Analytical and numerical stability of nonlinear neutral delay integro-differential equations☆
Introduction
In this paper, we study the analytical and numerical stability of nonlinear neutral delay integro-differential equations (NDIDEs) of the form:where . Functions , are smooth enough, and initial function is continuously differentiable. The delay function satisfies the following hypotheses (see [33]):
- (H1)
is a continuous function and there exists a constant such that , .
- (H2)
is strictly increasing for all and divergent as .
This class of equations play an important role in modeling phenomena of the real world. They arise widely in scientific and engineering fields such as biology, physics, control theory, ecology and so on (see [4], [5], [40]). So it is valuable to investigate the properties of the solution of these equations. Since most of these equations cannot be solved exactly, it is necessary to study efficient numerical methods to solve these equations. An important problem in this context is the investigation of the stability of numerical methods.
In recent years, many papers discussed analytical stability and numerical stability of delay differential equations (DDEs) (see [3], [11], [12], [13], [14], [17], [18], [20], [32], [33] and the references therein), neutral delay differential equations (NDDEs) (see [1], [2], [9], [19], [22], [23], [24], [25], [26], [38] and the references therein) and delay integro-differential equations (DIDEs) (see [6], [10], [15], [16], [34], [35] and the references therein). A significant number of important results have already been found. But for neutral delay integro-differential equations (NDIDEs), only a few of results have been presented in the literature. Zhao et al. [39] considered the linear neutral Volterra delay integro-differential system:where , and the matrix A may be singular. They gave a sufficient condition such that the system is asymptotically stable, and proved that every linear with and A-stable BDF methods preserve the delay-independent stability of the analytical solution. Later, Xu et al. [29] further considered Runge–Kutta methods for system (1.2). It is proved that every A-stable natural Runge–Kutta method preserves the delay-independent stability of the analytical solution. In 2008, Zhang et al. [36] investigated the asymptotic stability of exact and discrete solutions of neutral multidelay-integro-differential equations. A sufficient condition for the asymptotic stability of the analytical solution has been derived. And the asymptotic stability criteria of Runge–Kutta methods and linear multistep methods were established. Wu et al. [28] considered a test equation for one-dimension NDIDEs and got some delay-dependent stability results.
In the nonlinear case, this topic has also attracted the attention of some authors. Yu et al. [31] and Zhang et al. [37] investigated the stability of Runge–Kutta methods for the equation:For the numerical stability analysis of general nonlinear NDIDEs with a constant delay, we refer the reader to [30], where many methods, including one-leg methods, Runge–Kutta methods, general linear methods and linear , are considered. For the analytical stability analysis of nonlinear NDIDEs, we mention the paper by Wen et al. [27] and the doctoral thesis by Wang [21]. In [27], by applying a generalized Halanay inequality to Eq. (1.3), some dissipativity and asymptotic stability results were obtained. In [21], the analytical stability of more general neutral functional differential equations (NFDEs) was studied and the results obtained there are also applicable to NDIDEs.
In this paper, we further investigate the analytical and numerical stability of NDIDEs. We first study the analytical stability of problem (1.1). The approach we use is based on the theory of stability analysis which was first introduced by Torelli [20] and developed by Zennaro [33], Bellen et al. [1] and Wang et al. [22]. We derive some new sufficient conditions for boundedness stability and asymptotic stability of the exact solution. Then, we focus our attention on the nonlinear stability of the numerical solutions produced by linear multistep methods. By using the equivalence between A-stability and G-stability, which was established by Dahlquist [8], it is shown that every A-stable linear multistep method can preserve the asymptotic stability of a class of NDIDEs with a constant delay. At last, we give some numerical experiments to confirm our theoretical results.
Section snippets
Analytical stability of nonlinear NDIDEs
Let be an inner product on and be the corresponding norm. We consider the following initial value problem:which is a corresponding perturbed problem of Eq. (1.1) with a different continuously differentiable initial function. In this paper we assume that the functions f and g satisfy the following one-sided Lipschitz or global Lipschitz conditions:
Linear multistep methods for nonlinear NDIDEs
In this section, we consider linear multistep methods for NDIDEs. Let stepsize h satisfy , with m a positive integer, and let tn=nh. A linear multistep method for problem (2.43) is given bywhere E is the translation operator defined by and the argument is an approximation to the exact solution . And are generating polynomials of the method, which are assumed to have real coefficients and no common divisor.
Numerical stability analysis
In this section, we investigate the asymptotic stability of A-stable linear multistep methods for nonlinear NDIDEs. We will make use of the conclusion that A-stability is equivalent to G-stability which was established by Dahlquist [8] in 1978.
