A survey on shooting arguments for boundary value problems (Q2475001)

This is an expository paper devoted to the applications of the shooting method to some boundary value problem. The authors first develop the method for the heat equation with absorption \[ u''+\left(\frac{N-1}{x}+\frac{x}{2}\right)u'+\frac{\alpha}{2}u-u^p=0,\quad 0<x<\infty \] subject to the boundary conditions \[ u'(0)=0,\,u>0,\;u(\infty)=0 \] where \(p>1\) and \(\alpha>N.\) Then the following modified Burgers equation is discussed \[ f''+2\eta f'+\frac{2}{\alpha}f-2^{\frac{3}{2}}f^\alpha f'=0 \] together with the conditions \[ f'(0)=0,\;\lim_{\eta\to\pm\infty}f(\eta)=0\; \text{ and }\, f>0\,\text{ on }\,(-\infty,\infty). \] Finally, the authors give an account of some other related works. With 24 references, the paper presents a good survey of the classical shooting method.