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

Jan Awrejcewicz, Vadim A. Krysko, Anton V. Krysko

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 languageEnglish
Pages (from-to)377-410
Number of pages34
JournalMathematical Problems in Engineering
Volume2004
Issue number4
DOIs
Publication statusPublished - 28 Oct 2004
Externally publishedYes

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Linear equations
Algebraic Equation
Boundary value problems
Linear equation
Shell
Newton-Raphson method
Boundary element method
Boundary Value Problem
Linearization
Nonlinear equations
Partial differential equations
Parallelepiped
Nonlinear Algebraic Equations
Differential equations
Exact Method
Boundary conditions
Heat transfer
Newton Methods
Hyperplane
Boundary Elements

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 journalArticle

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