Perturbed Delay Differential Equations Research Articles

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

Quartic B-Spline Method for Non-Linear Second Order Singularly Perturbed Delay Differential Equations

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme.The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

Non-polynomial spline approach for solving system of singularly perturbed delay differential equations of large delay

ABSTRACT A computational technique for analysing coupled system of singularly perturbed convection-diffusion delay differential equations with large delay is presented in this study. We solve this problem using non-polynomial spline technique and arrive at a system that can potentially be solved. To show the theoretical investigation, an example is solved for different perturbation parameter values, and the computational results are shown in tables. The model we have proposed is of first order convergent, and this technique indicated that the acquired findings were more accurate than earlier results.

Singular Perturbations and Large Time Delays Through Accelerated Spline-Based Compression Technique

In the quest to solve the singularly perturbed delay differential equations (SPDDEs) involving large delay with integral boundary condition, the cubic spline in compression technique is explored for the study of dynamical systems to capture complex temporal phenomena in a wide range of scientific disciplines. The integral boundary condition is handled using Simpson's 1/3 rule and the scheme's applicability is validated by numerically experimenting with some problems at different values of mesh size and perturbation parameter. Numerical data are tabulated to show that the suggested approach is more accurate and is an improvement over the methods used in the literature. The insights gained from this research paper provide a foundation for further exploration and utilization of SPDDEs in understanding and predicting the behavior of complex systems across diverse scientific domains.

Dynamics of one-dimensional maps and Gurtin–Maccamy's population model. Part I. Asymptotically constant solutions

UDC 517.9 Motivated by the recent work by Ma and Magal [Proc. Amer. Math. Soc. (2021); https://doi.org/10.1090/proc/15629] on the global stability property of the Gurtin–MacCamy's population model, we consider a family of scalar nonlinear convolution equations with unimodal nonlinearities. In particular, we relate the Ivanov and Sharkovsky analysis of singularly perturbed delay differential equations in [https://doi.org/10.1007/978-3-642-61243-5_5] with the asymptotic behavior of solutions of the Gurtin–MacCamy's system. According the classification proposed in [https://doi.org/10.1007/978-3-642-61243-5_5], we can distinguish three fundamental kinds of continuous solutions of our equations, namely, solutions of the asymptotically constant type, relaxation type and turbulent type. We present various conditions assuring that all solutions belong to the first of these three classes. In the setting of unimodal convolution equations, these conditions suggest a generalized version of the famous Wright's conjecture.

Tension spline approach to singularly perturbed delay differential equations involving large delay

To solve the singlularly perturbed delay differential equations of convection type, spline in tension technique is used. Taylor series approximation is used to tackle the first order derivative term involved in the problem and finally a tridiagonal scheme is obtained, which is then solved by Thomas algorithm. An example is illustrated to show the method's applicability and it is solved numerically for different values of perturbation parameter. The maximum absolute errors and rate of convergence are given in tables and it is seen that our method is of first order convergent.

A deep learning feed-forward neural network framework for the solutions to singularly perturbed delay differential equations

In this paper, we explore a deep learning feedforward artificial neural network (ANN) framework as a numerical tool for approximating the solutions to singularly perturbed delay differential equations (SPDDE). Our approach trains the network with fewer uniform data points along with linear interpolation as a dependent variable. More importantly, the exact solution is not used for training the network. A mean square or Euclidean norm type total loss function is utilized to assess the deep learning network’s performance. In the training and testing phase, our network adjusts itself to fit the tangent and the curvature by minimizing the total loss function and fine-tuning the network’s hyperparameters during the backpropagation stage. Our focus in this contribution is to investigate an answer to the question- “whether neural networks can learn to solve the differential equations with both delay and boundary layer behavior”? Traditional numerical methods are known to perform poorly in approximating the solutions to SPDDE due to boundary layer behavior, which is the sharp change in the gradient of the solution inside a region of very small width. Our proposed network architecture is used to investigate the above question for three varieties of SPDDE. The numerical results demonstrate that the fine-tuned adaptive deep learning architecture can effectively approximate the solution to SPDDE for varieties of delay and perturbation parameters. It is expected that the current method can be extendable to approximate 2D convection-diffusion partial differential equations with a boundary layer.

Numerical solution of a time dependent singularly perturbed delay differential equation on an exponentially graded mesh

Numerical solution of a time dependent singularly perturbed delay differential equation on an exponentially graded mesh

Integrated Stochastic Investigation of Singularly Perturbed Delay Differential Equations for the Neuronal Variability Model

The proposed research utilizes a computational approach to attain a numerical solution for the singularly perturbed delay differential equation (SPDDE) problem arising in the neuronal variability model through artificial neural networks (ANNs) with different solvers. The log-sigmoid function is used to construct the fitness function. The implementation of ANN on SPDDE problems is formulated for different solvers and trained with different weights. The optimization solvers such as the genetic algorithm (GA), sequential quadratic programming (SQP), and pattern search (PS) are hybridized with the active set technique (AST) and the interior-point technique (IPT) and is used to check the accuracy and rapid convergence of the numerical results of the SPDDE model. The numerical outcomes demonstrate that the system is easy to handle and efficient to solve with boundary conditions. Moreover, we used the mean residual error for one hundred runs for each solver to validate the accuracy of the proposed scheme.

