Bounds for the solution of certain nonlinear boundary value problems of sturm-liouville type

Abstract

Upper and lower pointwise bounds are presented for the solution of certain nonlinear Sturm-Liouville problems. Our method requires an inclusion domain for the solution and an integral equation formulation in terms of a nonlinear Hammerstein equation with associated contraction operator. Several different sets of bounds are discussed, including a simple first order set and two new second order sets. The Barnsley-Robinson embedding procedure is used as a basis for the second order bounds. The relative merits and efficiency of the various bounding results are examined in calculations for a problem arising in the theory of electric current flow in excitable cells, in which estimates are obtained for the current and the potential at a point.

Keywords:

• Sturm-Liouville problem
• Hammerstein equation
• Inclusion domain
• Green's function
• Hilbert-Schmidt
• Compact operator
• Ridge regression
• Contraction operator
• Pointwise bounds
• Bivariational method cell membrane problem
• Jacobian elliptic function.

^∗Present address: Department of Electronics, University of York, Heslington, York, U.K.

^∗Present address: Department of Electronics, University of York, Heslington, York, U.K.

Notes

^∗Present address: Department of Electronics, University of York, Heslington, York, U.K.

Additional information

Notes on contributors

A.M. Arthurs