Methods For Delay Differential Equations Research Articles

Bridging machine learning and weighted residual methods for delay differential equations of fractional order

In this research, an attempt has been made to build a bridge between a machine learning model called Least-Squares Support Vector Regression (LS-SVR) and Weighted Residual Methods (WRMs) to solve delay differential equations in fractional derivative order. Our machine learning algorithm, which is influenced by the collocation method, uses Legendre polynomials as a kernel function to approximate the solution of linear and nonlinear delay differential equations, including the Lane–Emden pantograph equation and some fractional cases. In the proposed algorithm, the estimated solution of delay differential equations is approximated as a linear combination of M degrees of Legendre polynomials and a set of weights that are learned during the fitting process. The roots of the Legendre functions are utilized as training data to develop the algorithm. The effectiveness of our method is demonstrated through the use of numerical graphs and tables. A comparative analysis of the numerical results has shown that the proposed method yields better accuracy and convergence compared to other numerical techniques.

Numerical Methods for Delay Differential Equations with Threshold State-Dependent Delay

Threshold delays arise naturally in a wide variety of dynamical systems. Standard routines for performing numerical analysis of delay differential equations (DDEs) typically handle discrete constant or state-dependent delays, but are not directly applicable to distributed delay problems. We show how to reformulate the threshold delay problem as a discrete delay DDE, with the delay as an extra variable, by differentiating the threshold condition. This allows initial value problems to be solved in Matlab using ddesd. We also describe two implementations of the threshold delay DDE in DDE-BIFTOOL to perform numerical bifurcation analysis. One approach enables us to determine steady state bifurcations via a correction of characteristic values from a modified but related problem with discrete delays. The second approach involves introducing dummy constant delays which are used to discretize the integral in the threshold condition. This allows the threshold delay to be solved for directly within DDE-BIFTOOL, and enables the full functionality of DDE-BIFTOOL, so we can compute periodic orbits as well as steady states, along with their stability and bifurcations. We illustrate these techniques on a gene regulation model where the delay, driven by a transport process, is implicitly defined by a threshold condition. The speed of the transport process depends on the state of the system making the delay both state-dependent and distributed.

Collocation method based on radial basis functions via symmetric variable shape parameter for solving a particular class of delay differential equations

In this article, we use the collocation method based on the radial basis functions with sym- metric variable shape parameter (SVSP) to obtain numerical solutions of neutral-type functional- differential equations with proportional delays. We used Gaussian radial basis functions with SVSP. Using non uniform collocation points, we achieved a system and solving this system yielded the prob- lem solutions. Several examples are given to illustrate the efficiency and accuracy of the introduced method in comparison with the same method with the constant shape parameter (CSP) as well as other analytical and numerical methods. Comparison of the obtained numerical results shows the considerable superiority of the collocation method based on RBFs with SVSP in accuracy and convergence over the collocation method based on the RBFs with CSP and other analytical and numerical methods for delay differential equations (DDEs). Finally, numerical rate of convergence analysis of the numerical approximation was also obtained. It is observed that by comparing be- tween the obtained ROC values of error norms by the SVSP and CSP method, SVSP results were considered acceptable.

A New Convergent Pseudospectral Method for Delay Differential Equations

Delay differential equations have a wide range of applications in science and engineering. When these equations are nonlinear, we cannot usually obtain an exact solution. Hence, we must utilize an efficient numerical method with high convergence rate and low error to approximate the solution. In this paper, we propose a new shifted pseudospectral method to solve nonlinear delay differential equations. First, we convert the problem into an equivalent continuous-time optimization problem and then use a pseudospectral method to discretize the problem. By solving the obtained discrete-time problem, we achieve an approximate solution for the main delay differential equation. Here, we analyze the convergence of the method and solve some practical delay differential equations to show the efficiency of the method.

Dynamical Behaviours of a Lotka Volterra Model with Katugampola Fractional Derivative

In this paper, we study the dynamical behaviours of a very special type of delay differential equation known as Lotka Volterra model (predator-prey model) via Katugampola fractional derivative. The existence and uniqueness results are obtained using the Krasnoselskii’s fixed point theorem. Moreover, we derived fractional Euler’s method for delay differential equations. Illustrative examples are solved to show the applicability of theory results.

