Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method

Abdelhakem, M.; Alaa-Eldeen, Toqa; Băleanu, Dumitru; Alshehri, Maryam G.; El-Kady, M.

doi:10.3390/fractalfract5040165

Cited by 26 publications

(12 citation statements)

References 45 publications

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“…Numerical Tests for the CNLSE. In this section, plenty of designated numerical examples are conducted, as recently followed elsewhere [42][43][44][45][46][47], to examine how efficient, fast, and accurate the proposed numerical techniques are, especially when compared with the exact analytical solution.…”

Section: The Error and Convergence Discussionmentioning

confidence: 99%

Extended Split-Step Fourier Transform Approach for Accurate Characterization of Soliton Propagation in a Lossy Optical Fiber

Farag

ElTanboly

El–Azab

et al. 2022

Journal of Function Spaces

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In this paper, we present a novel extension of the well-known split-step Fourier transform (SSFT) approach for solving the one-dimensional nonlinear Schrödinger equation (NLSE), which incorporates the fiber loss term. While this essential equation governs the pulse propagation in a lossy optical fiber, it is not supported by an exact analytical solution. In this regard, extended versions of the Fourier pseudospectral method (FPSM) and Hopscotch method (HSM) are effectively established as well to cope with the fiber losses effects associated with the pulses’ propagation through the fiber optics, and thus, numerous comparisons are exhaustively conducted among these three compelling numerical approaches to validate their reliability, stability, and accuracy. Based on this, the MATLAB numerical findings bolster that the extended version of the SSFT approach demonstrates superior performance over the other suggested schemes in simulating the solitons propagation in a lossy optical fiber.

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Section: The Error and Convergence Discussionmentioning

confidence: 99%

Extended Split-Step Fourier Transform Approach for Accurate Characterization of Soliton Propagation in a Lossy Optical Fiber

Farag

ElTanboly

El–Azab

et al. 2022

Journal of Function Spaces

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“…In addition, they employed this class of polynomials to treat certain types of even-order BVPs. The authors in [33,34] handled some BVPs and IVP using the Chebyshev polynomials' first derivative.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

Abd-Elhameed

2023

Fractal Fract

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The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new formulas concerned with these generalized polynomials are established. These generalized orthogonal polynomials are employed to treat the multi-term linear fractional differential equations (FDEs) that include some specific problems that arise in many applications. The basic idea behind the derivation of our proposed algorithm is built on utilizing a new power form representation of the shifted generalized Chebyshev polynomials along with the application of the spectral Galerkin method to transform the FDE governed by its initial conditions into a system of linear equations that can be efficiently solved via a suitable numerical solver. Some illustrative examples accompanied by comparisons with some other methods are presented to show that the presented algorithm is useful and effective.

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“…The Chebyshev polynomials of the first kind were employed to treat different types of differential and integral equations (see, for example, [25][26][27][28]). Some types of boundaryvalue problems were handled on the basis of employing the first kind of Chebyshev derivative polynomials in [29]. The second kind of Chebyshev polynomials were utilized in [30] to treat third-order Emden-Fowler singular differential equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials

et al. 2023

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This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.

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Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method

Cited by 26 publications

References 45 publications

Extended Split-Step Fourier Transform Approach for Accurate Characterization of Soliton Propagation in a Lossy Optical Fiber

Extended Split-Step Fourier Transform Approach for Accurate Characterization of Soliton Propagation in a Lossy Optical Fiber

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

Numerical Contrivance for Kawahara-Type Differential Equations Based on Fifth-Kind Chebyshev Polynomials

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