×

A nonlinear Bessel differential equation associated with Cauchy conditions. (English) Zbl 0855.34007

This paper is concerned with the existence and uniqueness of solutions of Bessel’s nonlinear differential equation with Cauchy conditions, \(y''(x) + ({1 \over x}) y'(x) + y(x) = f(y(x))\), \(x > 0\), \(y(0) = 1\), \(y'(0) = y_0'\), where \(f \in C^1 (\mathbb{R},\mathbb{R})\), \(f(0) = 0\).
By using a fixed point technique, the existence and uniqueness of solutions of Bessel’s nonlinear problem is proved. In the case \(f(y) = y^2\), a case of some interest in applications, the authors show that \(y(x)\), the unique solution of Bessel’s nonlinear differential equation, together with \(y'(x)\) and \(y''(x)\) tend to zero as \(x \to \infty\). At last, a numerical example of the above Cauchy problem on a very large interval is presented.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34D05 Asymptotic properties of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Synge, J. L., On a certain nonlinear differential equation, (Proc. Royal Irish Acad., 62 (1962)), 17-412 · Zbl 0104.31501
[2] Nehari, Z., On a nonlinear differential equation arising in nuclear physics, (Proc. Royal Irish Acad., 62 (1962)), 117-135 · Zbl 0124.30204
[3] Onumanyi, P.; Ortiz, E. L.; Samara, H., Software for a method of finite approximations for the numerical solution of differential equations, Appl. Math. Modelling, 5, 117-135 (1981) · Zbl 0464.65055
[4] Ortiz, E. L., The tau method, SIAM J. Numer. Anal., 6, 480-492 (1969) · Zbl 0195.45701
[5] Pham Ngoc Dinh, A., Existence and uniqueness of solution of Bessel’s nonlinear differential equation, Mathl. Comput. Modelling, 11, 676-678 (1988678)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.