A numerical method on the mixed solution of matrix equation ∑i=1tAiXiBi=E with sub-matrix constraints and its application

In this paper, we use the collocation method together with Chebyshev polynomials to solve system of Lane–Emden type (SLE) equations. We first transform the given SLE equation to a matrix equation by means of a truncated Chebyshev series with unknown coefficients. Then, the numerical method reduces each SLE equation to a nonlinear system of algebraic equations. The solution of this matrix equation yields the unknown coefficients of the solution function. Hence, an approximate solution is obtained by means of a truncated Chebyshev series. Also, to show the applicability, usefulness, and accuracy of the method, some examples are solved numerically and the errors of the solutions are compared with existing solutions.

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Two-stage Algorithm for solving Arbitrary Trapezoidal Fully Fuzzy Sylvester Matrix Equations

10.20944/preprints202112.0147.v1 ◽

2021 ◽

Author(s):

Ahmed AbdelAziz Elsayed ◽

Bassem saassouh ◽

Nazihah Ahmad ◽

Ghassan Malkawi

Keyword(s):

Numerical Method ◽

Matrix Equation ◽

Fuzzy Numbers ◽

Linear Equations ◽

Sylvester Matrix Equation ◽

Two Stage ◽

Trapezoidal Fuzzy Numbers ◽

Sylvester Matrix ◽

Sylvester Matrix Equations ◽

Fuzzy Solutions

Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.

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Examination of Extra Electron Diffraction Spots Observed in Deuterium-Irradiated Silicon by Using Parallel Electron Beam Illumination

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100136052 ◽

1990 ◽

Vol 48 (2) ◽

pp. 490-491

Author(s):

Ryuichiro Oshima ◽

Shoichiro Honda ◽

Tetsuo Tanabe

Keyword(s):

Electron Microscopy ◽

Transmission Electron Microscopy ◽

Silicon Single Crystal ◽

Mixed Solution ◽

Parallel Beam ◽

Defect Clusters ◽

Float Zone ◽

Transmission Electron ◽

Extra Electron ◽

Diffraction Patterns

In order to examine the origin of extra diffraction spots and streaks observed in selected area diffraction patterns of deuterium irradiated silicon, systematic diffraction experiments have been carried out by using parallel beam illumination.Disc specimens 3mm in diameter and 0.5mm thick were prepared from a float zone silicon single crystal(B doped, 7kΩm), and were chemically thinned in a mixed solution of nitric acid and hydrogen fluoride to make a small hole at the center for transmission electron microscopy. The pre-thinned samples were irradiated with deuterium ions at temperatures between 300-673K at 20keV to a dose of 1022ions/m2, and induced lattice defects were examined under a JEOL 200CX electron microscope operated at 160kV.No indication of formation of amorphous was obtained in the present experiments. Figure 1 shows an example of defects induced by irradiation at 300K with a dose of 2xl021ions/m2. A large number of defect clusters are seen in the micrograph.

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