scholarly journals On the numerical solution of system of linear algebraic equations with ill-conditioned matrices

2019 ◽  
Vol 6(64) (4) ◽  
pp. 619-626
Author(s):  
Anastasia V. Lebedeva ◽  
◽  
Victor M. Ryabov ◽  
Keyword(s):  
Numerical Solution ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Ill Conditioned Matrices

scholarly journals Solving Some Differential Equations Arising in Electric Engineering Using Hermite Polynomials

2020 ◽  
Vol 12 (4) ◽  
pp. 517-523
Author(s):  
G. Singh ◽  
I. Singh
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Hermite Polynomials ◽  
Electric Circuit ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Electric Engineering ◽  
Collocation Points ◽  
Linear Algebraic Equations

In this paper, a collocation method based on Hermite polynomials is presented for the numerical solution of the electric circuit equations arising in many branches of sciences and engineering. By using collocation points and Hermite polynomials, electric circuit equations are transformed into a system of linear algebraic equations with unknown Hermite coefficients. These unknown Hermite coefficients have been computed by solving such algebraic equations. To illustrate the accuracy of the proposed method some numerical examples are presented.

scholarly journals Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Qingxue Huang ◽  
Fuqiang Zhao ◽  
Jiaquan Xie ◽  
Lifeng Ma ◽  
Jianmei Wang ◽  
...  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Differential Equations ◽  
Operational Matrix ◽  
Numerical Approach ◽  
Principal Characteristic ◽  
Legendre Functions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.

scholarly journals Numerical solution of a weakly singular integral equation by the method of orthogonal polynomials

2021 ◽  
pp. 98-103
Author(s):  
Sergei M. Sheshko
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Singular Integral ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
Analytical Expressions

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

scholarly journals Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Khadijeh Sadri ◽  
Kamyar Hosseini ◽  
Dumitru Baleanu ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Polynomials ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Variable Order ◽  
Algebraic Equations ◽  
Weakly Singular Kernel ◽  
Weakly Singular ◽  
Linear Algebraic Equations

AbstractThe shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

scholarly journals Stochastic model for multi-term time-fractional diffusion equations with noise

2021 ◽  
Vol 25 (Spec. issue 2) ◽  
pp. 287-293
Author(s):  
Vahid Hosseini ◽  
Mohamad Remazani ◽  
Wennan Zou ◽  
Seddigheh Banihashemii
Keyword(s):  
Numerical Solution ◽  
Numerical Approach ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Data Driven ◽  
Algebraic Equations ◽  
Fractional Diffusion Equations ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Collocation Approach

This paper studies a spectral collocation approach for evaluating the numerical solution of the stochastic multi-term time-fractional diffusion equations associated with noisy data driven by Brownian motion. This model describes the symmetry breaking in molecular vibrations. The numerical solution of the stochastic multi-term time-fractional diffusion equations is proposed by means of collocation points method based on sixth-kind Chebyshev polynomial approach. For this purpose, the problem under consideration is reduced to a system of linear algebraic equations. Two examples highlight the robustness and accuracy of the proposed numerical approach.

scholarly journals Sound Transmission in a Duct with Sudden Area Expansion, Extended Inlet, and Lined Walls in Overlapping Region

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Ahmet Demir
Keyword(s):  
Fourier Transform ◽  
Numerical Solution ◽  
Sound Transmission ◽  
Algebraic Equations ◽  
Hopf Equation ◽  
Linear Algebraic Equations ◽  
Expansion Coefficients ◽  
The Fourier Transform ◽  
Area Expansion ◽  
Overlapping Region

The transmission of sound in a duct with sudden area expansion and extended inlet is investigated in the case where the walls of the duct lie in the finite overlapping region lined with acoustically absorbent materials. By using the series expansion in the overlap region and using the Fourier transform technique elsewhere we obtain a Wiener-Hopf equation whose solution involves a set of infinitely many unknown expansion coefficients satisfying a system of linear algebraic equations. Numerical solution of this system is obtained for various values of the problem parameters, whereby the effects of these parameters on the sound transmission are studied.

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