Reconstructed Elzaki Transform Method for Delay Differential Equations with Mamadu-Njoseh Polynomials

Reconstructed Elzaki Transform Method for Delay Differential Equations with Mamadu-Njoseh Polynomials

Combined functional continuous method for delay differential equations

Combined functional continuous method for delay differential equations

Combined functional continuous method for delay differential equations

Combined functional continuous method for delay differential equations

Efficient Collocation Operational Matrix Method for Delay Differential Equations of Fractional Order

In this paper, an efficient approximation of two common delay fractional-order differential equations is developed. The proposed method applies the collocation method with generalized Laguerre polynomials basis. Operational differential generalized Laguerre matrix with an optimization technique is developed to obtain the unknowns. The convergence of the approximation method is discussed. We present numerical results and comparisons of previous treatments to demonstrate the efficiency and applicability of the proposed method. The resulting solution is close to the exact one in the linear case, and the error is very small in the nonlinear one.

Nonlinear roll oscillation of semisubmersible system and its control

The control and dynamic stability analysis of roll oscillation of the strongly nonlinear semisubmersible system are not widely investigated in the frequency domain. In particular, the suppression of undesirable nonlinear responses like period doubling route to chaos, subharmonic oscillation etc. of such a system under harmonic excitation has not been attempted so far. The nonlinear semisubmersible system is characterized by a fairly strong quintic nonlinearity arising out of the restoring action. In the present study, the incremental harmonic balance method along with arc-length continuation technique (IHBC) is employed to identify the primary and higher order subharmonic responses of the system. Stability of the responses is checked by the Floquet’s theory. A reformulation of the IHBC is then presented to control the responses of the semisubmersible system with quintic nonlinearity under state feedback control with time delay. The stability of the controlled responses is examined by applying the semi-discretization method for delay differential equation. It is shown that by considering appropriate feedback gains and time delay, (i) appreciable reduction of the amplitudes of primary responses, (ii) exclusion of all higher order subharmonics 2T, 3T, 4T, 8T, and (iii) the reduction of the extent of domain of various types of instability phenomena like, bifurcation of solutions, jump phenomena, chaotic responses etc. are possible. Further, it is found that the negative velocity feedback gain is much more effective than the full state feedback gains or varying the time delay parameter for better control of the roll oscillation of the system.

Convergence and Stability Analysis of Exponential General Linear Methods for Delay Differential Equations

Convergence and Stability Analysis of Exponential General Linear Methods for Delay Differential Equations

Response of a Harmonically Forced Dry Friction Damped System Under Time-Delayed State Feedback

Nonlinear dynamics, control, and stability analysis of dry friction damped system under state feedback control with time delay are investigated. The dry friction damped system is harmonically excited, and the nonlinearities in the equation of motion arise due to nonlinear damping and spring force. In this paper, a frequency domain-based method, viz., incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary responses which may be present in such system. The IHBC is then reformulated in a manner to treat the dry friction damped system under state feedback control with time delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses is obtained by Floquet's theory using Hsu' scheme, and the stability of the controlled responses is obtained by applying a semidiscretization method for delay differential equation (DDE).

On FBDF5 Method for Delay Differential Equations of Fractional Order with Periodic and Anti-Periodic Conditions

In this paper, we study the fractional backward differential formula (FBDF) for the numerical solution of fractional delay differential equations (FDDEs) of the following form: \(\lambda _n {}_0^C D_t^{\alpha _n } y(t - \tau ) + \lambda _{n - 1} {}_0^C D_t^{\alpha _{n - 1} } y(t - \tau ) + \cdots + \lambda _1 {}_0^C D_t^{\alpha _1 } y(t - \tau ) + \lambda _{n + 1} y(t) = f(t), t \in [0,T]\), where \( \lambda _i \in \)\(\mathbb {R}\,(i = 1,\ldots ,n + 1)\,,\,\lambda _{n + 1} \ne 0,\,\, 0 \leqslant \alpha _1 0,\) in Caputo sense. We find the Green’s functions for this equation corresponding to periodic/anti-periodic conditions in term of the Mittag-Leffler type. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Finally, some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis

Sensitivity via the complex-step method for delay differential equations with non-smooth initial data

In this report, we use the complex-step derivative approximation technique to compute sensitivities for delay differential equations (DDEs) with non-smooth (discontinuous and even distributional) history functions. We compare the results with exact derivatives and with those computed using the classical sensitivity equations whenever possible. Our results demonstrate that the implementation of the complex-step method using the method of steps and the Matlab solver dde23 provides a very good approximation of sensitivities as long as discontinuities in the initial data do not cause loss of smoothness in the solution: that is, even when the underlying smoothness with respect to the initial data for the Cauchy-Riemann derivation of the method does not hold. We conclude with remarks on our findings regarding the complex-step method for computing sensitivities for simpler ordinary differential equation systems in the event of lack of smoothness with respect to parameters.

