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

• Main Authors: Awrejcewicz Jan, Krysko Vadim A., Krysko Anton V.
• Format: Article
• Language: English
• Published: Hindawi Limited 2004-01-01
• Series: Mathematical Problems in Engineering
• Online Access: http://www.hindawi.net/access/get.aspx?journal=mpe&volume=2004&pii=S1024123X04403093

<p>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 <math alttext="$mathbb{R}^3$"> <mrow> <msup> <mi>&Ropf;</mi> <mn>3</mn> </msup> </mrow> </math> 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 <math alttext="$O(h_{x_1}^2 +h_{x_2}^2)$"> <mrow> <mi>O</mi><mrow><mo>(</mo> <mrow> <msubsup> <mi>h</mi> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </msubsup> <mo>+</mo><msubsup> <mi>h</mi> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </msubsup> </mrow> <mo>)</mo></mrow> </mrow> </math>. 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.</p>