### Abstract

The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ^{3} is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(h_{x1}^{2} + h_{x2}^{2}). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

Original language | English |
---|---|

Pages (from-to) | 377-410 |

Number of pages | 34 |

Journal | Mathematical Problems in Engineering |

Volume | 2004 |

Issue number | 4 |

DOIs | |

Publication status | Published - 28 Oct 2004 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

*Mathematical Problems in Engineering*,

*2004*(4), 377-410. https://doi.org/10.1155/S1024123X04403093

**On the economical solution method for a system of linear algebraic equations.** / Awrejcewicz, Jan; Krysko, Vadim A.; Krysko, Anton V.

Research output: Contribution to journal › Article

*Mathematical Problems in Engineering*, vol. 2004, no. 4, pp. 377-410. https://doi.org/10.1155/S1024123X04403093

}

TY - JOUR

T1 - On the economical solution method for a system of linear algebraic equations

AU - Awrejcewicz, Jan

AU - Krysko, Vadim A.

AU - Krysko, Anton V.

PY - 2004/10/28

Y1 - 2004/10/28

N2 - The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12 + hx22). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

AB - The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped in ℝ3 is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error of O(hx12 + hx22). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

UR - http://www.scopus.com/inward/record.url?scp=12444333515&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=12444333515&partnerID=8YFLogxK

U2 - 10.1155/S1024123X04403093

DO - 10.1155/S1024123X04403093

M3 - Article

VL - 2004

SP - 377

EP - 410

JO - Mathematical Problems in Engineering

JF - Mathematical Problems in Engineering

SN - 1024-123X

IS - 4

ER -