### 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 |
---|---|

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.

}

_{xx}- ∈U

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

^{4}) finite difference method',

*Numerical Methods for Partial Differential Equations*, vol. 16, no. 4, pp. 395-407.

**Solving the equation - u _{xx} - ∈U_{yy} = f(x, y, u) by an O(h^{4}) finite difference method.** / Lungu, E.; Motsumi, T.; Styś, Tadeusz.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Solving the equation - uxx - ∈Uyy = f(x, y, u) by an O(h4) finite difference method

AU - Lungu, E.

AU - Motsumi, T.

AU - Styś, Tadeusz

PY - 2000/7

Y1 - 2000/7

N2 - The semi-linear equation - uxx - ∈uyy = f(x,y,u) with Dirichlet boundary conditions is solved by an O(h4) finite difference method, which has local truncation error O(h2) at the mesh points neighboring the boundary and O(h4) at most interior mesh points. It is proved that the finite difference method is O(h4) 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.

AB - The semi-linear equation - uxx - ∈uyy = f(x,y,u) with Dirichlet boundary conditions is solved by an O(h4) finite difference method, which has local truncation error O(h2) at the mesh points neighboring the boundary and O(h4) at most interior mesh points. It is proved that the finite difference method is O(h4) 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.

UR - http://www.scopus.com/inward/record.url?scp=0041688077&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041688077&partnerID=8YFLogxK

M3 - Article

VL - 16

SP - 395

EP - 407

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 4

ER -

_{xx}- ∈U

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

^{4}) finite difference method. Numerical Methods for Partial Differential Equations. 2000 Jul;16(4):395-407.