Estimation of the quality of approximate solutions of a system of linear algebraic equations

1965 ◽  
Vol 5 (1) ◽  
pp. 151-154
Author(s):  
A.Ya. Belostotskii
Keyword(s):  
Approximate Solutions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

scholarly journals Solving of Linear Volterra-Fredholm Integral Equations via Modification of Block Pulse Functions

2021 ◽  
Vol 17 (1) ◽  
pp. 33
Author(s):  
Ayyubi Ahmad
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Triangular System ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Linear Algebraic Equations

A computational method based on modification of block pulse functions is proposed for solving numerically the linear Volterra-Fredholm integral equations. We obtain integration operational matrix of modification of block pulse functions on interval [0,T). A modification of block pulse functions and their integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of  linear Volterra-Fredholm integral equations. Furthermore, the rate of convergence is  O(h) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

Solutions of a singular integro-differential equation related to the adhesive contact problems of elasticity theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nugzar Shavlakadze ◽  
Otar Jokhadze
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Contact Problems ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Algebraic Equations ◽  
Adhesive Interaction ◽  
Normal Contact ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations

Abstract Exact and approximate solutions of a some type singular integro-differential equation related to problems of adhesive interaction between elastic thin half-infinite or finite homogeneous patch and elastic plate are investigated. For the patch loaded with vertical forces, there holds a standard model in which vertical elastic displacements are assumed to be constant. Using the theory of analytic functions, integral transforms and orthogonal polynomials, the singular integro-differential equation is reduced to a different boundary value problem of the theory of analytic functions or to an infinite system of linear algebraic equations. Exact or approximate solutions of such problems and asymptotic estimates of normal contact stresses are obtained.

scholarly journals A MUNTZ-LEGENDRE APPROACH TO OBTAIN SOLUTIONS OF SINGULAR PERTURBED PROBLEMS

2020 ◽  
Vol 20 (3) ◽  
pp. 537-544
Author(s):  
SUAYIP YUZBASI ◽  
EMRAH GOK ◽  
MEHMET SEZER
Keyword(s):  
Error Function ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Algebraic Equations ◽  
Residual Function ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Collocation Approach ◽  
Estimated Error

Singularly perturbed differential equations are encountered in mathematical modelling of processes in physics and engineering. Aim of this study is to give a collocation approach for solutions of singularly perturbed two-point boundary value problems. The method provides obtaining the approximate solutions in the form of Müntz-Legendre polynomials by using collocation points and matrix relations. Singularly perturbed problem is transformed into a system of linear algebraic equations. By solving this system, the approximate solution is computed. Also, an error estimation is done using the residual function and the approximate solutions are improved by means of the estimated error function. Two numerical examples are given to show the applicability of the method.

scholarly journals An Efficient Algorithm for Solving Integro-Differential Equation

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Hassan Hamad AL-Nasrawy ◽  
Abdul Khaleq O. Al-Jubory ◽  
Kasim Abbas Hussaina
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Difference Equations ◽  
Matrix Equation ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Approximate Methods ◽  
Linear Algebraic Equations ◽  
Linear Delay

In this paper, we study and modify an approximate method as well as a new collocation method, which is based on orthonormal Bernstein polynomials to find approximate solutions of mixed linear delay Fredholm integro-differential-difference equations under the mixed conditions. The main purpose of this paper is to study and develop some approximate methods to solve the mixed linear delay Fredholm integro-differential-difference equations. We employ a new algorithm to find approximate solution via perpendicular Bernstein polynomials on the interval [0,1], and we construct a new matrix of derivatives that will be used to find an approximate solution of matrix equation, that will reduce it to the systems of linear algebraic equations. We study the convergence approximate solutions to the exact solutions. Finally, two examples are given and their results are shown in figures to illustrate the efficiency and accuracy of this method. All the computations are implemented using Math14.