Numerical Simulation of Singularly Perturbed Delay Differential Equations With Large Delay Using an Exponential Spline

In this study, numerical solution of a differential-difference equation with a boundary layer at one end of the domain is suggested using an exponential spline. The numerical scheme is developed using an exponential spline with a special type of mesh. A fitting parameter is inserted in the scheme to improve the accuracy and to control the oscillations in the solution due to large delay. Convergence of the method is examined. The error profiles are represented by tabulating the maximum absolute errors in the solution. Graphs are being used to show that how the fitting parameter influence the layer structure.

Analysis of a Hybrid Numerical Scheme for Singularly Perturbed Convection-Diffusion Type Delay Problems

In this paper, a class of singularly perturbed delay differential equations of convection-diffusion (C–D) type is considered. The solutions for these problems exhibit boundary as well as interior layer due to the presence of delay. A few a priori estimates are given on the exact solution and its derivatives. We propose a numerical technique consisting of hybrid finite difference scheme on a Shishkin-type mesh. The proposed technique is analyzed for its consistency, stability and convergence. To show the effectiveness of the present approach, we conduct some numerical experiments. An almost second-order [Formula: see text]-uniform convergence is established.

Solving singularly perturbed delay differential equations via fitted mesh and exact difference method

In this paper, different numerical schemes are developed and investigated for solving singularly perturbed delay differential equations (DDEs). The solution of the considered equations exhibits a strong boundary layer when the perturbation parameter approaches zero. Standard numerical methods developed for solving regular problems fail to treat accurately the considered equations. We developed three numerical schemes for treating the considered equations. The first scheme (Scheme I) uses a non-standard mid-point upwind finite difference method on uniform mesh; the second scheme (Scheme II) uses a standard mid-point upwind finite difference method on Shiskin mesh and the third scheme (Scheme III) uses a non-standard mid-point upwind finite difference method on Shiskin mesh. The existence of a unique discrete solution is investigated using the discrete maximum principle. The stability and uniform convergence of the schemes are investigated and proved. Scheme III gives better accuracy and order of convergence than Scheme I. It has a boundary layer resolving behaviour which is the main drawback of Scheme II. Numerical examples are considered and treated to validate the theoretical finding of the schemes for different values of the perturbation parameter and delay parameter.

Fitted Parameter Exponential Spline Method for Singularly Perturbed Delay Differential Equations with a Large Delay

In this paper, we present a new computational method based on an exponential spline for solving a class of delay differential equations with a negative shift in the differentiated term. When the shift parameter is O ε , the proposed method works well and also controls the oscillations in the solution’s layer region. To accomplish this, we included a parameter in the proposed numerical scheme that is based on a special type of mesh, and the parameter is evaluated using the theory of singular perturbation. Maximum absolute errors and convergences of numerical solutions are tabulated to demonstrate the efficiency of the proposed computational method and to support the convergence analysis of the presented scheme.

Second order singularly perturbed delay differential equations with non-local boundary condition

This paper is concerned with the singularly perturbed delay differential equations with integral boundary conditions. To solve the problem, a finite difference scheme with an appropriate piecewise Shishkin type mesh is suggested. We prove that the proposed approach is almost first order convergent. Utilizing the discrete norm, an error estimate is also calculated. Finally, a simulation examples is given to demonstrate the effectiveness and to show the superiority of the theoretical results.

Analysis of Exponential–Polynomial Single Step Method for Singularly Perturbed Delay Differential Equations

This research presents Exponential – Polynomial Single Step Method (EPSSM), which uses non-linear polynomial interpolating function to solve Singularly Perturbed Delay Differential Equations (SPPDDEs). The stability polynomial of the method is obtained. The Local Grid Search Algorithm termed LGSA is applied in determining the stability region. The efficiency of these methods is compared with the other two numerical methods, namely Runge-Kutta Method (RKM) and Rational Single Step Method (RSM) through two examples of Singularly Perturbed Delay Differential Equations (SPPDDEs).

Numerical solution of delay differential equation using two-derivative Runge-Kutta type method with Newton interpolation

Numerical approach of two-derivative Runge-Kutta type method with three-stage fifth-order (TDRKT3(5)) is developed and proposed for solving a special type of third-order delay differential equations (DDEs) with constant delay. An algorithm based on Newton interpolation and hybrid with the TDRKT method is built to approximate the solution of third-order DDEs. In this paper, three-stage fifth-order called TDRKT3(5) method with single third derivative and multiple evaluations of the fourth derivative is highlighted to solve third-order pantograph type delay differential equations directly with the aid of the Newton interpolation method. Stability analysis of TDRKT3(5) method is investigated. The numerical experiments illustrate high efficiency and validity of the new method for solving a special class of third-order DDEs and some future works are recommended by extending proposed method to solve fractional and singularly perturbed delay differential equations.

Numerical analysis of a system of semilinear singularly perturbed first‐order differential equations on an adaptive grid

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method.

Renormalization group approach to a class of singularly perturbed delay differential equations

In this paper, we present a systematic renormalization group method to investigate a class of singularly perturbed delay differential equations. The uniformly valid approximate solution can be obtained, and we give a rigorous proof of the error estimate of the approximate solution. In addition, some numerical comparisons between the exact solution, the result by the averaging method and our result are given for two examples.

Robust mid-point upwind scheme for singularly perturbed delay differential equations

Robust mid-point upwind scheme for singularly perturbed delay differential equations

Uniformly convergent fitted operator method for singularly perturbed delay differential equations

This paper deals with numerical treatment of singularly perturbed delay differential equations having delay on first derivative term. The solution of the considered problem exhibits boundary layer behaviour on left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting to asymptotically equivalent singularly perturbed boundary value problem. Uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis.