Piecewise spectral collocation method for system of Volterra integral equations

Gu, Zhi‐Gang

doi:10.1007/s10444-016-9490-z

Cited by 10 publications

(4 citation statements)

References 34 publications

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“…and |.| refers to componentwise absolute value. Applying Gronwall's inequality [22] in the inequality (39) we can conclude…”

Section: Convergence Analysismentioning

confidence: 94%

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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“…and |.| refers to componentwise absolute value. Applying Gronwall's inequality [22] in the inequality (39) we can conclude…”

Section: Convergence Analysismentioning

confidence: 94%

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

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“…To find the approximate solution of the system of VIEs of the first kind given in Equation ( 1), the unknown function v q (k) is replaced with the Bernstein basis function of degree r defined in Equation (3). Then, for p = 0, .…”

Section: Bernstein's Approximation For the Vies Of The First Kindmentioning

confidence: 99%

“…Mutaz [2] proposed the Biorthogonal wavelet-based method to find the numerical results of the system of VIEs. Gu [3] presented the piecewise collocation approach for the system of VIEs. The main purpose of this research was to compare the convergence rate with the global spectrum collocation method.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis

2022

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In this paper, a new numerical technique is introduced to find the solution of the system of Volterra integral equations based on symmetric Bernstein polynomials. The use of Bernstein polynomials to find the numerical solutions of differential and integral equations increased due to its fast convergence. Here, the numerical solution of the system of Volterra integral equations on any finite interval [m,n] is obtained by replacing the unknown functions with the generalized Bernstein basis functions. The proposed technique converts the given system of equations into the system of algebraic equations which can be solved by using any standard rule. Further, Hyers–Ulam stability criteria are used to check the stability of the given technique. The comparison between exact and numerical solution for the distinct nodes is demonstrated to show its fast convergence.

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“…However, if the function does not belongs to H 1 (a, b), then the derivative can be defined as In papers [15,16], authors proposed predictor-corrector schema equipped with the approximation of f (t, y(t)) by the linear or quadratic interpolation to approximate the solution of the Volterra integral equation which is equivalent to that of the fractional order differential equation (1) with a singular fractional derivative such as the Caputo fractional derivative. And many authors developed numerical methods for solving Volterra integral equations to evade dealing with singular kernel in the fractional differential equation (1) [17,18,19,20,21,22,23,24,25]. Comparably, for the Caputo-Fabrizio fractional derivative, it can be converted into the integral formulation as…”

Section: Introductionmentioning

confidence: 99%

Efficient and fast predictor-corrector method for solving nonlinear fractional differential equations with non-singular kernel

Lee,

Kim

et al. 2020

Preprint

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Efficient and fast predictor-corrector methods are proposed to deal with nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods achieve a uniform accuracy order with the second-order scheme for linear interpolation and the third-order scheme for quadratic interpolation. The convergence analysis is proved by using the discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed fast algorithm requires approximately O(N ) arithmetic operations while O(N 2 ) is required in case of the regular predictor-corrector schemes, where N is the total number of time step.The following numerical examples demonstrate the accuracy of the proposed methods as well as the efficiency; nonlinear fractional differential equations, time-fraction sub-diffusion, and time-fractional advectiondiffusion equation. The theoretical convergence rates are also verified by numerical experiments.

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Piecewise spectral collocation method for system of Volterra integral equations

Cited by 10 publications

References 34 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

Symmetric Bernstein Polynomial Approach for the System of Volterra Integral Equations on Arbitrary Interval and Its Convergence Analysis

Efficient and fast predictor-corrector method for solving nonlinear fractional differential equations with non-singular kernel

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