Exact and Numerical Solution of Abel Integral Equations by Orthonormal Bernoulli Polynomials

Singh, Mithilesh; Singhal, Shivani; Handa, Nidhi

doi:10.1007/s40819-019-0734-8

Cited by 3 publications

(2 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[20] obtained numerical approximation for generalized pantograph equation using Bernoulli matrix method, Tohidi and Khorsand [21] to solve second-order linear system of partial differential equations, Mohsenyzadeh [22] used Bernoulli polynomials to solve Volterra type integral equations. Recently, Singh et al [23] used Bernoulli polynomials to develop a trigonal operational matrix to solve Abel-Volterra type integral equations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new approach to find an approximate solution of linear initial value problems

Singh¹

2020

Preprint

View full text Add to dashboard Cite

This work investigates a new approach to find closed form analytical approximate solution of linear initial value problems. Classical Bernoulli polynomials have been used to derive a finite set of orthonormal polynomials and a finite operational matrix to simplify derivatives of dependent variable. These orthonormal polynomials together with the operational matrix of relevant order provides a good approximation to the solution of a linear initial value problem. Depending upon the nature of a problem, a series form approximation or numerical approximation can be obtained. The technique has been demonstrated through three problems. Approximate solutions have been compared with available exact or other numerical solutions. High degree of accuracy has been noted in numerical values of solutions for considered problems.

show abstract

Section: Introductionmentioning

confidence: 99%

“…In this work, it is proposed to solve linear initial value problems using orthogonal polynomials derived from Bernoulli polynomials with a modified operational matrix [23].…”

Section: Introductionmentioning

confidence: 99%

A new approach to find an approximate solution of linear initial value problems

Singh¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

A method for singular weakly linear Volterra-Integro-differential equations by Euler polynomials

Singh¹,

Handa

Singhal

2022

AIP Conference Proceedings

View full text Add to dashboard Cite

Shifted Legendre Fractional Pseudo-Spectral Integration Matrices for Solving Fractional Volterra Integro-Differential Equations and Abel’s Integral Equations

ABDELHAKEM

2023

Fractals

View full text Add to dashboard Cite

Shifted Legendre polynomials (SLPs) with the Riemann–Liouville fractional integral operator have been used to create a novel fractional integration tool. This tool will be called the fractional shifted Legendre integration matrix (FSL B-matrix). Two algorithms depending on this matrix are designed to solve two different types of integral equations. The first algorithm is to solve fractional Volterra integro-differential equations (VIDEs) with a non-singular kernel. The second algorithm is for Abel’s integral equations. In addition, error analysis for the spectral expansion has been proven to ensure the expansion’s convergence. Finally, several examples have been illustrated, including an application for the population model.

show abstract

Exact and Numerical Solution of Abel Integral Equations by Orthonormal Bernoulli Polynomials

Cited by 3 publications

References 26 publications

A new approach to find an approximate solution of linear initial value problems

A new approach to find an approximate solution of linear initial value problems

A method for singular weakly linear Volterra-Integro-differential equations by Euler polynomials

Shifted Legendre Fractional Pseudo-Spectral Integration Matrices for Solving Fractional Volterra Integro-Differential Equations and Abel’s Integral Equations

Contact Info

Product

Resources

About