Exact analytic solutions of unsolvable classes of first- and second-order nonlinear ODEs (Part II: Emden–Fowler and relative equations)

Panayotounakos, D.E.; Sotiropoulos, N.

doi:10.1016/j.aml.2004.09.005

Cited by 12 publications

(9 citation statements)

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“…For example, Panayotounakos and Sotiropoulos [7] determined the exact solution for both the Lane-Emden and generalized Emden-Fowler type equations by reducing them to the Abel s equation of the second kind by means of admissible functional transformations. The approximate solutions of equation (3) were presented by Shawagfeh [3] using the Adomian decomposition method [6].…”

Section: Introductionmentioning

confidence: 99%

An efficient computational approach to solving singular initial value problems for Lane–Emden type equations

Šmarda

Khan

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: Z.Šmarda, Y. Khan, An efficient computational approach to solving singular initial value problems for Lane-Emden type equations, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractIn the paper, we present the new computational approach for solving singular initial problems for the LaneEmden type equations with nonlinear terms based on the differential transformation method (DTM) and the modified general formula for the Adomian polynomials with differential tranformation components. To illustrate the capability and efficiency of the method three examples are given. The results obtained in this work are also compared with numerical results of Heydari [31], Wazwaz [26], Singh [32] and the standard DTM as well. These results show that the proposed technique, without linearization or small perturbation, is very effective and convenient for solving of linear and nonlinear differential equations.

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Section: Introductionmentioning

confidence: 99%

An efficient computational approach to solving singular initial value problems for Lane–Emden type equations

Šmarda

Khan

2015

Journal of Computational and Applied Mathematics

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“…Abel Ordinary Differential Equations (ODE) of the second kind [1], (1) [b 0 (t) + b 1 (t)x]ẋ = a 0 (t) + a 1 (t)x + a 2 (t)x 2 , a i (t), b i (t) ∈ C([0, T ]), can be regarded as a generalization of Riccati's equation [1]. This family of equations deserves special interest in the applied mathematics field because it appears in different contexts, running from control problems [2] to mathematical physics and nonlinear mechanics issues [1,3]. It is also remarkable that a class of Abel equations of the first kind [1] can be written as (1) with the change of variables x → x −1 .…”

Section: Introductionmentioning

confidence: 99%

Periodic solutions with nonconstant sign in Abel equations of the second kind

Olm

Ros‐Oton

Seara

2011

Journal of Mathematical Analysis and Applications

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The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation xmaps tox−1, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.Peer ReviewedPostprint (updated version

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“…In the same vein as in [2,[25][26][27]29,31] we shall use, in the first part of the present section, an admissible functional transformation to reduce problem (1), (2) to an Abel equation, with one parameter, and give some connections between the present problem and some problems in boundary-layer equations.…”

Section: Reduction To An Abel Equationmentioning

confidence: 97%

Multiple positive solutions to a singular boundary value problem for a superlinear Emden–Fowler equation

Guedda

2009

Journal of Mathematical Analysis and Applications

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A multiplicity result for the singular ordinary differential equation y + λx −2 y σ = 0, posed in the interval (0, 1), with the boundary conditions y(0) = 0 and y(1) = γ , where σ > 1, λ > 0 and γ 0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ ∈ (0, σ/2] such that a solution to the above problem is possible if and only if λγ σ −1 Σ . For 0 < λγ σ −1 < Σ , there are multiple positive solutions, while if γ = (λ −1 Σ ) 1/(σ −1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x → 0 + is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem − u = d(x) −2 u σ in Ω, where Ω ⊂ R N , N 2, is a smooth bounded domain and d(x) = dist(x, ∂Ω).

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Exact analytic solutions of unsolvable classes of first- and second-order nonlinear ODEs (Part II: Emden–Fowler and relative equations)

Cited by 12 publications

References 1 publication

An efficient computational approach to solving singular initial value problems for Lane–Emden type equations

An efficient computational approach to solving singular initial value problems for Lane–Emden type equations

Periodic solutions with nonconstant sign in Abel equations of the second kind

Multiple positive solutions to a singular boundary value problem for a superlinear Emden–Fowler equation

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