A complete asymptotic analysis of an oscillation free nonlinear equation of Bessel type with a pole in the dependent variable

• Дата: 2010
• Автори: Mingarelli, A.B., Pacheco-Castelao, J.M., Melkonian, S.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2010
• Назва видання: Нелінійні коливання
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/174926
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: A complete asymptotic analysis of an oscillation free nonlinear equation of Bessel type with a pole in the dependent variable / A.B. Mingarelli, J.M. Pacheco-Castelao, S. Melkonian // Нелінійні коливання. — 2010. — Т. 13, № 2. — С. 206-239. — Бібліогр.: 31 назв. — англ.

We characterize the solution set of a nonlinear perturbation of Bessel’s equation of order zero on a half- line where the nonlinearity is analytic in the independent variable, algebraic in the dependent variable and, indeed, admits a pole in this variable. We show that the equation fails the Painleve´ test and that there are no points in [0,∞) where blow-up occurs. Although we cannot find even one closed-form solution, it is shown that there are only four families of solutions: those that are asymptotically linear and increasing, solutions that are asymptotically linear and decreasing, another set of solutions that are asymptotically constant, and a final set of solutions that admit singularities at finite points on [0,∞). As a consequence, we deduce that every solution with or without singularities on [0,∞) is non-oscillatory and, in fact, has at most two zeros. We also show that the plane Π of real initial conditions (y(0), y'(0)) can be decomposed into a union of connected regions, in each of which the solutions are exactly one of the types mentioned above. Furthermore, we obtain that the set of those initial conditions leading to asymptotically constant solutions is a piecewise differentiable curve in Π, one that can be estimated theoretically to a high degree of precision. In addition, the asymptotic behavior of solutions near a finite singularity is obtained. Esti- mates relating the growth of solutions to their initial conditions are also described and numerical examples are presented to illustrate the theory. Finally, we observe that every solution of our equation has finite si- ngularities when viewed as a solution on the whole line.