quasilinearization Method for Functional Differential Equations with Delayed Arguments

We apply the quasilinearization method to boundary value problems for first order functional differential equations with delayed arguments. We formulate sufficient conditions for semi-quadratic or quadratic convergence of corresponding monotone sequences to a unique solution.

Development of the direct Lyapunov method for functional-differential equations with infinite delay

We propose new sufficient conditions for the uniform asymptotic stability of the zero solution of a retarded functional-differential equation with unbounded (infinite) delay. The equation can be nonlinear and nonautonomous. The conditions are formulated in terms of Razumikhin-type functions, and in this case, a function is coupled with a functional related to this function by a certain dependence. In the results presented here, because of additional restrictions imposed on the right-hand side of the equation and the use of the limiting equation techniques, the classical requirements stating that the function and its derivative must be of fixed sign along the solution are weakened to the requirements that the function and its derivative must be of constant signs.

An iterative method for functional differential equations

A suitable periodic boundary conditions for a functional differential equations x ̇ (t)=f(t,x,x t) are conditions of the form x 0( θ)= x 2 π ( θ). In this paper we use the notion of upper and lower solutions coupled with monotone iterative method to prove the existence of solutions of this periodic boundary value problem.

Robust Control Using Interval Analysis

The synthesis procedure of a control law that guarantees properties of robust stability with respect to structured parameter perturbations is proposed. The solution of the considered problem is based on the Razumikhin's method for functional differential equations generalized for parameter perturbation systems with time delay. The extension is obtained by using interval Lyapunov functions. The robust control law is represented through a solution of an interval matrix Riccati type equation.

Monotone iterative method for functional differential equations

Monotone iterative method for functional differential equations

Numerical Modelling of Control Time-Delay System

Discrete-time control schemes are often used for realization of positional control strategies for continuous time control problems (Krasovskii and Subbotin, 1974; Osipov, 1971) . However for general systems with delays (whicb are often called functional differential equations (FDE)) exact solutions in both continuous and discrete schemes can be found in exceptional cases. So, as rule, it is necessary to solve corresponding control problems using suitable numerical algorithms. In surveys (Cryer and Tavernini, 1972; Hall and watt, 1976; Bellen, 1985) different numerical methods for FDE are discussed.In this paper positional implicit Runge-Kutta-like numerical methods of solving general FDE are presented. The elaborated methods are direct analogs of the corresponding numerical methods of ODE case. Also we present new theorem (which generalizes the results of (Kim and Pimenov, 1997; Kim and Pimenov, 1998)) on convergence order.The obtained results are based on distinguishing of finite dimensional and infinite dimensional components in the structure of FDE, on interpolation and extrapolation of the discrete prehistory of the model, on application of constructions of i- smooth analysis (Kim, 1996) .The notion of an approximation order for an optimal strategy in the discrete scheme of realization is introduced. The conditions, which guarantee required approximation order of the optimal strategy, are presented.

Multistep numerical methods for functional differential equations

Different numerical methods are developed for solving retarded differential equations [1, 9]. Multistep numerical methods for general functional differential equations (FDE) were elaborated in [5, 10]. In contrast to those works presented in this paper, multistep numerical methods are based on the interpolation of discrete model, but not on the approximation of functionals (in the right-hand side of FDE). Basic attention is given to investigating of convergence orders of the methods. In stable multistep numerical method of solving ordinary differential equations (ODE) the convergence order is defined only by approximation order and starting procedure order. In case of FDE the order of convergence of stable multistep numerical method depends in addition on two parameters: the approximation orders of interpolation and extrapolation of the numerical model pre-history.

Periodic boundary value problems and monotone iterative methods for functional differential equations

Periodic boundary value problems and monotone iterative methods for functional differential equations

Linear multistep methods for functional-differential equations

A new way to define linear multistep methods for functional differential equations is presented, and some of their properties are analyzed. The asymptotic behavior of the global discretization error is investigated. Finally, Milne's device is generalized to functional differen- tial equations. The effect of the nonsmoothness of the exact solution is taken into account. 1. Introduction. Consider the functional differential equation (FDE) (1) x'(t) = F(t, xt) (to < t < T), xto = O, where x(t) E RI, 0 E C((-T,0), Rn), FE C((to,T) x C((-T,0), Rn), RI) and xt: 9 -* x(t + 9) for 0 E ( - T, 0), t E (to, T). It is well known that the solution x of (1) is usually not smooth, that is, even if F and 4 are C? functions, x may have jump discontinuities in its derivatives. The occurrence of these jump discontinuities may lead to order-breakdown for numerical methods if no special provisions are made. It seems that all available techniques require information about the location of the jumps. If the delay is not state-dependent, then these locations may be known a priori. If not, then they may be calculated numerically (see Feldstein and Neves (6)). When the location of a jump discontinuity is known, two different techniques are available: (i) Make the location of the jump a grid point, and restart. In practice, this

Linear Multistep Methods for Functional Differential Equations

Linear Multistep Methods for Functional Differential Equations

Asymptotic stability analysis of ?-methods for functional differential equations

Stability analysis of ?-methods for functional differential equations based on the test equation $$\begin{gathered} y'(t) = ay(t - \lambda ) + by(t),t > 0 \hfill \\ y(t) = \psi (t),t \in \left[ { - \lambda ,0} \right] \hfill \\ \end{gathered} $$ ?>0, is presented. It is known thaty(t)?0 ast?? if and only if |b|<?a and we investigate whether this property is inherited by the numerical solution approximatingy.

