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 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.
| 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
On the economical solution method for a system of linear algebraic equations. / Awrejcewicz, Jan; Krysko, Vadim A.; Krysko, Anton V.
In: Mathematical Problems in Engineering, Vol. 2004, No. 4, 28.10.2004, p. 377-410.Research output: Contribution to journal › Article
}
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.
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U2 - 10.1155/S1024123X04403093
DO - 10.1155/S1024123X04403093
M3 - Article
AN - SCOPUS:12444333515
VL - 2004
SP - 377
EP - 410
JO - Mathematical Problems in Engineering
JF - Mathematical Problems in Engineering
SN - 1024-123X
IS - 4
ER -