### Abstract

The semi-linear equation - u_{xx} - ∈u_{yy} = f(x,y,u) with Dirichlet boundary conditions is solved by an O(h^{4}) finite difference method, which has local truncation error O(h^{2}) at the mesh points neighboring the boundary and O(h^{4}) at most interior mesh points. It is proved that the finite difference method is O(h^{4}) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved.

Original language | English |
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Pages (from-to) | 395-407 |

Number of pages | 13 |

Journal | Numerical Methods for Partial Differential Equations |

Volume | 16 |

Issue number | 4 |

Publication status | Published - Jul 2000 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics
- Computational Mathematics

### Cite this

_{xx}- ∈U

_{yy}= f(x, y, u) by an O(h

^{4}) finite difference method.

*Numerical Methods for Partial Differential Equations*,

*16*(4), 395-407.