A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Erdoğan, Fevzi; Sakar, Mehmet Giyas; Saldır, Onur

doi:10.2478/amns.2020.1.00040

Cited by 30 publications

(19 citation statements)

References 24 publications

(26 reference statements)

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“…The DD model has been implemented in extensive applications, such as transport systems, engineering fields, communication networks, population dynamics and economic studies [17][18][19][20]. Many researchers solved the DD model by considering its significance in various ways; e.g., Bildik et al [21] applied a perturbation iteration scheme, Aziz et al [22] used the Haar wavelet, Tomasiello [23] introduced the fuzzy transform approach, Sabir et al [24] applied heuristic as well as swarm approaches, Erdogan et al [25] presented a finite difference approach, and some other recent related investigations are found in references [26][27][28]. The PD model was recently introduced and its literature form is given as [29]:…”

Section: Introductionmentioning

confidence: 99%

A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model

et al. 2022

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In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and local search genetic algorithms (GAs) and the active-set method (ASM), i.e., ANN-GAASM. The novel NLE-PDSM was derived from the standard LE and the PDSM along with the details of singular points, prediction terms and shape factors. The modeling strength of ANN was implemented to create a merit function based on the second kind of NLE-PDSM using the mean squared error, and optimization was performed through the GAASM. The corroboration, validation and excellence of the ANN-GAASM for three distinct problems were established through relative studies from exact solutions on the basis of stability, convergence and robustness. Furthermore, explanations through statistical investigations confirmed the worth of the proposed scheme.

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Section: Introductionmentioning

confidence: 99%

A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model

et al. 2022

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“…Due to their contributions significantly to solve the real-life problems. These equations have numerous applications in many biological models, as well as scientific phenomena such as communication system models, dynamical population models, economic systems, engineering systems, and transport models, so they have attracted a lot of attention from the research community [19,20], and various numerical/analytical techniques are introduced to solve them [21,22].…”

Section: Introductionmentioning

confidence: 99%

A new numerical strategy for solving nonlinear singular Emden-Fowler delay differential models with variable order

Ahmed

Melad

2022

Math Sci

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The present study is related to the numerical solutions of new mathematical models based on the variable order Emden-Fowler delay differential equations. The shifted fractional Gegenbauer, $$C_{S,j}^{(\alpha ,\mu )}(t),$$ C S , j ( α , μ ) ( t ) , operational matrices (OMs) of VO differentiation, in conjunction with the spectral collocation method are used to solve aforementioned models numerically. The VO operator of differentiation will be used in the Caputo sense. The proposed technique simplifies these models by reducing them to systems of algebraic equations that are easy to solve. To determine the effectiveness and accuracy of the sugested technique, the absolute errors and maximum absolute errors for four realistic models are studied and illustrated by several tables and graphs at different values of the VO and the SFG parameters; $$\alpha$$ α and $$\mu.$$ μ . Also numerical comparisons between the suggested technique with other numerical methods in the existing literature are held. The numerical results confirm that the suggested technique is accurate, computationally efficient and easy to implement.

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“…In Reference 15, authors coined a second‐order uniformly convergent central difference scheme for the numerical solution of a system of coupled reaction‐diffusion equations. In recent years, many authors proposed several types of adaptive mesh analysis and its convergent analysis for delay and nondelay differential equations, to cite a very few: References 16‐27.…”

Section: Introductionmentioning

confidence: 99%

An asymptotic streamline diffusion finite element method for singularly perturbed convection‐diffusion delay differential equations with point source

Sethurathinam

Subburayan

Arasamudi

et al. 2021

Comp and Math Methods

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In this article, we presented an asymptotic SDFEM for singularly perturbed convection diffusion type differential difference equations with point source term. First, the solution is decomposed into two functions, among them one is the solution of delay differential equation and other one is the solution of differential equation with point source. Furthermore, using the asymptotic expansion approximation, the delay differential equation is modified as a nondelay differential equations. Streamline diffusion finite element methods are applied to approximate the solutions of the two problems. We prove that the present method gives an almost second‐order convergence in maximum norm and square integrable norm, whereas first‐order convergence in H1 norm. Numerical results are presented to validate the theoretical results.

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A finite difference method on layer-adapted mesh for singularly perturbed delay differential equations

Cited by 30 publications

References 24 publications

A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model

A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model

A new numerical strategy for solving nonlinear singular Emden-Fowler delay differential models with variable order

An asymptotic streamline diffusion finite element method for singularly perturbed convection‐diffusion delay differential equations with point source

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