Sinc collocation approximation of non-smooth solution of a nonlinear weakly singular Volterra integral equation

Zakeri, Gholam-Ali; Navab, Mitra

doi:10.1016/j.jcp.2010.05.010

Cited by 17 publications

(8 citation statements)

References 25 publications

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“…Nonlinear FIEs are another area of interest in literature [25,7]. The structure of the non-linear equation is defined by equation (12)…”

Section: Nonlinear Fiementioning

confidence: 99%

“…The examples examined are taken from [25] where polynomial nonlinearities are used and exact solutions are available. We note that all examples in [25] in the "Numerical results" are wrong. The stated equations miss factors of √ π.…”

Section: Nonlinear Fiementioning

confidence: 99%

“…In a similar manner as in [25] we can set up different other non-linear FIE. For further demonstrations we choose the following equation ; −3x In conclusion the Sinc method shows excellent convergence for linear and non-linear FIEs.…”

Section: Nonlinear Fiementioning

confidence: 99%

“…Stenger's Sinc methods have among other features a very fast convergence of exponential order and in addition effectively deals with singularities. Fractional integrals and derivatives always are dealing by definition with so called weak singularities [25,26]. The idea of Sinc methods is to use a Sinc function as a base functions which by itself is an approximation of a "singular function" a distribution.…”

Section: Introductionmentioning

confidence: 99%

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Fractional calculus and Sinc methods

Baumann

Stenger

2011

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Fractional integrals, fractional derivatives, fractional integral equations, and fractional differential equations are numerically solved by Sinc methods. Sinc methods are able to deal with singularities of the weakly singular integral equations of Riemann-Liouville and Caputo type. The convergence of the numerical method is numerically examined and shows exponential behavior. Different examples are used to demonstrate the effective derivation of numerical solutions for different types of fractional differential and integral equations, linear and non-linear ones. Equations of mixed ordinary and fractional derivatives, integro-differential equations are solved using Sinc methods. We demonstrate that the numerical calculation needed in fractional calculus can be gained with high accuracy using Sinc methods.MSC 2010 : Primary 65-XX, 45-XX, 97-XX: Secondary 65D15, 45E10, 44A35, 97N50

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“…Nonlinear FIEs are another area of interest in literature [25,7]. The structure of the non-linear equation is defined by equation (12)…”

Section: Nonlinear Fiementioning

confidence: 99%

Section: Nonlinear Fiementioning

confidence: 99%

Section: Nonlinear Fiementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fractional calculus and Sinc methods

Baumann

Stenger

2011

fcaa

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“…It is well known that there are many numerical methods for solving second kind Volterra integral equations such as the Runge-Kutta method and the collocation method based on piecewise polynomials; see, for example, Brunner [1] and references therein. For more information of the progress on the study of the problem, we refer the readers to [2][3][4][5][6][7][8]. Recently, a few works touched the spectral approximation to Volterra integral equations.…”

Section: Introductionmentioning

confidence: 99%

A Jacobi-Collocation Method for Second Kind Volterra Integral Equations with a Smooth Kernel

Guo

Cai

Xin

2014

Abstract and Applied Analysis

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The purpose of this paper is to provide a Jacobi-collocation method for solving second kind Volterra integral equations with a smooth kernel. This method leads to a fully discrete integral operator. First, it is shown that the fully discrete integral operator is stable in bothL∞and weightedL2norms. Then, the proposed approach is proved to arrive at an optimal (the most possible) convergent order in both norms. One numerical example demonstrates the efficiency and accuracy of the proposed method.

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Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

Eshaghi

Adibi

Kazem

2015

Math Methods in App Sciences

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In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.

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Sinc collocation approximation of non-smooth solution of a nonlinear weakly singular Volterra integral equation

Cited by 17 publications

References 25 publications

Fractional calculus and Sinc methods

Fractional calculus and Sinc methods

A Jacobi-Collocation Method for Second Kind Volterra Integral Equations with a Smooth Kernel

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

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