The combined reproducing kernel method and Taylor series to solve nonlinear Abel’s integral equations with weakly singular kernel

Alvandi, Azizallah; Paripour, Mahmoud

doi:10.1080/23311835.2016.1250705

Cited by 9 publications

(6 citation statements)

References 22 publications

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“…Example 5.2 Consider the following singular Volterra integral equation (of the Abel's type, see also Alvandi and Paripour 2016;Cao et al 2003), with…”

Section: Numerical Examplesmentioning

confidence: 99%

Approximate solutions of Volterra integral equations by an interpolation method based on ramp functions

Costarelli

2019

Comp. Appl. Math.

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In the present paper we develop a numerical collocation scheme for integral equations based on the interpolation method which uses the so-called neural network operators activated by ramp functions. The proposed collocation scheme allows to solve both linear and nonlinear Volterra integral equations of the second kind; the convergence of the numerical method has been proved under standard assumptions. Numerical examples have been provided and discussed; the proposed method has been compared with the classical piecewice polynomials collocation method and a method based on sigmoidal functions, such as the Heaviside and the logistic functions.

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“…Example 5.2 Consider the following singular Volterra integral equation (of the Abel's type, see also Alvandi and Paripour 2016;Cao et al 2003), with…”

Section: Numerical Examplesmentioning

confidence: 99%

Approximate solutions of Volterra integral equations by an interpolation method based on ramp functions

Costarelli

2019

Comp. Appl. Math.

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“…Now, the approximate solution of Equation (1) can be obtained by the n-term intercept of Equation (13) and Proof:…”

Section: Lemma 31mentioning

confidence: 99%

“…Recently, the reproducing kernel method to solve a variety of singular or nonsingular integral equations was presented. Alvandi and Paripour solved nonlinear Abel's integral equations with weakly singular kernel in the reproducing kernel space and removed the singularity of the equation considered [13]. The same authors presented a simple and efficient method to solve linear Volterra integro-differential equations [14], nonlinear Volterra-Fredholm integro-differential equations [15], and see [16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Reproducing kernel method for a class of weakly singular Fredholm integral equations

Alvandi

Paripour

2018

Journal of Taibah University for Science

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Numerical methods for solving integral equations have been the focus of much research, including reproducing kernel methods. We present a new algorithm to solve weakly singular Fredholm integral equations (WSFIEs). The advantage of this method is that it is possible to pick any point in the interval of integration and also the approximate solution. The advantage of this method is used to remove singularity and reproducing kernel functions are used as a basis. The convergence of approximation solution to the exact solution is also proved. Some examples are displayed to demonstrate that the method is accurate and efficient for WSFIEs.

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“…Li et al [6] applied the method for numerical solutions of fractional Riccati differential equations. For more details see [7]- [9].…”

Section: Introductionmentioning

confidence: 99%

On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation

Yildirim¹,

Akgül²

2020

Fundamental Journal of Mathematics and Applications

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Higher order differential equations (ODE) has an important role in the modelling process. It is also much significant which the method is used for the solution. In this study, in order to get the approximate solution of a nonhomogeneous initial value problem, reproducing kernel Hilbert space method is used. Reproducing kernel functions have been obtained and the given problem transformed to the homogeneous form. The results have been presented with the graphics. Absolute errors and relative errors have been given in the tables.

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The combined reproducing kernel method and Taylor series to solve nonlinear Abel’s integral equations with weakly singular kernel

Cited by 9 publications

References 22 publications

Approximate solutions of Volterra integral equations by an interpolation method based on ramp functions

Approximate solutions of Volterra integral equations by an interpolation method based on ramp functions

Reproducing kernel method for a class of weakly singular Fredholm integral equations

On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation

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