State feedback control of surge oscillations of two-point mooring system

Stability analysis of surge oscillations of two-point mooring system under state feedback control with time-delay is investigated. The two-point mooring system is harmonically excited and essentially represents a strongly nonlinear Duffing oscillator. In this paper, a frequency domain based method viz. incremental harmonic balance method along with arc-length continuation technique (IHBC) is first employed to identify the primary and higher order subharmonic responses which may be present in such system. The IHBC is then reformulated in a manner to treat two-point mooring system under state feedback control with time-delay and is applied to obtain control of responses in an efficient and systematic way. The stability of uncontrolled responses for primary and higher order subharmonic oscillations is obtained by Floquet’s theory using Hsu’ scheme; whereas the stability of controlled responses is obtained by applying semi-discretization method for delay differential equation. The study focussed on the controlling primary, higher order subharmonics and chaotic responses by considering appropriate feedback gains and delay by way of (i) appreciable reduction of primary, subharmonic responses, (ii) exclusion of all higher order subharmonics 2T, 3T, 5T and 9T (1/n subharmonics or period-n solutions), and (iii) reduction of the extent of domain of all instability phenomena represented by various type of bifurcation of solutions, jump phenomena, chaotic responses etc. In the study, negative velocity feedback is observed to be much effective than state feedback for better controlling of surge oscillation of two-point mooring system. Also, the effect of larger gain values is investigated by an extensive parametric study for vibration control with different delay values.

Radial basis function networks for delay differential equation

In this article, a numerical technique is developed for solving delay differential equations. The proposed method combines the method of steps with the radial basis function networks. A delay differential equation is transformed to an ordinary differential equation and then the radial basis function collocationmethod is implemented to find the solution of the ordinary differential equation. The procedure for implementation of the radial basis function collocation method and the proposed method for delay differential equation is presented in detail. Numerical experiments are carried out on a number of examples to show the advantages of the proposed technique over the radial basis function collocation method.

Local Dynamics of a Logistic Equation with Delay

We considered a logistic equation with delay and studied its local dynamics. The critical cases have been found in the problem of the equilibrium state stability. We applied standard Andronov-Hopf biffurcation methods for delay differential equations and an asymptotic method, developed by one of the authors, based on the construction of special evolution equations that define the local dynamics equations with delay. It is shown that all solutions of the equation tend to an equilibrium state or result in a single stable cycle. The results of numerical modelling are presented in this paper. The study has proved that analytical and numerical modeling results have a good correlation.

Modified Laguerre Wavelets Method for delay differential equations of fractional-order

In this article, Laguerre Wavelets Method (LWM) is proposed and combined with steps Method to solve linear and nonlinear delay differential equations of fractional-order. Computational work is fully supportive of compatibility of proposed algorithm and hence the same may be extended to other physical problems also. A very high level of accuracy explicitly reflects the reliability of this scheme for such problems.

A Stochastic Collocation Method for Delay Differential Equations with Random Input

AbstractIn this work, we concern with the numerical approach for delay differential equations with random coefficients. We first show that the exact solution of the problem considered admits good regularity in the random space, provided that the given data satisfy some reasonable assumptions. A stochastic collocation method is proposed to approximate the solution in the random space, and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations. Convergence property of the proposed method is analyzed. It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space. Numerical examples are given to illustrate the theoretical results.

Formula omitted] error estimates of discontinuous Galerkin methods for delay differential equations

In this paper, we investigate the convergence behavior of discontinuous Galerkin methods for solving a class of delay differential equations. Although discontinuities may occur in various orders of the derivative of the solutions, we show that the m-degree DG solutions have (m+1)th order accuracy in L∞ norm. Numerical experiments confirm the theoretical results of the methods.