An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

Didgar, M.; Vahidi, Alireza

doi:10.1088/0253-6102/70/2/145

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…Now, by substituting ( 32) and ( 33) in (31) and using Definition 1 and relation (29) one can rewrite…”

Section: Construction Of Recursive Tau-approximate Solutionmentioning

confidence: 99%

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

Shahmorad¹,

Mokhtary²,

Talaei³

et al. 2021

Preprint

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This paper provides an efficient recursive approach of the spectral Tau method, to approximate the solution of system of generalized Abel-Volterra integral equations. In this regards, we first investigate the existence, uniqueness as well as smoothness of the solutions under various assumptions on the given data. Next, from a numerical perspective, we express approximated solution as a linear combination of suitable canonical polynomials which are constructed by an easy to use recursive formula. Mostly, the unknown parameters are calculated by solving a low dimensional algebraic systems independent of degree of approximation which prevent from high computational costs. Obviously, due to singular behavior of the exact solution, using classical polynomials to construct canonical polynomials, leads to low accuracy results. In this regards, we develop a new fractional order canonical polynomials using Müntz-Legendre polynomials which have a same asymptotic behavior with the solution of underlying problem. The convergence analysis is discussed, and the familiar spectral accuracy is achieved in L ∞ -norm. Finally, the reliability of the method is evaluated using various problems.Keywords The recursive approach of the Tau method • System of generalized Abel-Volterra integral equations • Müntz-Legendre polynomials • Fractional vector canonical polynomials • convergence analysis.

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“…Now, by substituting ( 32) and ( 33) in (31) and using Definition 1 and relation (29) one can rewrite…”

Section: Construction Of Recursive Tau-approximate Solutionmentioning

confidence: 99%

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

Shahmorad¹,

Mokhtary²,

Talaei³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Furthermore, Shahmorad and Ahdiaghdam [14] proposed Chebyshev polynomials approximation for the numerical solution of a system of Cauchy-type singular integral equations of the first kind on a finite segment. Taylor Expansion method as an approximate approach for systems of singular Volterra integral equations is proposed by Didgar and Vahidi [15]. There are many methods developed for one-dimensional CSIEs (see [16][17][18][19][20][21][22][23]).…”

Section: Introductionmentioning

confidence: 99%

“…There are many methods developed for one-dimensional CSIEs (see [16][17][18][19][20][21][22][23]). However, not many researchers have researched the system of CSIEs [10][11][12][13][14][15]. Nevertheless, the HPM for the system of CSIEs has rarely been applied, and very few articles have been published.…”

Section: Introductionmentioning

confidence: 99%

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

Mamatova,

Eshkuvatov,

Ismail

2023

Mathematics

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In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted Lp space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others.

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New fractional Lanczos vector polynomials and their application to system of Abel–Volterra integral equations and fractional differential equations

Conte

Shahmorad

Talaei

2020

Journal of Computational and Applied Mathematics

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An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

Cited by 3 publications

References 31 publications

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

New fractional Lanczos vector polynomials and their application to system of Abel–Volterra integral equations and fractional differential equations

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