Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel’s equations)

Panayotounakos, D.E.

doi:10.1016/j.aml.2004.09.004

Cited by 40 publications

(38 citation statements)

References 4 publications

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“…Our goal is the development of the construction of exact analytic solutions of the above equations based on a mathematical technique leading to the derivation of exact analytic solutions of the Abel equation of the second kind of the normal form (see Refs. [6], [7]). …”

Section: Discussionmentioning

confidence: 99%

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Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

Panayotounakos

Sotiropoulou

Sotiropoulos

et al. 2007

International Journal of Non-Linear Mechanics

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Two kinds of second-order nonlinear ordinary differential equations (ODEs) appearing in mathematical physics and nonlinear mechanics are analyzed in this paper. The one concerns the Kidder equation in porous media and the second the gas pressure diffusion equation. Both these equations are strongly nonlinear including quadratic first order derivatives (damping terms). By a series of admissible functional transformations we reduce the prescribed equations to Abel's equations of the second kind of the normal form that they do not admit exact analytic solutions in terms of known (tabulated) functions. According to a mathematical methodology recently developed concerning the construction of exact analytic solutions of the above class of Abel's equations, we succeed in performing the exact analytic solutions of both Kidder's and gas pressure diffusion equations. The boundary and initial data being used in the above constructions are in according with each specific problem under considerations.

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Section: Discussionmentioning

confidence: 99%

“…[6,7], we will provide the exact analytic solution of the Kidder equation (2.1). Exactly the same mathematical technique follows the gas pressure diffusion equation (2.14).…”

Section: The Reduction Procedures A) Kidder's Equationmentioning

confidence: 99%

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

Panayotounakos

Sotiropoulou

Sotiropoulos

et al. 2007

International Journal of Non-Linear Mechanics

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“…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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The expected time to end the tug-of-war in a wedge

DeBlassie

Smits

2011

Probab. Theory Relat. Fields

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Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I: Abel’s equations)

Cited by 40 publications

References 4 publications

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

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

The expected time to end the tug-of-war in a wedge

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