scholarly journals Comparison of the Orthogonal Polynomial Solutions for Fractional Integral Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 59
Author(s):  
Ayşegül Daşcıoğlu ◽  
Serpil Salınan
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Integral Equations ◽  
Collocation Method ◽  
Fractional Integral ◽  
The Other ◽  
Numerical Examples ◽  
Polynomial Solutions

In this paper, a collocation method based on the orthogonal polynomials is presented to solve the fractional integral equations. Six numerical examples are given to illustrate the method. The results are compared with the other methods in the literature, and the results obtained by different kinds of polynomials are compared.

scholarly journals Spectral approximations to the fractional integral and derivative

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Changpin Li ◽  
Fanhai Zeng ◽  
Fawang Liu
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Caputo Derivative ◽  
Initial Value Problems ◽  
Fractional Integral ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
Initial Value ◽  
Numerical Examples ◽  
Spectral Approximations

AbstractIn this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

Jacobi Collocation Approximation for Solving Multi-dimensional Volterra Integral Equations

2017 ◽  
Vol 18 (5) ◽  
pp. 411-425 ◽  
Author(s):  
Mohamed A. Abdelkawy ◽  
Ahmed Z. M. Amin ◽  
Ali H. Bhrawy ◽  
José A. Tenreiro Machado ◽  
António M. Lopes
Keyword(s):  
Error Analysis ◽  
Integral Equations ◽  
Collocation Method ◽  
Jacobi Polynomials ◽  
Volterra Integral Equations ◽  
The Novel ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Novel Technique ◽  
Shifted Jacobi Polynomials

AbstractThis paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel technique takes advantage of the properties of shifted Jacobi polynomials and is applied for solving multi-dimensional VIEs. Several numerical examples demonstrate the efficiency of the method and an error analysis verifies the correctness and feasibility of the proposed method when solving VIE.

scholarly journals A parallel Numerical Algorithm For Solving Some Fractional Integral Equations

2019 ◽  
Vol 32 (1) ◽  
pp. 184
Author(s):  
Khalid Mindeel Mohammed
Keyword(s):  
Neural Network ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Fractional Integral ◽  
High Accuracy ◽  
Numerical Results ◽  
New Method ◽  
Training Algorithm ◽  
Numerical Examples ◽  
Fractional Equations

In this study, He's parallel numerical algorithm by neural network is applied to type of integration of fractional equations is Abel’s integral equations of the 1st and 2nd kinds. Using a Levenberge – Marquaradt training algorithm as a tool to train the network. To show the efficiency of the method, some type of Abel’s integral equations is solved as numerical examples. Numerical results show that the new method is very efficient problems with high accuracy.

On Linear Ordinary Differential Equation With Orthogonal Polynomial Solutions

2018 ◽  
Vol 1 (20) ◽  
pp. 559-572
Author(s):  
ولدان وليد محمود
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Differential Operators ◽  
Linear Ordinary Differential Equation ◽  
Large Piece ◽  
Polynomial Solutions ◽  
The Subject

: The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. In this paper we give a  survey of the orthogonal polynomial solutions of second-order and fourth-order linear ordinary differential equation, on the generated self-adjoint differential operators .

scholarly journals New coincidence point results for generalized graph-preserving multivalued mappings with applications

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hasanen A. Hammad ◽  
Manuel De la Sen ◽  
Praveen Agarwal
Keyword(s):  
Integral Equations ◽  
Fractional Integral ◽  
Metric Spaces ◽  
Coincidence Point ◽  
The Other ◽  
Multivalued Mappings ◽  
Nonlinear Integral Equations ◽  
Existence Of A Solution ◽  
Multivalued Contraction ◽  
Theoretical Results

AbstractThis research aims to investigate a novel coincidence point (cp) of generalized multivalued contraction (gmc) mapping involved a directed graph in b-metric spaces (b-ms). An example and some corollaries are derived to strengthen our main theoretical results. We end the manuscript with two important applications, one of them is interested in finding a solution to the system of nonlinear integral equations (nie) and the other one relies on the existence of a solution to fractional integral equations (fie).

scholarly journals A COMPUTATIONAL METHOD FOR SOLVING TWO-DIMENSIONAL LINEAR FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND

2008 ◽  
Vol 49 (4) ◽  
pp. 543-549 ◽  
Author(s):  
A. TARI ◽  
S. SHAHMORAD
Keyword(s):  
Orthogonal Polynomials ◽  
Integral Equations ◽  
Numerical Solutions ◽  
Expansion Method ◽  
Computational Method ◽  
Fredholm Integral Equations ◽  
Two Dimensional ◽  
Numerical Examples

AbstractIn this paper an expansion method, based on Legendre or any orthogonal polynomials, is developed to find numerical solutions of two-dimensional linear Fredholm integral equations. We estimate the error of the method, and present some numerical examples to demonstrate the accuracy of the method.

The Krall Polynomials as Solutions to a Second Order Differential Equation

1983 ◽  
Vol 26 (4) ◽  
pp. 410-417 ◽  
Author(s):  
Lance L. Littlejohn
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Polynomial Solutions ◽  
Second Order Differential Equations ◽  
Image Position

AbstractA popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form1.1

scholarly journals Applications of Continuous Fractions in Orthogonal Polynomials

2018 ◽  
Vol 6 (12) ◽  
pp. 284-295
Author(s):  
Ana Cláudia Marassá Roza Boso ◽  
Luís Roberto Almeida Gabriel Filho ◽  
Camila Pires Cremasco Gabriel ◽  
Bruno César Góes ◽  
Fernando Ferrari Putti
Keyword(s):  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Integral Equations ◽  
Theoretical Studies ◽  
Approximation Of Functions ◽  
Irrational Numbers ◽  
Electric Circuits ◽  
Continuous Fraction ◽  
The Way

Several applications of continuous fractions are restricted to theoretical studies, such as problems associated with the approximation of functions, determination of rational and irrational numbers, applications in physics in determining the resistance of electric circuits and integral equations and in several other areas of mathematics. This work aimed to study the results that open the way for the connection of continuous fractions with the orthogonal polynomials. As support, we will study the general case, where the applications of the Wallis formulas in a monolithic orthogonal polynomial, which generates a continuous fraction of the Jacobi type. It will be allowed applications with relations of recurrence of three terms in the polynomials of Tchebyshev and Legendre, through the results found, establishing connection between them with the continuous fractions. And finally, will be presented the "Number of gold", that is an application of this theory.

scholarly journals An efficient method for the numerical solution of the nonlinear Hammerstein fractional integral equations

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 419-429
Author(s):  
Ahmadabadi Nili ◽  
M.R. Velayati
Keyword(s):  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Efficient Method ◽  
Fractional Integral ◽  
Singular Integrals ◽  
Computational Cost ◽  
Picard Iteration ◽  
Numerical Examples ◽  
Weakly Singular

In this paper, we present a numerical method for solving nonlinear Hammerstein fractional integral equations. The method approximates the solution by Picard iteration together with a numerical integration designed for weakly singular integrals. Error analysis of the proposed method is also investigated. Numerical examples approve its efficiency in terms of accuracy and computational cost.

Sign in / Sign up

Export Citation Format

Share Document