Exact analytic solutions of the Abel, Emden–Fowler and generalized Emden–Fowler nonlinear ODEs

Panayotounakos, D.E.; Kravvaritis, Dimitrios

doi:10.1016/j.nonrwa.2005.03.025

Cited by 13 publications

(18 citation statements)

References 16 publications

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“…4.8 and 4.21). Using the solution methodology recently developed in [6,7] we extracted analytic solutions in parametric form (Eqs. 4.9 and 4.32) of the original buckling nonlinear ODE (2.4) concerning large deflections of elastic bars (Fig.…”

Section: Discussionmentioning

confidence: 99%

“…It was recently proved in [6,7] Summarizing the results of [6,7], we perform the solution of Eq. (4.21) as follows Here C 1 is an integration constant, while the functionN (s * ) is given implicitly in terms of the function G(s * ) and the second member ± 2 λs * 2 of Eq.…”

Section: Approximations: Existence Of Exact Analytic Solutionsmentioning

confidence: 99%

“…In addition, according to several subsidiary relations involved in [6,7], one evaluates the derivativeN s (s * ) by the formulaN …”

Section: Approximations: Existence Of Exact Analytic Solutionsmentioning

confidence: 99%

“…In the limits of large or small values of the slope θ of the deflected elastica up to O cos 2 θ or up to O sin 2 θ , we expand asymptotically the prescribed integrodifferential equation to nonlinear ODEs of the Emden-Fowler or Abel nonlinear type (see [8]). In these cases, using the solution methodology recently developed in [6,7], we construct exact implicit analytic solutions in parametric form of these equations and thus approximate implicit analytic solutions to the original elastica buckling nonlinear ODE.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

2006

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It is shown that by a series of admissible functional transformations the already derived (third-order) strongly nonlinear ordinary differential equation (ODE), describing the elastica buckling analysis of a straight bar under its own weight [Int. J. Solids Struct. 24(12), 1179-1192, 1988, The Theory of Elastic Stability, McGraw-Hill, New York, 1961], is reduced to a first-order nonlinear integrodifferential equation. The absence of exact analytic solutions of the reduced equation leads to the conclusion that there are no exact analytic solutions in terms of known (tabulated) functions of this elastica buckling problem. In the limits of large or small values of the slope of the deflected elastica, we expand asymptotically the above integrodifferential equation to nonlinear ODEs of the Emden-Fowler or Abel nonlinear type. In these cases, using the solution methodology recently developed in Panayotounakos [Appl. Math. Lett. 18:155-162, 2005] and Panayotounakos and Kravvaritis [Nonlin. Anal. Real World Appl., 7(2):634-650, 2006], we construct exact implicit analytic solutions in parametric form of these types of equations and thus approximate implicit analytic solutions of the original elastica buckling nonlinear ODE.

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Section: Discussionmentioning

confidence: 99%

Section: Approximations: Existence Of Exact Analytic Solutionsmentioning

confidence: 99%

“…In addition, according to several subsidiary relations involved in [6,7], one evaluates the derivativeN s (s * ) by the formulaN …”

Section: Approximations: Existence Of Exact Analytic Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

2006

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“…Thus, X cannot be considered as a Lie system and we conclude that the first-order nonlinear oscillator (22) describing integral curves of the time-dependent vector field (23) (which is not a Lie system) can be described by means of the quasi-Lie scheme S(W, V ).…”

Section: Nonlinear Oscillatorsmentioning

confidence: 97%

Quasi-Lie schemes: theory and applications

Cariñena¹,

Grabowski²,

Lucas³

2009

J. Phys. A: Math. Theor.

115

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A novel numerical approach for the third order Emden–Fowler type equations

Sakar,

Saldır,

Aydın

et al. 2024

Z Angew Math Mech

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This article aims to achieve robust numerical results by applying the Chebyshev reproducing kernel method without homogenizing the initial‐boundary conditions of the Emden–Fowler (E‐F) equation, thereby introducing a new perspective to the literature. A novel numerical approach is presented for solving the initial‐boundary value problem of third‐order E‐F equations using Chebyshev reproducing kernel theory. Unlike previous applications, which were confined to homogeneous initial‐boundary value problems or required homogenization, the proposed method is effective for both homogeneous and nonhomogeneous cases. To handle the initial‐boundary conditions of the E‐F equations, additional basis functions are introduced rather than imposing conditions on the reproducing kernel Hilbert space. The method's effectiveness is demonstrated through five examples, which validate the theoretical analysis. Overall, the results emphasize the method's efficiency.

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Exact analytic solutions of the Abel, Emden–Fowler and generalized Emden–Fowler nonlinear ODEs

Cited by 13 publications

References 16 publications

Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

Nonlinear asymptotic analysis in elastica of straight bars—analytical parametric solutions

Quasi-Lie schemes: theory and applications

A novel numerical approach for the third order Emden–Fowler type equations

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