Legendre‐Spectral Algorithms for Solving Some Fractional Differential Equations

Youssri, Y. H.; Abd-Elhameed, W. M.

doi:10.1002/9781119654223.ch8

Cited by 7 publications

(6 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Previous investigations have employed the presented scheme to approximate various differential or integral problems, a few of which will be mentioned. In [ 36 ], researchers successfully characterized the telegraph fraction model leveraging the SGLA. Similarly, in [ 35 ], the SGLA was utilized to find approximation of Fredholm fraction integrodifferential problems.…”

Section: Foundationsmentioning

confidence: 99%

Well-posedness analysis and pseudo-Galerkin approximations using Tau Legendre algorithm for fractional systems of delay differential models regarding Hilfer (α,β)-framework set

Sweis,

Abu Arqub,

Shawagfeh

2024

PLoS ONE

View full text Add to dashboard Cite

Fractional calculus serves as a versatile and potent tool for the modeling and control of intricate systems. This discussion debates the system of DFDEs with two regimes; theoretically and numerically. For theoretical analysis, we have established the EUE by leveraging the definition of Hilfer (α,β)-framework. Our investigation involved the examination of the possessions of the FRD, FCD, and FHD, utilizing their forcefulness and qualifications to convert the concerning delay system into an equivalent one of fractional DVIEs. By employing the CMT, we have successfully demonstrated the prescribed requirements. For numerical analysis, the Galerkin algorithm was implemented by leveraging OSLPs as a base function. This algorithm allows us to estimate the solution to the concerning system by transforming it into a series of algebraic equations. By employing the software MATHEMATICA 11, we have effortlessly demonstrated the requirements estimation of the nodal values. One of the key advantages of the deployed algorithm is its ability to achieve accurate results with fewer iterations compared to alternative methods. To validate the effectiveness and precision of our analysis, we conducted a comprehensive evaluation through various linear and nonlinear numerical applications. The results of these tests, accompanied by figures and tables, further support the superiority of our algorithm. Finally, an analysis of the numerical algorithm employed was provided along with insightful suggestions for potential future research directions.

show abstract

Section: Foundationsmentioning

confidence: 99%

Well-posedness analysis and pseudo-Galerkin approximations using Tau Legendre algorithm for fractional systems of delay differential models regarding Hilfer (α,β)-framework set

Sweis,

Abu Arqub,

Shawagfeh

2024

PLoS ONE

View full text Add to dashboard Cite

show abstract

“…Practically most of the pre-existing approaches have several strengths and limitations, and the choice of the scheme used depends on the specific shape concerning the problem. Likewise, one of the significant advantages of this algorithm is its capability of handling both linear/nonlinear fractional models, resulting in an approximation solution that closely matches the exact one exploiting only a few OSLPs, and the GSLM follows a straightforward numerical procedure, making it easy to obtain the required approximation solution [ 25 , 26 , [38] , [39] , [40] ].…”

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness, and galerkin shifted Legendre's approximation of time delays integrodifferential models by adapting the Hilfer fractional attitude

Sweis,

Shawagfeh,

Abu Arqub

2024

Heliyon

View full text Add to dashboard Cite

“…A fairly large number of works on approximate solutions, for the above linear problem, can be found in the literature using Legendre polynomials, the Chebyshev Tau method, decomposition method, homotopy perturbation method, He's variational iteration method, Rothe-Wavelet-Galerkin method, Haar wavelets, numerical spectral Legendre-Galerkin algorithm, finite difference scheme approximation, Sinc-collocation techniques, Bernstein polynomials operational matrices, the Caputo fractional difference formula and Grünwald difference, and the natural transform decomposition method [32][33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

Othmani

Tatar

2023

Asian Journal of Control

View full text Add to dashboard Cite

The ordinary diffusion models in continuous media are derived from basic principles: Conservation of energy and momentum. The classical telegraph equation is one of the earliest and most commonly used equation to describe the transmission of electromagnetic waves. In fractal and disordered media, to capture the complex diffusion feature, we consider rather fractional models. Indeed, the mean square displacement in anomalous media is no longer proportional to the time but rather proportional to a power of time whose exponent may be between zero and one or between one and two. This is the case for the present fractional telegraph equation in the presence of non‐negligible voltage wave. Here, the well‐posedness and stability of a telegraph problem with two time‐fractional derivatives is discussed. In addition to being a noninteger problem, the equation in the model is subject to a nonlinear source as well as a nonlinear lower order fractional derivative term. First, the well‐posedness in an appropriate underlying space is established by the use of resolvent operators. Then, it is proved that, despite the presence of these nonlinearities, solutions are Mittag–Leffler stable. This confirms (and extends) the fact that the lower order term plays the role of a damper in the fractional case. To this end, we combine the energy method with some properties of the Caputo fractional derivative.

show abstract

Legendre‐Spectral Algorithms for Solving Some Fractional Differential Equations

Cited by 7 publications

References 42 publications

Well-posedness analysis and pseudo-Galerkin approximations using Tau Legendre algorithm for fractional systems of delay differential models regarding Hilfer (α,β)-framework set

Well-posedness analysis and pseudo-Galerkin approximations using Tau Legendre algorithm for fractional systems of delay differential models regarding Hilfer (α,β)-framework set

Existence, uniqueness, and galerkin shifted Legendre's approximation of time delays integrodifferential models by adapting the Hilfer fractional attitude

Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

Contact Info

Product

Resources

About