Numerical solution of singular nonlinear boundary value problems. (English) Zbl 0843.65055

Baxley, John V.

Bainov, D. (ed.) et al., Proceedings of the third international colloquium on numerical analysis, Plovdiv, Bulgaria, August 13-17, 1994. Utrecht: VSP. 15-24 (1995).

Summary: We discuss numerical procedures for boundary value problems for nonlinear ordinary differential equations with singularities at the endpoints, of the type \[ y''+ {\phi(t)\over y^\lambda}= 0,\quad 0< t< 1,\quad y(0)= 0,\quad y(1)= 0, \] where \(\phi(t)\) is positive and continuous on \(0< t< 1\) and \(\lambda> 0\). Our methods are based on using prior knowledge of existence and uniqueness theorems for such problems as well as knowledge of the asymptotic behavior of the solutions near the endpoints to shoot from (near) the endpoints toward the middle of the interval, adjusting shooting parameters via a modified Newton’s method in order that the solution and its derivative match at the midpoint.
For the entire collection see [Zbl 0835.00028].

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

Keywords:

singular nonlinear boundary value problems; shooting; Newton’s method