Singular problem for a third-order nonlinear ordinary differential equation arising in fluid dynamics

Dyshko, A. L.; Konyukhova, N. B.; Sukov, A. I.

doi:10.1134/s0965542507070044

Cited by 7 publications

(21 citation statements)

References 2 publications

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“…(σ, n, a, u 0 ) exact value g (0.5) approximate value g (0.5) 4, −7, − 1 7 , 1 0.48328975663 0.48328975662 (0, −3, −1, 1) 0.707106781187 0.707106781191 2, −5, − 1 5 , 1 0.491296596926 0.491296596911 (−1, −2, −1, 3)…”

Section: 4mentioning

confidence: 99%

“…This kind of problems has many practical applications, for example, in fluid mechanics and plasma physics. With this purpose, we shall apply results obtained by other authors for similar problems (see [1]- [3]), as well as general results from the stability theory for ordinary differential equations ( [5]). The obtained families will be used to compute the solutions of the two singular boundary value problems considered above.…”

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confidence: 99%

“…This problem, which has a unique solution, according to [7], Corollary 7.5, is singular at u = u 0 and, depending on σ, may also have a singularity at zero. In [1] and [2], one-parameter families of solutions were found for Cauchy problem (2.1)∧(2.3), in the cases a = 1 n , σ = 1, and u 0 = 1. In [3] lower and upper solutions were determined and some numerical results were obtained for u 0 = 1.…”

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confidence: 99%

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Analysis of singular boundary value problems for an Emden-Fowler equation

Lima¹,

Morgado²

2006

Communications on Pure &Amp; Applied Analysis

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In this work we are concerned about a second order nonlinear ordinary differential equation. Our main purpose is to describe one-parameter families of solutions of this equation which satisfy certain boundary conditions. These one-parameter families of solutions are obtained in the form of asymptotic or convergent series. The series expansions are then used to approximate the solutions of two boundary value problems. We are specially interested in the cases where these problems are degenerate with respect to the unknown function and/or to the independent variable. Lower and upper solutions for each of the considered boundary value problems are obtained and, in certain particular cases, a closed formula for the exact solution is derived. Numerical results are presented and discussed.2000 Mathematics Subject Classification. 65L05. Key words and phrases. Singular boundary value problems, one-parameter families of solutions, convergent Lyapunov series, shooting methods.

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Section: 4mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Analysis of singular boundary value problems for an Emden-Fowler equation

Lima¹,

Morgado²

2006

Communications on Pure &Amp; Applied Analysis

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“…Note that in this model we have always p = 2, even when n < −1, which corresponds to non-newtonian fluids. In [9][10][11] and [12] the authors also considered the case p = 2, but with f (r, g) = ar σ g n , σ > −1, a > 0. In these works the asymptotic behavior of the solutions near the singularity at r = 1 has been analyzed and numerical methods were introduced, which take into account this behavior.…”

Section: Introductionmentioning

confidence: 99%

Finite difference solution of a singular boundary value problem for the p-Laplace operator

Morgado

Lima

2010

Numer Algor

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We are concerned about a singular boundary value problem for a second order nonlinear ordinary differential equation. The differential operator of this equation is the radial part of the so-called N-dimensional p-Laplacian (where p > 1), which reduces to the classical Laplacian when p = 2. We introduce a finite difference method to obtain a numerical solution and, in order to improve the accuracy of this method, we use a smoothing variable substitution that takes into account the behavior of the solution in the neighborhood of the singular points.

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“…Some mathematical models in fluid dynamics, in particular, in boundary layer theory, lead to the Emden-Fowler equation g (u) = au σ g n (u) , 0 < u < u 0 , (1.1) as described, for example, in [1][2][3]8,12] and the references therein. In [9], a singular boundary value problem for Eq.…”

Section: Introductionmentioning

confidence: 99%

Analytical-numerical investigation of a singular boundary value problem for a generalized Emden–Fowler equation

Lima¹,

Morgado²

2009

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional p-laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods.

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Singular problem for a third-order nonlinear ordinary differential equation arising in fluid dynamics

Cited by 7 publications

References 2 publications

Analysis of singular boundary value problems for an Emden-Fowler equation

Analysis of singular boundary value problems for an Emden-Fowler equation

Finite difference solution of a singular boundary value problem for the p-Laplace operator

Analytical-numerical investigation of a singular boundary value problem for a generalized Emden–Fowler equation

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