New methods for boundary value problems

Abstract

Nonlinear two-point boundary value problems abound in the theory of the urine concentrating mechanism. Systems of order 50 are commonly considered. Various approaches to these problems have been employed including quasilinearization. Three difficulties have been noted with quasilinearization: 1) Large Jacobian matrices need to be evaluated at each integration step. 2) A large amount of memory is needed to store the temporary particular and complementary solution values. 3) The differential equations may be stiff and therefore difficult to integrate.A recent conceptual advance has made possible the automatic and exact evaluation of the gradient functions. This advance makes coding of the derivatives less prone to error and enables the derivatives to be evaluated efficiently. The memory constraint is overcome by storing the temporary function values on the disk. Since access to these values is sequential, the amount of time spent storing and retrieving the values from disk is minimal. The difficulty in integrating the differential equations can be solved by using an invariant imbedding technique, which has proven useful in the past for solving stiff counterflow problems.