Numerical solution of some systems of nonlinear algebraic equations

2021 ◽  
pp. 1-20
Author(s):  
Stefan M. Stefanov
Keyword(s):  
Numerical Solution ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations

scholarly journals Numerical solution of two-dimensional nonlinear fractional order reaction-advection-diffusion equation by using collocation method

2021 ◽  
Vol 29 (2) ◽  
pp. 211-230
Author(s):  
Manpal Singh ◽  
S. Das ◽  
Rajeev ◽  
E-M. Craciun
Keyword(s):  
Numerical Solution ◽  
Collocation Method ◽  
Operational Matrix ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Existing Problems ◽  
Advection Diffusion Equation ◽  
Error Analyses ◽  
Nonlinear Algebraic Equations ◽  
Accurate Numerical Solution

Abstract In this article, two-dimensional nonlinear and multi-term time fractional diffusion equations are solved numerically by collocation method, which is used with the help of Lucas operational matrix. In the proposed method solutions of the problems are expressed in terms of Lucas polynomial as basis function. To determine the unknowns, the residual, initial and boundary conditions are collocated at the chosen points, which produce a system of nonlinear algebraic equations those have been solved numerically. The concerned method provides the highly accurate numerical solution. The accuracy of the approximate solution of the problem can be increased by expanding the terms of the polynomial. The accuracy and efficiency of the concerned method have been authenticated through the error analyses with some existing problems whose solutions are already known.

Numerical Solution of Fractional Sine-Gordon Equation Using Spectral Method and Homogenization

2019 ◽  
Vol 20 (7-8) ◽  
pp. 811-822
Author(s):  
Maryam Hasanpour ◽  
Mahmoud Behroozifar ◽  
Nazanin Tafakhori
Keyword(s):  
Numerical Solution ◽  
Jacobi Polynomials ◽  
Second Step ◽  
Test Problems ◽  
Algebraic Equations ◽  
Newton’S Iterative Method ◽  
Main Equation ◽  
Dimensional Approximation ◽  
Nonlinear Algebraic Equations ◽  
Sg Equation

AbstractIn this paper, a new method for the numerical solution of fractional sine-Gordon (SG) equation is presented. Our method consists of two steps, in first step: the main equation is converted to a homogeneous one using interpolation. In second step: two-dimensional approximation of functions by shifted Jacobi polynomials is used to reduce the problem into a system of nonlinear algebraic equations. The archived system is solved by Newton’s iterative method. Our method is stated in general case on rectangular [a,b] × [0,T] which is based upon Jacobi polynomial by parameters (α,β). Several test problems are employed and results of numerical experiments are presented and also compared with analytical solutions. Also, we verify the numerical stability of the method, by applying a disturbance in the problem. The obtained results confirm the acceptable accuracy and stability of the presented method.

scholarly journals Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
S. H. Behiry
Keyword(s):  
Numerical Solution ◽  
Numerical Method ◽  
Present Method ◽  
Bernstein Polynomials ◽  
Integrodifferential Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions ◽  
Nonlinear Algebraic Equations

A numerical method for solving nonlinear Fredholm integrodifferential equations is proposed. The method is based on hybrid functions approximate. The properties of hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

scholarly journals Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

2005 ◽  
Vol 12 (6) ◽  
pp. 425-434 ◽  
Author(s):  
Menglin Lou ◽  
Qiuhua Duan ◽  
Genda Chen
Keyword(s):  
Perturbation Method ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Solution Process ◽  
Equation Of Motion ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Shear Distortion

Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

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