Systems Of Neutral Delay Differential Equations Research Articles

Numerical solution of neutral delay differential equations using orthogonal neural network

In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order 2^{-n}, where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials.

Emergence of quasiperiodic regimes in a neutral delay model of flute-like instruments: Influence of the detuning between resonance frequencies

<p style='text-indent:20px;'>Musical instruments display a wealth of dynamics, from equilibria (where no sound is produced) to a wide diversity of periodic and non-periodic sound regimes. We focus here on two types of flute-like instruments, namely a recorder and a pre-hispanic Chilean flute. A recent experimental study showed that they both produce quasiperiodic sound regimes which are avoided or played on purpose depending on the instrument. We investigate the generic model of sound production in flute-like musical instruments, a system of neutral delay-differential equations. Using time-domain simulations, we show that it produces stable quasiperiodic oscillations in good agreement with experimental observations. A numerical bifurcation analysis is performed, where both the delay time (related to a control parameter) and the detuning between the resonance frequencies of the instrument – a key parameter for instrument makers – are considered as bifurcation parameters. This demonstrates that the large detuning that is characteristic of prehispanic Chilean flutes plays a crucial role in the emergence of stable quasiperiodic oscillations.</p>

Analytical Algorithm for Systems of Neutral Delay Differential Equations

Delay differential equations (DDEs), as well as neutral delay differential equations (NDDEs), are often used as a fundamental tool to model problems arising from various areas of sciences and engineering. However, NDDEs particularly the systems of these equations are special transcendental in nature; it has therefore, become a challenging task or times almost impossible to obtain a convergent approximate analytical solution of such equation. Therefore, this study introduced an analytical method to obtain solution of linear and nonlinear systems of NDDEs. The proposed technique is a combination of Homotopy analysis method (HAM) and natural transform method, and the He’s polynomial is modified to compute the series of nonlinear terms. The presented technique gives solution in a series form which converges to the exact solution or approximate solution. The convergence analysis and the maximum estimated error of the approach are also given. Some illustrative examples are given, and comparison for the accuracy of the results obtained is made with the existing ones as well as the exact solutions. The results reveal the reliability and efficiency of the method in solving systems of NDDEs and can also be used in various types of linear and nonlinear problems.

Linear stability of general linear methods for systems of neutral delay differential equations

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.

Estimation of numerically stable step-size for neutral delay-differential equations via spectral radius

We derive the estimates of numerically stable step-size for systems of neutral delay-differential equations (NDDEs), which only need to be calculated the spectral radius of the corresponding matrices. The stable step-size for numerical integration of NDDEs can be easily selected by means of the estimates. The stability regions of both linear multistep methods and explicit Runge-Kutta methods are presented.

Stability analysis of numerical methods for systems of neutral delay-differential equations

Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicitA-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.