A method for fractional Volterra integro-differential equations by Laguerre polynomials

Bayram, Dilek Varol; Daşçıoğlu, Ayşegül

doi:10.1186/s13662-018-1924-0

Cited by 13 publications

(3 citation statements)

References 54 publications

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“…Different approaches have been studied to solve FIE. Ghosh [8] employed Katugampola fractional operator to investigate analytical approach for the fractional-order Hepatitis B model, Ghosh & Kumar [9] investigated the accuracy of fractional Covid-19 model via spectral collocation method, Jani et al [10] investigated the accuracy of numerical solution of FIE with non-local conditions using Bernstein polynomials, Zaky [11] proposed improved tau method to investigate the accuracy of multi-dimensional fractional Rayleigh-Stokes problem, Wang & Zhu [12] proposed Euler wavelet operational matrix method for solving non-linear Volterra integro-differential equations, Huang et al [13] tested the effectiveness of Taylor expansion method on the solution of FIE, Bayram & Das ¸cioglu [14,15] investigated the accuracy of fractional linear Volterra-Fredholm IDEs via Laguerre polynomials as an approximation, Mittal & Nigam [16] investigated the accuracy of Adomian decomposition method (ADM) in the solution of FIE. Convergence of Jacobi spectral collocation method was investigated in the solution of FIE by Huang et al [17,18], FIE with weakly singular kernel was studied using legendre wavelets method by Yi et al [19], spline collocation method was used to solve fractional weakly singular IDEs by Pedas et al [20].…”

Section: Introductionmentioning

confidence: 99%

Approximate analytical solution of fractional-order generalized integro-differential equations via fractional derivative of shifted Vieta-Lucas polynomial

Issa,

A. Bello,

Jos Abubakar

2024

J. Nig. Soc. Phys. Sci.

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In this paper, we extend fractional-order derivative for the shifted Vieta-Lucas polynomial to generalized-fractional integro-differential equations involving non-local boundary conditions using Galerkin method as transformation technique and obtained N - \delta + 1 system of linear algebraic equations with \lambda i, i = 0, . . . , N unknowns, together with \delta non-local boundary conditions, we obtained (N + 1)- linear equations. The accuracy and effectiveness of the scheme was tested on some selected problems from the literature. Judging from the table of results and figures, we observed that the approximate solution corresponding to the problem that has exact solution in polynomial form gives a closed form solution while problem with non-polynomial exact solution gives better accuracy compared to the existing results.

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Section: Introductionmentioning

confidence: 99%

Approximate analytical solution of fractional-order generalized integro-differential equations via fractional derivative of shifted Vieta-Lucas polynomial

Issa,

A. Bello,

Jos Abubakar

2024

J. Nig. Soc. Phys. Sci.

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“…Fractional integro-differential equations are sometimes difficult to solve analytically, requiring the construction of effective approximation solutions. The Jacobi spectral method [20], Runge Kutta method [24], Chebyshev collocation method [3], Laplace Power Series Method [1], rationalized Haar functions method [15], Galerkin methods with hybrid functions [14] and Laguerre collocation method [5] are just a few of the numerical techniques that have been used to solve such equations.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

Saad,

Khirallah

2024

Eur. J. Pure Appl. Math.

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In this paper, a numerical method for solving the fractional order Fredholm integro-differential equations via the Caputo-Liouville derivative is presented. The method uses the well-known shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson’s 1/3, and Simpson’s 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton’s method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program.

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“…Preliminarily, Bayram et al [12] have applied the Sinc-collocation method, and Daşcıoğlu et al [13] have used a collocation method based upon the Laguerre polynomials to attain the solutions of the linear fractional IDEs in the conformable sense. This method mentioned in [13] is an improvement of the method that had been used for the solutions of the linear Caputo fractional IDEs of the Volterra type [14] and Caputo fractional linear IDEs of the Fredholm type [15]. However, for the conformable fractional Fredholm IDEs, there has not been a method in the literature in the sense of Chebyshev polynomials.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations

Bayram

2023

Journal of New Theory

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In this study, Chebyshev polynomials have been applied to construct an approximation method to attain the solutions of the linear fractional Fredholm integro-differential equations (IDEs). By this approximation method, the fractional IDE has been transformed into a linear algebraic equations system with the aid of the collocation points. In the method, the conformable fractional derivatives of the Chebyshev polynomials have been calculated in terms of the Chebyshev polynomials. Using the results of these calculations, the matrix relation for the conformable fractional derivatives of Chebyshev polynomials was attained for the first time in the literature. After that, the matrix forms have been replaced with the corresponding terms in the given fractional integro-differential equation, and the collocation points have been used to have a linear algebraic system. Furthermore, some numerical examples have been presented to demonstrate the preciseness of the method. It is inferable from these examples that the solutions have been obtained as the exact solutions or approximate solutions with minimum errors.

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A method for fractional Volterra integro-differential equations by Laguerre polynomials

Cited by 13 publications

References 54 publications

Approximate analytical solution of fractional-order generalized integro-differential equations via fractional derivative of shifted Vieta-Lucas polynomial

Approximate analytical solution of fractional-order generalized integro-differential equations via fractional derivative of shifted Vieta-Lucas polynomial

A Spectral Collocation Method for Solving Caputo-Liouville Fractional Order Fredholm Integro-differential Equations

Chebyshev Collocation Method for the Fractional Fredholm Integro-Differential Equations

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