Order of Convergence of Linear Multistep Methods for Functional Differential Equations

The general convergence theory of [1] and a stability functional introduced in [2] are used to analyze the stability and order of consistency of linear multistep methods for Volterra functional differential equations. The techniques used require weaker continuity conditions than have been used before. For example, considering the equation \[ y'(t) = f(t,y(t),y(\alpha (t)),\int_0^t {w(t,s,y(s))ds} ), \]$0 \leqq \alpha (t) \leqq t \leqq 1$, where f is Lipschitz continuous in its second, third and fourth arguments; and using a linear pth order multistep method that satisfies the weak stability root condition, it is proved that the method converges if $y'(t)$ and $w(t,s,y(s))$ are Riemann integrable, and has order $O(h^p )$ convergence if $y^{(p - 1)} (t)$ and $({\partial / {\partial s}})^{p - 2} w(t,s,y(s))$ are the integrals of functions of bounded variation. There is no requirement that either f, $\alpha $ or w be continuous functions of t. The results apply for a function y taking on values in any real vector space, finite- or infinite-dimensional.

Special stability problems for functional differential equations

Stability properties of numerical methods for functional differential equations, similar to A-stability for ordinary differential equations, are considered. Definitions are proposed for categories of numerical stability. Some of the backward differentiation methods are shown to have these stability properties.

On Some Functional Differential Equations: Existence of Solutions and Difference Approximations

Previous article Next article On Some Functional Differential Equations: Existence of Solutions and Difference ApproximationsR. J. ThompsonR. J. Thompsonhttps://doi.org/10.1137/0705038PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] L. Bobisud, On a class of partial differential equations with time lag, J. Math. Anal. Appl., 18 (1967), 115–128 MR0206479 0148.10901 CrossrefISIGoogle Scholar[2] Sherwood C. Chu and , J. B. Diaz, Remarks on a generalization of Banach's principle of contraction mappings, J. Math. Anal. Appl., 11 (1965), 440–446 10.1016/0022-247X(65)90096-X MR0184218 0129.38301 CrossrefISIGoogle Scholar[3] Rodney D. Driver, Existence and continuous dependence of solutions of a neutral functional-differential equation, Arch. Rational Mech. Anal., 19 (1965), 149–166 MR0179406 0148.05703 CrossrefGoogle Scholar[4] L. È. Èl'sgol'c, Introduction to the theory of differential equations with deviating arguments, Translated from the Russian by Robert J. McLaughlin, Holden-Day Inc., San Francisco, Calif., 1966vi+109 MR0192154 Google Scholar[5] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966xix+592 MR0203473 0148.12601 CrossrefGoogle Scholar[6] P. D. Lax and , R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), 267–293 MR0079204 0072.08903 CrossrefISIGoogle Scholar[7] Robert D. Richtmyer, Difference methods for initial-value problems, Interscience tracts in pure and applied mathematics. Iract 4, Interscience Publishers, Inc., New. York, 1957xii+238 MR0093918 0079.33702 Google Scholar[8] Robert J. Thompson, Difference approximations for inhomogeneous and quasi-linear equations, J. Soc. Indust. Appl. Math., 12 (1964), 189–199 10.1137/0112018 MR0162369 0124.07203 LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails A method of lines for a nonlinear abstract functional evolution equation1 January 1984 | Transactions of the American Mathematical Society, Vol. 286, No. 1 Cross Ref Existence of solutions in a closed set for delay differential equations in Banach spacesNonlinear Analysis: Theory, Methods & Applications, Vol. 2, No. 1 Cross Ref Finite Difference Approximations for a Class of Semilinear Volterra Evolution ProblemsLucio Tavernini14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 14, No. 5AbstractPDF (1604 KB)The linear quadratic optimal control problem for hereditary differential systems: Theory and numerical solutionApplied Mathematics & Optimization, Vol. 3, No. 2-3 Cross Ref Autonomous nonlinear functional differential equations and nonlinear semigroupsJournal of Mathematical Analysis and Applications, Vol. 46, No. 1 Cross Ref Difference approximations for some functional differential equations23 August 2006 Cross Ref The Numerical Solution of Volterra Functional Differential Equations by Euler’s MethodColin W. Cryer and Lucio Tavernini1 August 2006 | SIAM Journal on Numerical Analysis, Vol. 9, No. 1AbstractPDF (1936 KB)NUMERICAL METHODS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS Cross Ref Volume 5, Issue 3| 1968SIAM Journal on Numerical Analysis History Submitted:05 January 1968Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705038Article page range:pp. 475-487ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics