A brief survey on numerical methods for solving singularly perturbed problems

Kadalbajoo, Mohan K.; Gupta, Vikas

doi:10.1016/j.amc.2010.09.059

Cited by 126 publications

(80 citation statements)

References 586 publications

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“…where the asymptotic sequence of order functions δ i (ε) an asymptotic sequence such that δ i+1 (ε) = o(δ i (ε)), as ε → 0 and all the functions φ i (x, ε) are of strict order 1, φ i = O S (1). Under these conditions the approximation (5) …”

Section: Description Of the Methodsmentioning

confidence: 99%

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SCEM approach for singularly perturbed linear turning mid-point problems with an interior layer

Atay¹,

Cengizci²,

Eryılmaz³

2016

ntmsci

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Abstract:In this work, we study an important class of singular perturbation problems: singularly perturbed two-point turning point problems (TPP's) in ordinary differential equations (ODE's) exhibiting interior layer at the mid-point of their intervals. We consider this type of problems as two different singular perturbation problems which have a boundary layer at an end point of their intervals. This study is devoted to the Successive complementary expansion method (SCEM) to solve these two sub-problems. Two test problems are considered to check the efficiency and accuracy of the proposed method. Our numerical experiments show that SCEM approximations are in good agreement with exact and previously obtained solutions.

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Section: Description Of the Methodsmentioning

confidence: 99%

“…Kadalbajoo and Gupta made a great survey and investigated the development of the conventional and novel methods to solve singular perturbation problems in [1]. In another great study [2], Parul gave some examples to these kinds of problems occuring in almost all science branches and briefly examined the conventional methods.…”

Section: Introductionmentioning

confidence: 99%

SCEM approach for singularly perturbed linear turning mid-point problems with an interior layer

Atay¹,

Cengizci²,

Eryılmaz³

2016

ntmsci

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“…A lot of numerical methods to treat the singular perturbation problems can be seen (Bender & Orszag, 1978;Shampine & Gear, 1979;Nayfeh, 1981;Kevorkian & Cole, 1981O'Malley 1991;De Jager & Jiang, 1996), and the survey was provided by Kadalbajoo & Patidar (2002) and Kadalbajoo & Gupta (2010). For some recently developed numerical methods to solve the SPBVPs, the reader may refer (Liu, 2006a(Liu, , 2006bLiu, 2012a;Khuri & Sayfy, 2013;Dogan, Erturk & Akin, 2012).…”

Section: Introductionmentioning

confidence: 99%

Solving Third-Order Singularly Perturbed Problems: Exponentially and Polynomially Fitted Trial Functions

Liu¹

2016

JMR

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For the third-order linearly singularly perturbed problems under four different types boundary conditions, we develop a weak-form integral equation method (WFIEM) to find the singular solution. In the WFIEM the exponentially and polynomially fitted trial functions are designed to satisfy the boundary conditions automatically, while the test functions satisfy the adjoint boundary conditions exactly. The WFIEM provides accurate and stable solutions to the highly singular third-order problems.

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“…In 2002, Kadalbajoo and Patidar [2] presented another surveyonthework done by numerous researchers in the area of singular perturbation from 1984 to2000. Kadalbajoo and Patidar [3] presented are view of singularly perturbed partial differential equations in 2003.In [4] Kadalbajoo and Vikasgave brief survey on the computational techniques used to solve different classes of singular perturbation problems. In 2013, Sharma et a review on singular perturbed differential equations with turning point and interior layers.…”

Section: Introductionmentioning

confidence: 99%

“…view of singularly perturbed partial differential equations in 2003.In [4] Kadalbajoo and Vikasgave brief survey on the computational used to solve different classes of singular al. [5] presented a ew on singular perturbed differential equations with turning point and interior layers.…”

Section: Introductionmentioning

confidence: 99%