Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations

Hammachukiattikul, Porpattama; Sekar, Elango; Tamilselvan, A.; Vadivel, R.; Gunasekaran, Nallappan; Agarwal, Praveen

doi:10.1155/2021/6636607

Cited by 13 publications

(5 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Most of the recent studies in the domain of singularly perturbed advanced-delay differential or partial differential equations are obtained under weaker requirements and involve a positive real parameter ϵ > 0. For an overview on recent numerical approaches developed for problems both singularly perturbed and advanced-delay and for statements on their error bounds, we mention [22,23].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

Malek

2023

Abstract and Applied Analysis

View full text Add to dashboard Cite

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuchsian type in time t and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter ϵ known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic-type functions in time and are bounded holomorphic in space. A set of logarithmic-type solutions are shaped by means of Laplace transforms relatively to t and ϵ and Fourier integrals in space. Furthermore, a formal logarithmic-type solution is modeled which represents the common asymptotic expansion of the Gevrey type of the genuine solutions with respect to ϵ on bounded sectors at the origin.

show abstract

Section: Introductionmentioning

confidence: 99%

“…to (23) which is subjected to the next two features (i) The formal series û and ŵ are asymptotically equivalent in the sense that for any A > 0, there exists…”

Section: Introductionmentioning

confidence: 99%

Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

Malek

2023

Abstract and Applied Analysis

View full text Add to dashboard Cite

show abstract

“…Lalu and Phaneendra [10] developed a difference scheme for nonlinear singularly perturbed differential equation using the trigonometric spline technique and obtained a uniformly convergent method. In [11], singularly perturbed differential equations involving mixed large delays of convection-diffusion type are solved by developing a fitted mesh numerical method. The authors used finite and hybrid difference schemes to treat the problem on piece-wise uniform Shishkin mesh.…”

Section: Introductionmentioning

confidence: 99%

“…Though various forms of perturbation problems have been treated in literature, very few attentions are given for singularly perturbed differential equations with mixed large shifts. In this paper, motivated by the fitted mesh methods developed in [11,12], we inspired to develop a fitted operator difference scheme for the reaction-diffusion type singularly perturbed ordinary differential equations with mixed large shifts. To tackle the influence of the shift arguments, we have chosen a special uniform mesh in which the shift arguments lie on the nodal points.…”

Section: Introductionmentioning

confidence: 99%

Parameter-Robust Numerical Scheme for Computing Singularly Perturbed Differential Equations with Mixed Large Shifts

Ejere

2023

Preprint

View full text Add to dashboard Cite

This study focuses on the formulation and analysis of a numerical method to compute singularly perturbed functional differential equations (SP-FDEs) involving large mixed shifts. SP-FDEs are a class of differential equations with both shifts and small perturbation parameter. Such equations arise in various scientific fields to model phenomena with memory effects and perturbed dynamics. In this particular study, the considered SP-FDEs involve large negative shift (delay) and large positive shift (advance). Dealing with large mixed shifts in SP-FDEs gives rise to challenges in terms of stability analysis, numerical methods, and convergence of a solution. In this paper, the influence of the large mixed shifts is treated by choosing a uniform mesh discretized in such a way that the term with the shifts lie on the nodal points. Then, using the Numerov’s finite difference method, the continuous problem is transformed into a finite difference scheme and the influence of the perturbation parameter is handled by introducing a fitting factor in the difference scheme. Stability estimate and convergence analysis of the developed scheme are investigated and proved. Numerical experiments are carried out to demonstrate the validity and applicability of the developed scheme. It is shown that the numerical scheme obtained in this work is parameter-robust of second order convergence rate.

show abstract

“…The Dhage approach and the Banach contraction standard are used to demonstrate the presence and uniqueness of the solutions to the specified boundary value issue. The authors of the paper [19] considered a class of singularly perturbed advanced DDEs of convection-diffusion type. They use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh.…”

Section: Introductionmentioning

confidence: 99%

Approximate Analytical Solution to Nonlinear Delay Differential Equations by Using Sumudu Iterative Method

Moltot

Deresse

2022

Advances in Mathematical Physics

View full text Add to dashboard Cite

In this study, an efficient analytical method called the Sumudu Iterative Method (SIM) is introduced to obtain the solutions for the nonlinear delay differential equation (NDDE). This technique is a mixture of the Sumudu transform method and the new iterative method. The Sumudu transform method is used in this approach to solve the equation’s linear portion, and the new iterative method’s successive iterative producers are used to solve the equation’s nonlinear portion. Some basic properties and theorems which help us to solve the governing problem using the suggested approach are revised. The benefit of this approach is that it solves the equations directly and reliably, without the prerequisite for perturbations or linearization or extensive computer labor. Five sample instances from the DDEs are given to confirm the method’s reliability and effectiveness, and the outcomes are compared with the exact solution with the assistance of tables and graphs after taking the sum of the first eight iterations of the approximate solution. Furthermore, the findings indicate that the recommended strategy is encouraging for solving other types of nonlinear delay differential equations.

show abstract

Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations

Cited by 13 publications

References 26 publications

Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

Parameter-Robust Numerical Scheme for Computing Singularly Perturbed Differential Equations with Mixed Large Shifts

Approximate Analytical Solution to Nonlinear Delay Differential Equations by Using Sumudu Iterative Method

Contact Info

Product

Resources

About