Summary
Considerf‴+α ff″+β (1−f′2)+γ f″=0 together with the boundary conditionsf(0)=f′(0)=0,f′ (∞)=1. Ifα=−1,β>0,γ arbitrary there is at least one solution which satisfies 0<f′<1 on (0, ∞). By the additional conditionf′>0 on (0, ∞) or, alternately 0<β≤1, the uniqueness of the solution is demonstrated.
Ifα=1,β<0,γ arbitrary the existence of solutions for which −1<f′<0 in some initial interval (0,t) and satisfying generallyf′>1 is established. In both problems, bounds forf″ (0) and qualitative behavior of the solutions are shown.
Sommario
Si consideri il problema definito dall'equazionef‴+α f f″+β (1−f′2)+γ f″=0 e dalle condizioni al contornof(0)=f′ (0)=0,f′(∞)=1. Assumendoα=−1,β>0,γ arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f′<1 nell'intervallo (0, ∞). Se in aggiunta si ipotizzaf′>0 in (0, ∞), oppure 0<β=≤1, l'unicità délia soluzione è assicurata.
Successivamente si considéra il problema di valori al contorno conα=1,β<0,γ arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano −1<f′<0 in un intorno dell'origine e tali chef′>1, in generale.
Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref″(0).
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Gabutti, B. On the equation of similar profiles. Z. angew. Math. Phys. 35, 265–281 (1984). https://doi.org/10.1007/BF00944877
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DOI: https://doi.org/10.1007/BF00944877