Two spectral Legendre's derivative algorithms for Lane-Emden, Bratu equations, and singular perturbed problems

Abdelhakem, M.; Youssri, Y. H.

doi:10.1016/j.apnum.2021.07.006

Cited by 38 publications

(25 citation statements)

References 37 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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“…The author in [30] introduced the idea of using the derivative of the orthogonal polynomials. Recently, the first complete contribution for the Legendre's derivative was introduced in [31]. Thus, we will continue this novelty in this work by investigating Chebyshev's derivative as a new base function.…”

Section: Introductionmentioning

confidence: 89%

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

Abdelhakem

Alaa-Eldeen²,

Băleanu

et al. 2021

Fractal Fract

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An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems.

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“…Example We consider the following nonlinear Lane‐Emden problem 35 :

y^{''} false(x false) + \frac{1}{x} y^{'} false(x false) + e^{y false(x false)} = 0,

with the BCs

y^{'} false(0 false) = 0, y false(1 false) = 0 .

…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The obtained numerical results are compared those obtained by other numerical techniques given in previous works. 17,20,21,35 It is shown that the proposed technique provides very accurate results even with less number of mesh points and is of O(h 6 ) accuracy. The computational time of the method is provided.…”

Section: Introductionmentioning

confidence: 97%

A high‐order efficient technique and its convergence analysis for Bratu‐type and Lane‐Emden‐type problems

Roul

Goura

2022

Math Methods in App Sciences

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The present work deals with the construction of a high‐order efficient technique for solving one‐dimensional nonlinear Bratu‐type and Lane‐Emden‐type problems describing various physical, chemical and biological phenomena in applied science and engineering. Convergence analysis of the method is studied. Numerical experiments are carried out to demonstrate the efficiency, accuracy and performance of proposed method. The method is shown to have sixth order convergence. The computed results are compared with those obtained by other numerical techniques to justify the advantage of present scheme. It is shown that the proposed method is fast convergent and provides accurate solutions for Bratu‐type and Lane‐Emden‐type problems with low computational cost.

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Two spectral Legendre's derivative algorithms for Lane-Emden, Bratu equations, and singular perturbed problems

Cited by 38 publications

References 37 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

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

A high‐order efficient technique and its convergence analysis for Bratu‐type and Lane‐Emden‐type problems

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