scholarly journals New Numerical Approach for Solving Abel’s Integral Equations

2021 ◽  
Vol 46 (3) ◽  
pp. 255-271
Author(s):  
Ayşe Anapalı Şenel ◽  
Yalçın Öztürk ◽  
Mustafa Gülsu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Efficient Method ◽  
Fractional Derivatives ◽  
Numerical Approach ◽  
Main Character ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Taylor Expansions

Abstract In this article, we present an efficient method for solving Abel’s integral equations. This important equation is consisting of an integral equation that is modeling many problems in literature. Our proposed method is based on first taking the truncated Taylor expansions of the solution function and fractional derivatives, then substituting their matrix forms into the equation. The main character behind this technique’s approach is that it reduces such problems to solving a system of algebraic equations, thus greatly simplifying the problem. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. Figures and tables are demonstrated to solutions impress. Also, all numerical examples are solved with the aid of Maple.

scholarly journals Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Berna Bülbül ◽  
Mehmet Sezer
Keyword(s):  
Matrix Method ◽  
Numerical Approach ◽  
Duffing Equation ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

scholarly journals Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

2019 ◽  
Vol 24 (2) ◽  
pp. 176-188 ◽  
Author(s):  
Eid H. Doha ◽  
Mohamed A. Abdelkawy ◽  
Ahmed Z.M. Amin ◽  
Dumitru Baleanu
Keyword(s):  
Chebyshev Polynomials ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Variable Order ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Riccati Differential Equations ◽  
Spectral Technique ◽  
Expansion Coefficients

In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.

APPROXIMATE SOLUTIONS OF NONLINEAR VOLTERRA INTEGRAL EQUATION SYSTEMS

2010 ◽  
Vol 24 (32) ◽  
pp. 6235-6258 ◽  
Author(s):  
SALIH YALÇINBAŞ ◽  
KÜBRA ERDEM
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Nonlinear Volterra Integral Equation ◽  
Solution Function ◽  
Taylor Coefficients ◽  
Linear Algebraic Equations ◽  
New System

The purpose of this study is to implement a new approximate method for solving system of nonlinear Volterra integral equations. The technique is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Some numerical results are also given to illustrate the efficiency of the method.

AN EFFICIENT NUMERICAL APPROACH FOR SOLVING ABEL’S INTEGRAL EQUATIONS BY USING MODIFIED TAYLOR WAVELETS

2021 ◽  
Vol 10 (5) ◽  
pp. 2285-2294
Author(s):  
A. Kumar ◽  
S. R. Verma
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Numerical Approach ◽  
Algebraic Equations ◽  
Wavelet Method

In this paper, a modified Taylor wavelet method (MTWM) is developed for numerical solutions of various types of Abel's integral equations. This method is based on the modified Taylor wavelet (MTW) approximation. The purpose behind using the MTW approximation is to transform the introduction problems into an equivalent set of algebraic equations. To check the accuracy and applicability of the proposed method, some examples have been solved and compared with other existing methods.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

Application of Bernoulli wavelet method to solve a system of fuzzy integral equations

2018 ◽  
Vol 9 (1-2) ◽  
pp. 16-27 ◽  
Author(s):  
Mohamed Abdel- Latif Ramadan ◽  
Mohamed R. Ali
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
Bernoulli Polynomials ◽  
Approximate Solutions ◽  
Fredholm Integral Equations ◽  
Fuzzy Integral ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Bernoulli Wavelet

In this paper, an efficient numerical method to solve a system of linear fuzzy Fredholm integral equations of the second kind based on Bernoulli wavelet method (BWM) is proposed. Bernoulli wavelets have been generated by dilation and translation of Bernoulli polynomials. The aim of this paper is to apply Bernoulli wavelet method to obtain approximate solutions of a system of linear Fredholm fuzzy integral equations. First we introduce properties of Bernoulli wavelets and Bernoulli polynomials, then we used it to transform the integral equations to the system of algebraic equations. The error estimates of the proposed method is given and compared by solving some numerical examples.

scholarly journals Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 200
Author(s):  
Ji-Huan He ◽  
Mahmoud H. Taha ◽  
Mohamed A. Ramadan ◽  
Galal M. Moatimid
Keyword(s):  
Linear System ◽  
Computer Program ◽  
Integral Equations ◽  
Operational Matrix ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Block Pulse Functions ◽  
Operational Matrix Of Integration

The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified into a linear system of algebraic equations through the use of IBPFs in addition to the operational matrix of integration. Typically, the classical alterations have enhanced the time taken by the computer program to solve the system of algebraic equations. The current modification works perfectly and has improved the efficiency over the regular block–pulse basis functions (BPF). Additionally, the paper handles the uniqueness plus the convergence theorems of the solution. Numerical examples have been presented to illustrate the efficiency as well as the accuracy of the method. Furthermore, tables and graphs are used to show and confirm how the method is highly efficient.

scholarly journals PROJECTION METHODS FOR INTEGRAL EQUATIONS IN EPIDEMIC

2002 ◽  
Vol 7 (2) ◽  
pp. 229-240 ◽  
Author(s):  
L. Hacia
Keyword(s):  
Mathematical Modeling ◽  
Numerical Methods ◽  
General Theory ◽  
Integral Equations ◽  
Projection Methods ◽  
Volterra Integral Equations ◽  
Temporal Development ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Spatio Temporal

In this paper numerical methods for mixed integral equations are presented. Studied equations arise in the mathematical modeling of the spatio‐temporal development of an epidemic. The general theory of these equations is given and used in the projection methods. Projection methods lead to a system of algebraic equations or to a system of Volterra integral equations. The considered theory is illustrated by numerical examples.

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 A Triangle Algorithm of Padé-Type Approximant for Two-Dimensional Fredholm Integral Equations of the Second Kind

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Meilan Sun ◽  
Chuanqing Gu
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Recursive Algorithm ◽  
High Order ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Initial Value ◽  
Numerical Examples ◽  
Type Approximation

The function-valued Padé-type approximation (2DFPTA) is used to solve two-dimensional Fredholm integral equation of the second kind. In order to compute 2DFPTA, a triangle recursive algorithm based on Sylvester identity is proposed. The advantage of this algorithm is that, in the process of calculating 2DFPTA to avoid the calculation of the determinant, it can start from the initial value, from low to high order, and gradually proceeds. Compared with the original two methods, the numerical examples show that the algorithm is effective.

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