A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer solutions

Cengizci, Süleyman; Atay, Mehmet Tarık; Eryılmaz, Aytekin

doi:10.1186/s40064-016-1865-6

Cited by 2 publications

(4 citation statements)

References 30 publications

(44 reference statements)

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“…is a uniformly valid approximation (UVA) [4][5][6]. For only one singularly perturbed differential equation, the uniformly valid SCEM approximation is in the regular form given by…”

Section: The Hybrid Methodsmentioning

confidence: 99%

“…Control theory, fluid mechanics, quantum mechanics, combustion theory, signal and image processing, and pharmacokinetics are some areas where singularly perturbed problems arise. One can find theoretical considerations regarding singular perturbation problems in [6,9,14,23,26,29] and various approximation methods in [4,5,7,12,13,16,17,30].…”

Section: Introductionmentioning

confidence: 99%

“…Some application areas may be given as control theory, fluid mechanics, quantum mechanics, combustion theory, signal and image processing, pharmacokinetics, etc. One can find theoretical considerations on singular perturbation problems in [10,25,27,31] and various approximation methods in [3,5,16,17,20,26,29].…”

Section: Introductionmentioning

confidence: 99%

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An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

Cengizci¹,

Natesan²,

Atay³

2019

Turk J Math

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This study concerns with singularly perturbed systems of second-order reaction-diffusion equations in ODE's. To handle this type of problems, a numerical-asymptotic hybrid method is employed. In this hybrid method, an efficient asymptotic method, the so-called Successive complementary expansion method (SCEM) is applied first and then, a numerical method based on finite differences is proposed to approximate the solution of the corresponding singularly perturbed reaction-diffusion systems. Two illustrative examples are provided to show the efficiency and easy-applicability of the present method with convergence properties.

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“…is a uniformly valid approximation (UVA) [4][5][6]. For only one singularly perturbed differential equation, the uniformly valid SCEM approximation is in the regular form given by…”

Section: The Hybrid Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

Cengizci¹,

Natesan²,

Atay³

2019

Turk J Math

Self Cite

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“…is clearly uniformly valid approximation (UVA) [28][29][30]. Now let us consider second-order singularly perturbed DDE in its general form (delay in the convection term):…”

Section: Description Of the Methodsmentioning

confidence: 99%

An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

Cengizci

2017

International Journal of Differential Equations

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In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

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A uniformly valid approximation algorithm for nonlinear ordinary singular perturbation problems with boundary layer solutions

Cited by 2 publications

References 30 publications

An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

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