### Abstract

In this paper, we present a new numerical method for the solution of linear two‐point boundary value problems of ordinary differential equations. After reducing the differential equation to a second kind integral equation, we discretize the latter via a high order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O (N·p^{2}) operations, where N is the number of nodes on the interval and p is the desired order of convergence. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end‐point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.

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

Pages (from-to) | 419-452 |

Number of pages | 34 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 44 |

Issue number | 4 |

DOIs | |

State | Published - 1991 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*44*(4), 419-452. https://doi.org/10.1002/cpa.3160440403

**On the numerical solution of two‐point boundary value problems.** / Greengard, Leslie; Rokhlin, V.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 44, no. 4, pp. 419-452. https://doi.org/10.1002/cpa.3160440403

}

TY - JOUR

T1 - On the numerical solution of two‐point boundary value problems

AU - Greengard, Leslie

AU - Rokhlin, V.

PY - 1991

Y1 - 1991

N2 - In this paper, we present a new numerical method for the solution of linear two‐point boundary value problems of ordinary differential equations. After reducing the differential equation to a second kind integral equation, we discretize the latter via a high order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O (N·p2) operations, where N is the number of nodes on the interval and p is the desired order of convergence. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end‐point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.

AB - In this paper, we present a new numerical method for the solution of linear two‐point boundary value problems of ordinary differential equations. After reducing the differential equation to a second kind integral equation, we discretize the latter via a high order Nyström scheme. A somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using O (N·p2) operations, where N is the number of nodes on the interval and p is the desired order of convergence. Thus, the advantages of the integral equation formulation (small condition number, insensitivity to boundary layers, insensitivity to end‐point singularities, etc.) are retained, while achieving a computational efficiency previously available only to finite difference or finite element methods.

UR - http://www.scopus.com/inward/record.url?scp=84990556194&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84990556194&partnerID=8YFLogxK

U2 - 10.1002/cpa.3160440403

DO - 10.1002/cpa.3160440403

M3 - Article

AN - SCOPUS:84990556194

VL - 44

SP - 419

EP - 452

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 4

ER -