scholarly journals A New Numerical Technique for Solving Volterra Integral Equations Using Chebyshev Spectral Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Integral Equations ◽  
Numerical Technique ◽  
Volterra Integral Equations ◽  
New Method ◽  
Algebraic Equations ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Method ◽  
Chebyshev Spectral Collocation Method

In this work, we propose a new method for solving Volterra integral equations. The technique is based on the Chebyshev spectral collocation method. The application of the proposed method leads Volterra integral equation to a system of algebraic equations that are easy to solve. Some examples are presented and compared with some methods in the literature to illustrate the ability of this technique. The results demonstrate that the new method is more efficient, convergent, and accurate to the exact solution.

scholarly journals THE METHOD OF NUMERICAL SOLUTION OF NONLINEAR VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND

2018 ◽  
pp. 10-18
Author(s):  
Karakeev T.T. ◽  
Mustafayeva N.T.
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Numerical Methods ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Initial Point ◽  
Nonlinear Problems ◽  
Volterra Integral Equations ◽  
Algebraic Equations

When considering systems of differential equations with very general boundary conditions, exact solution methods encounter great difficulties, which become insurmountable in the study of nonlinear problems. In this case it is necessary to apply to certain numerical methods. It is important to note that the use of numerical methods often allows you to abandon the simplified interpretation of the mathematical model of the process. The problems of numerical solution of nonlinear Volterra integral equations of the first kind with a differentiable kernel, which degenerates at the initial point of the diagonal, are studied in the paper. This equation is reduced to the Volterra integral equation of the third kind and a numerical method is developed on the basis of that regularized equation. The convergence of the numerical solution to the exact solution of the Volterra integral equation of the first kind is proved, an estimate of the permissible error and a recursive formula of the computational process are obtained. Keywords: nonlinear integral equation, system of nonlinear algebraic equations, error vectors, the Volterra equation, small parameter, numerical methods.

scholarly journals Numerical solution of linear stochastic Volterra integral equations via new basis functions

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5959-5966
Author(s):  
Tofigh Cheraghi ◽  
Morteza Khodabin ◽  
Reza Ezzati
Keyword(s):  
Linear System ◽  
Numerical Solution ◽  
Integral Equations ◽  
Orthogonal Basis ◽  
Volterra Integral Equations ◽  
Basis Functions ◽  
New Method ◽  
Algebraic Equations ◽  
Numerical Examples

In this article, we use a new method based on orthogonal basis functions for the numerical solution of stochastic Volterra integral equations of the second kind (SVIE). By using this method, a SVIE can be reduced to a linear system of algebraic equations. Finally, to show the efficiency of the proposed method, we give two numerical examples.

scholarly journals Numerical Solution of System of Three Nonlinear Volterra Integral Equations Using Implicit Trapezoidal

2017 ◽  
Vol 10 (1) ◽  
pp. 44
Author(s):  
Dalal Adnan Maturi ◽  
Honaida Mohammed Malaikah
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Volterra Integral Equations

In this project, we will be find numerical solution of Volterra Integral Equation of Second kind through using Implicit trapezoidal and that by using Maple 17 program, then we found that numerical solution was highly accurate when it was compared with exact solution.

scholarly journals Application of the Bernstein polynomials for solving Volterra integral equations with convolution kernels

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1045-1052 ◽  
Author(s):  
Ahmet Altürk
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Bernstein Polynomials ◽  
Simple Algorithm ◽  
Volterra Integral Equations ◽  
Convolution Product ◽  
Algebraic Equations ◽  
Convolution Kernels ◽  
The Difference ◽  
Linear Volterra Integral Equations

In this article, we consider the second-type linear Volterra integral equations whose kernels based upon the difference of the arguments. The aim is to convert the integral equation to an algebraic one. This is achieved by approximating functions appearing in the integral equation with the Bernstein polynomials. Since the kernel is of convolution type, the integral is represented as a convolution product. Taylor expansion of kernel along with the properties of convolution are used to represent the integral in terms of the Bernstein polynomials so that a set of algebraic equations is obtained. This set of algebraic equations is solved and approximate solution is obtained. We also provide a simple algorithm which depends both on the degree of the Bernstein polynomials and that of monomials. Illustrative examples are provided to show the validity and applicability of the method.

Chebyshev spectral-collocation method for a class of weakly singular Volterra integral equations with proportional delay

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
Z. Gu ◽  
Y. Chen
Keyword(s):  
Error Analysis ◽  
Integral Equations ◽  
Collocation Method ◽  
Volterra Integral Equations ◽  
Spectral Collocation Method ◽  
Proportional Delay ◽  
Spectral Collocation ◽  
Weakly Singular ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Collocation Method

Abstract-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence.

Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Harendra Singh
Keyword(s):  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
High Accuracy ◽  
Elliptic Boundary Value Problems ◽  
Strong Nonlinearity ◽  
Elliptic Boundary Value ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Method ◽  
Chebyshev Spectral Collocation Method

Abstract This paper deals with a class of Bratu’s type, Troesch’s and nonlocal elliptic boundary value problems. Due to strong nonlinearity and presence of parameter δ, it is very difficult to solve these problems. Here we solve these classes of important equations using the Chebyshev spectral collocation method. We have provided the convergence of the proposed approximate method. The trueness of the method is shown by applying it to some illustrative examples. Results are compared with some known methods to highlight its neglectable error and high accuracy.

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