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 inℝ3is 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 ofO(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.
Highlights
It is obvious that a vast number of problems in physics, mechanics, and technology is modelled through linear and nonlinear partial differential equations (PDEs, equations of mathematical physics)
It is worth noticing that the proposed novel approach becomes four times smaller, which enables a considerable decrease in the volume of the computer operating storage used to keep coordinates of points Xk
We briefly model the problem of stationary heat distribution in a certain volume G with surface Γ of three-dimensional space x = (x1, x2, x3)
Summary
It is obvious that a vast number of problems in physics, mechanics, and technology is modelled through linear and nonlinear partial differential equations (PDEs, equations of mathematical physics). It is worth noticing that the proposed novel approach becomes four times smaller, which enables a considerable decrease in the volume of the computer operating storage used to keep coordinates of points Xk. Consider SLAE II with a band matrix:. Solving the system of equations considered in [6, page 61], for the matrix conditioned by ω = 4.7 · 105, the proposed method gives exact decimal digits, that is, of one order less than the Gauss method This result is obtained owing to a particular symmetry of the fundamental computational formulas (2.13), (2.14) and a homogeneity in computations of all unknowns. The discussed algorithm for solving SLAE with a band matrix is suitable for solving boundary value problems using the method of finite differences in a rectangular parallelepiped since the matrix of SLAE possesses a band with a regular structure The latter property allows the use of one and only one set I for all equations.
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