Legendre-Gauss-Lobatto collocation method for solving multi-dimensional systems of mixed Volterra-Fredholm integral equations

Amin, A. Z. M.; Abdelkawy, M. A.; Amin, Amr Kamel; Lopes, António M.; Alluhaybi, Ahmad A.; Hashim, Ishak

doi:10.3934/math.20231063

Cited by 2 publications

(1 citation statement)

References 51 publications

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“…The essential step in all spectral approaches is to express the solution as a finite series of distinct functions. There are many different kinds of spectral approaches, such as collocation [22], tau [23], Galerkin [24], and Petrov-Galerkin [25]. After that, the coefficients will be selected to reduce the absolute error.…”

Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Amin,

Abdelkawy,

Solouma

et al. 2023

Fractal Fract

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One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.

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Section: Introductionmentioning

confidence: 99%

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Amin,

Abdelkawy,

Solouma

et al. 2023

Fractal Fract

View full text Add to dashboard Cite

show abstract

Spectral algorithm for two-dimensional fractional sine-Gordon and Klein–Gordon models

Abdelkawy,

Almadi,

Solouma

et al. 2024

Int. J. Mod. Phys. C

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Solitons and waves with memory effects are two examples of nonlinear wave phenomena that can be studied mathematically using the fractional nonlinear sine-Gordon equation. It is a modification of the traditional sine-Gordon equation that takes memory effects and nonlocal interactions into account by adding fractional derivatives. A more complex explanation of particle dynamics and interactions within relativistic quantum mechanics is made possible by the fractional Klein–Gordon model, a theoretical framework that expands the standard equation to include fractional derivatives. The study uses shifted Legendre–Gauss–Lobatto and shifted Legendre–Gauss–Radau collocation techniques to solve numerically two-dimensional sine-Gordon and Klein–Gordon models. The study handles two-dimensional sine-Gordon and Klein–Gordon models by extending a collocation approach using basis functions. It suggests a collocation technique that uses a suggested basis to automatically satisfy the conditions. The suggested methods’ spectral accuracy and efficiency are validated by numerical results.

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Legendre-Gauss-Lobatto collocation method for solving multi-dimensional systems of mixed Volterra-Fredholm integral equations

Cited by 2 publications

References 51 publications

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

A Spectral Collocation Method for Solving the Non-Linear Distributed-Order Fractional Bagley–Torvik Differential Equation

Spectral algorithm for two-dimensional fractional sine-Gordon and Klein–Gordon models

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