### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 855-857 |

Number of pages | 3 |

Journal | Mathematical and Computer Modelling |

Volume | 11 |

Issue number | C |

DOIs | |

State | Published - 1988 |

Externally published | Yes |

### Fingerprint

### Keywords

- automatic differentiation
- invariant imbedding
- Nonlinear boundary value problems
- quasilinearization

### ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Information Systems and Management
- Control and Systems Engineering
- Applied Mathematics
- Computational Mathematics
- Modeling and Simulation

### Cite this

*Mathematical and Computer Modelling*,

*11*(C), 855-857. https://doi.org/10.1016/0895-7177(88)90614-0

**New methods for boundary value problems.** / Kalaba, Robert E.; Wexler, Anthony S.

Research output: Contribution to journal › Article

*Mathematical and Computer Modelling*, vol. 11, no. C, pp. 855-857. https://doi.org/10.1016/0895-7177(88)90614-0

}

TY - JOUR

T1 - New methods for boundary value problems

AU - Kalaba, Robert E.

AU - Wexler, Anthony S.

PY - 1988

Y1 - 1988

N2 - 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.

AB - 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.

KW - automatic differentiation

KW - invariant imbedding

KW - Nonlinear boundary value problems

KW - quasilinearization

UR - http://www.scopus.com/inward/record.url?scp=45549113461&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45549113461&partnerID=8YFLogxK

U2 - 10.1016/0895-7177(88)90614-0

DO - 10.1016/0895-7177(88)90614-0

M3 - Article

VL - 11

SP - 855

EP - 857

JO - Mathematical and Computer Modelling

JF - Mathematical and Computer Modelling

SN - 0895-7177

IS - C

ER -