Approximate Solutions for Solving Fractional-order Painlevé Equations

2019 ◽  
Vol 1 (1) ◽  
pp. 12-24 ◽  
Author(s):  
Mohammad Izadi
Keyword(s):  
Approximate Solutions ◽  
Painlevé Equations ◽  
Algebraic Equations ◽  
Initial Value ◽  
Painleve Equations ◽  
Matrix Operations ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Collocation Scheme

In this work, Chebyshev orthogonal polynomials are employed as basis functions in the collocation scheme to solve the nonlinear Painlevé initial value problems known as the first and second Painlevé equations. Using the collocation points, representing the solution and its fractional derivative (in the Caputo sense) in matrix forms, and the matrix operations, the proposed technique transforms a solution of the initial-value problem for the Painlevé equations into a system of nonlinear algebraic equations. To get ride of nonlinearlity, the technique of quasi-linearization is also applied, which converts the equations into a sequence of linear algebraic equations. The accuracy and efficiency of the presented methods are investigated by some test examples and a comparison has been made with some existing available numerical schemes.

A shifted Legendre method for solving a population model and delay linear Volterra integro-differential equations

2017 ◽  
Vol 10 (07) ◽  
pp. 1750091 ◽  
Author(s):  
Şuayip Yüzbaşi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Matrix Equation ◽  
Population Model ◽  
Estimation Method ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations ◽  
The Matrix

In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polynomials. By using the required matrix operations and collocation points, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. The matrix equation corresponds to a system of linear algebraic equations. Also, an error estimation method for method and improvement of solutions is presented by using the residual function. Applications of population model and general delay integro-differential equation are given. The obtained results are compared with the known results.

scholarly journals Guaranteed a posteriori error bounds for low-rank tensor approximate solutions

2020 ◽  
Author(s):  
Sergey Dolgov ◽  
Tomáš Vejchodský
Keyword(s):  
Total Error ◽  
Approximation Error ◽  
Approximate Solutions ◽  
Posteriori Error ◽  
Energy Norm ◽  
Low Rank ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Rank Tensor ◽  
Linear Algebraic Equations

Abstract We propose a guaranteed and fully computable upper bound on the energy norm of the error in low-rank tensor train (TT) approximate solutions of (possibly) high-dimensional reaction–diffusion problems. The error bound is obtained from Euler–Lagrange equations for a complementary flux reconstruction problem, which are solved in the low-rank TT representation using the block alternating linear scheme. This bound is guaranteed to be above the energy norm of the total error, including the discretization error, the tensor approximation error and the error in the solver of linear algebraic equations, although quadrature errors, in general, can pollute its evaluation. Numerical examples with the Poisson equation and the Schrödinger equation with the Henon–Heiles potential in up to 40 dimensions are presented to illustrate the efficiency of this approach.

scholarly journals Approximate Solutions of Differential Equations by Using the Bernstein Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Y. Ordokhani ◽  
S. Davaei far
Keyword(s):  
Differential Equations ◽  
Bernstein Polynomial ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Bernstein Polynomial Basis

A numerical method for solving differential equations by approximating the solution in the Bernstein polynomial basis is proposed. At first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derive the operational matrices of integration and product of the Bernstein polynomials. Then, we employ them for solving differential equations. The method converts the differential equation to a system of linear algebraic equations. Finally some examples and their numerical solutions are given; comparing the results with the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

scholarly journals A numerical method for solving systems of higher order linear functional differential equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 15-25 ◽  
Author(s):  
Suayip Yüzbasi ◽  
Emrah Gök ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Legendre Polynomials ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Functional Differential

AbstractFunctional differential equations have importance in many areas of science such as mathematical physics. These systems are difficult to solve analytically.In this paper we consider the systems of linear functional differential equations [1-9] including the term y(αx + β) and advance-delay in derivatives of y .To obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. For this purpose, to obtain the approximate solutions of those systems, we present a matrix-collocation method by using Müntz-Legendre polynomials and the collocation points. This method transform the problem into a system of linear algebraic equations. The solutions of last system determine unknown co-efficients of original problem. Also, an error estimation technique is presented and the approximate solutions are improved by using it. The program of method is written in Matlab and the approximate solutions can be obtained easily. Also some examples are given to illustrate the validity of the method.

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