For any real symmetric positive definite k×k matrix , the norm on is defined by
The following result shows the asymptotic stability of linear multistep methods when applied to problem
Numerical experiment
In this section, we test some numerical methods to validate Theorem 4.1. We consider the following nonlinear NDIDE:and its corresponding perturbed problemHere we choose the parameters a=−8, b=0.1, c=0.5 and , and it is easy to get and so that i.e.,
Acknowledgments
The authors are indebted to the referees and the editors for their carefully reading of this paper and their valuable comments.
References (40)
- et al.
Numerical stability of nonlinear delay differential equations of neutral type
J. Comput. Appl. Math.
(2000) - et al.
Strong contractivity properties of numerical methods for ordinary and delay differential equations
Appl. Numer. Math.
(1992) - et al.
Numerical modelling in biosciences using delay differential equations
J. Comput. Appl. Math.
(2000) Stability of linear multistep methods for delay integro-differential equations
Comput. Math. Appl.
(2008)Asymptotic stability of multistep methods for nonlinear delay differential equations
Appl. Math. Comput.
(2008)- et al.
Stability and error analysis of one-leg methods for nonlinear delay differential equations
J. Comput. Appl. Math.
(1999) - et al.
Stability analysis of Runge–Kutta methods for non-linear delay differential equations
BIT
(1999) Stability of for delay integro-differential equations
J. Comput. Appl. Math.
(2003)- et al.
Delay-dependent criteria for absolute stability of uncertain time-delayed Lur’e dynamical systems
J. Franklin Inst.
(2010) - et al.
Delay-dependent stability analysis of trapezium rule for second order delay differential equations with three parameters
J. Franklin Inst.
(2010)
A delay-dependent stability criterion for linear neutral delay systems
J. Franklin Inst.
Stability of numerical methods for delay differential equations
J. Comput. Appl. Math.
Nonlinear stability of Runge–Kutta methods for neutral delay differential equations
J. Comput. Appl. Math.
Stability analysis of for nonlinear neutral functional differential equations
SIAM J. Sci. Comput.
Dissipativity and asymptotic stability of nonlinear neutral delay integro-differential equations
Nonlinear Anal.: Theory Methods Appl.
Analytical and numerical stability of neutral delay integro-differential equations and neutral delay partial differential equations
Comput. Math. Appl.
Stability of Runge–Kutta methods for neutral delay-integro-differential-algebraic system
Math. Comput. Simulation
Contractivity of Runge–Kutta methods with respect to forcing term
Appl. Numer. Math.
Cited by (9)
Asymptotic behavior of solutions to time fractional neutral functional differential equations
2021, Journal of Computational and Applied MathematicsCitation Excerpt :The stability of Volterra delay integro-differential equations was studied in [27]. Some other useful stability criteria for exact and discrete solutions of neutral multi delay-integro-differential equations are developed in [1,5,28–33]. Those results provide a basic framework for the theoretical and numerical analysis of functional differential equations.
Nonlinear stability of one-leg methods for neutral Volterra delay-integro-differential equations
2014, Mathematics and Computers in SimulationCitation Excerpt :Brunner and Vermiglio [6] considered the stability of continuous Runge–Kutta methods for NVDIDEs of the “Hale's form”. Several researchers investigated the stability of numerical method for nonlinear NVDIDEs where the kernel K(t, θ, y) does not depend on y′ and its linear version (see [31–33,37,13,38,29]). However, few studies have been done on the stability of numerical methods for nonlinear NVDIDEs (1.1) in which the kernel K(t, θ, y, y′) also depends on y′.
Polynomial based differential quadrature method for numerical solution of nonlinear Burgers equation
2011, Journal of the Franklin InstituteCitation Excerpt :Recently, in parallel with developments in computer speeds and numerical algorithms, lots of numerical and semi-numerical methods have been developed in order to solve partial differential equations [5–15].
Generalized difference simulation for coupled transport models of unsaturated soil water flow and solute
2011, Journal of the Franklin InstituteCitation Excerpt :Change of the solute in the soil water consists of physical, chemical and biochemical processes [1–5], and physical change processes are important for the solute, which are usually depicted by two equations (which appear in this issue, see [4]), i.e. the nonlinear equation of soil water flow and convection-dominated convection–diffusion equation of solute transport. Since the problem is described by a nonlinear system (see [6,7], which are nonlinear equations), it is impossible to obtain its analytical solution except for special cases. Therefore, numerical approximations [8–15] are typically used to solve the unsaturated water and solute transport equations.
Dynamic analysis of a class of neutral delay model based on the Runge-Kutta algorithm
2018, Eurasip Journal on Wireless Communications and NetworkingStability Analysis of One-Leg Methods for Nonlinear Neutral Delay Integrodifferential Equations
2015, Discrete Dynamics in Nature and Society
- ☆
This work was supported by the NSF of China (No. 10971077).