On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method

Mohamed, Emad M. H.; Mahdy, A. M. S.

doi:10.20852/ntmsci.2017.202

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…Finally, let y (x) the solution of ( 1) and ( 2) be approximated by (20) , then using lemmas (3), ( 4) and ( 5), we got the system of linear algebraic equations given by…”

Section: Transformation From Integral Equations Into Linear Algebraic...mentioning

confidence: 99%

Method for Solving Initial Value Problems of Linear Fractional Volterra Integro-differential Equations

Taparki

Samuel

Samson

2023

Preprint

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In this paper, a numerical approach is developed for solving initial value problem of linear fractional Volterra integro-differential equations. The approximate solution is substituted into the model equation and then collocated using shifted Chebyshev polynomial and Standard collocation points to obtain a system of linear algebraic equations, which is then solved by Newton-Rapson's method. Several numerical examples were solved to demonstrate the accuracy, reliability and efficiency of the method.

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“…Finally, let y (x) the solution of ( 1) and ( 2) be approximated by (20) , then using lemmas (3), ( 4) and ( 5), we got the system of linear algebraic equations given by…”

Section: Transformation From Integral Equations Into Linear Algebraic...mentioning

confidence: 99%

Method for Solving Initial Value Problems of Linear Fractional Volterra Integro-differential Equations

Taparki

Samuel

Samson

2023

Preprint

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“…Mohammed [5] utilized the approach of least squares in conjunction with a shifted Chebyshev polynomial in order to solve the FrLIoDE problem. In [6], Mahdy et al investigated the numerical solution of FrLIoDE by employing the least squares approach and supplementing it with a shifted Laguerre polynomial. In order to locate the numerical solution of FrLIoDE with the Caputo derivative, Nanware et al [7] used the Bernstein polynomial to solve it.…”

Section: Introductionmentioning

confidence: 99%

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method

Jan,

Abdou,

Basseem

2023

Fractal Fract

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In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space L2Ω×C0,T, T<1. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS).

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“…Tere has been a lot of interest in solving FI-DEs [26][27][28][29], specially using collocation methods. In [30], the authors used a wavelet collocation method to solve Fredholm FI-DEs.…”

Section: Introductionmentioning

confidence: 99%

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

2023

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In this paper, we develop an efficient numerical method to approximate the solution of fractional integro-differential equations (FI-DEs) of mixed Volterra−Fredholm type using spectral collocation method with shifted Chebyshev polynomials of the third kind (S-Cheb-3). The fractional derivative is described in the Caputo sense. A Chebyshev−Gauss quadrature is involved to evaluate integrals for more precision. Two types of equations are studied to obtain algebraic systems solvable using the Gauss elimination method for linear equations and the Newton algorithm for nonlinear ones. In addition, an error analysis is carried out. Six numerical examples are evaluated using different error values (maximum absolute error, root mean square error, and relative error) to compare the approximate and the exact solutions of each example. The experimental rate of convergence is calculated as well. The results validate the numerical approach’s efficiency, applicability, and performance.

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On the numerical solution of Hammerstein integral equations using shifted Chebyshev polynomials of the third kind method

Cited by 4 publications

References 13 publications

Method for Solving Initial Value Problems of Linear Fractional Volterra Integro-differential Equations

Method for Solving Initial Value Problems of Linear Fractional Volterra Integro-differential Equations

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method

Theoretical and Numerical Study for Volterra−Fredholm Fractional Integro-Differential Equations Based on Chebyshev Polynomials of the Third Kind

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