Novel spectral schemes to fractional problems with nonsmooth solutions

Atta, A. G.; Abd-Elhameed, W. M.; Moatimid, Galal M.; Youssri, Y. H.

doi:10.1002/mma.9343

Cited by 8 publications

(2 citation statements)

References 53 publications

(65 reference statements)

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“…However, a wider range of physical issues required more intricate mathematical differentiation operators. Fractional differentiation [21][22][23][24][25][26][27] and the notion of the fractal derivative have been combined to form an innovative differentiation concept. As a result, several mathematicians offered various types of fractional derivatives [28,29].…”

Section: Introductionmentioning

confidence: 99%

A dynamical behavior of the coupled Broer-Kaup-Kupershmidt equation using two efficient analytical techniques

Ansar,

Abbas,

Emadifar

et al. 2024

PLoS ONE

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The aim of the present study is to identify multiple soliton solutions to the nonlinear coupled Broer-Kaup-Kupershmidt (BKK) system, including beta, conformable, local-fractional, and M-truncated derivatives. The coupled Broer-Kaup-Kupershmidt system is employed for modelling nonlinear wave evolution in mathematical models of fluid dynamics, plasmic, optical, dispersive, and nonlinear long-gravity waves. The travelling wave solutions to the above model are found using the Unified and generalised Bernoulli sub-ODE techniques. By modifying certain parameter values, we may create bright soliton, squeezed bell-shaped wave, expanded v-shaped soliton, W-shaped wave, singular soliton, and periodic solutions. The four distinct kinds of derivatives are compared quite effectively using 2D line graphs. Also, contour plots and 3D graphics are given by using Mathematica 10. Lastly, any pair of propagating wave solutions has symmetrical geometrical forms.

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Section: Introductionmentioning

confidence: 99%

A dynamical behavior of the coupled Broer-Kaup-Kupershmidt equation using two efficient analytical techniques

Ansar,

Abbas,

Emadifar

et al. 2024

PLoS ONE

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“…Fractional partial differential equations have recently attracted the attention of a rising number of authors due to their utility in many areas of science and other disciplines [1][2][3][4][5]. One of the most important of these equations is the time-fractional heat conduction equation (TFHE).…”

Section: Introductionmentioning

confidence: 99%

Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions

Youssri,

Ismail,

Atta

2023

Phys. Scr.

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In this research paper, we address the time-fractional heat conduction equation in one spatial dimension, subject to nonlocal conditions in the temporal domain. To tackle this challenging problem, we propose a novel numerical approach, the ‘Rectified Chebyshev Petrov-Galerkin Procedure,’ which extends the classical Petrov-Galerkin method to efficiently handle the fractional temporal derivatives involved. Our method is characterized by several key contributions; We introduce a set of basis functions that inherently satisfy the homogeneous boundary conditions of the problem, simplifying the numerical treatment. Through careful mathematical derivations, we provide explicit expressions for the matrices involved in the Petrov-Galerkin method. These matrices are shown to be efficiently invertible, leading to a computationally tractable scheme. A comprehensive convergence analysis is presented, ensuring the reliability and accuracy of our approach. We demonstrate that our method converges to the true solution as the spatial and temporal discretization parameters are refined. The proposed Rectified Chebyshev Petrov-Galerkin Procedure is found to be robust, and capable of handling a wide range of problems with nonlocal temporal conditions. To illustrate the effectiveness of our method, we provide a series of numerical examples, including comparisons with existing techniques. These examples showcase the superiority of our approach in terms of accuracy and computational efficiency.

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A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

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Novel spectral schemes to fractional problems with nonsmooth solutions

Cited by 8 publications

References 53 publications

A dynamical behavior of the coupled Broer-Kaup-Kupershmidt equation using two efficient analytical techniques

A dynamical behavior of the coupled Broer-Kaup-Kupershmidt equation using two efficient analytical techniques

Chebyshev Petrov-Galerkin procedure for the time-fractional heat equation with nonlocal conditions

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

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