Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

Heydari, Mohammad Hossein; Avazzadeh, Z.; Cattani, Carlo

doi:10.1002/mma.6926

Cited by 13 publications

(4 citation statements)

References 39 publications

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“…The advantage of this method is that it provides a global solution to the problem. At the same time, the accuracy and effectiveness of the method are proved by numerical problems 30 . Hosseininia et al 31 introduced the two‐dimensional Kuramot‐Sivashinsky equation of variable fractional order and used the two‐dimensional Chebyshev cardinal functions (CCFs) to give a semi‐discrete method to solve the equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Cao

Chen

Wang

et al. 2021

Math Methods in App Sciences

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An innovative numerical procedure for solving the viscoelastic arch problem based on variable fractional rheological models, directly in time domain, is proposed and investigated. First, the nonlinear integral‐differential governing equation is established according to the variable fractional constitutive relation and geometrical relationship. Second, the nonlinear integral‐differential governing equation is transformed into algebraic equations and solved by using the shifted Legendre polynomials. Furthermore, the accuracy and effectiveness of the algorithm are verified according to the mathematical example. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic arch is investigated to determine the displacement at different times and positions. The displacement of the viscoelastic arch is compared under various loading (uniformly distributed load and linear load). The displacement of the viscoelastic arch of different materials under the same load conditions is also investigated. The results in the paper show the efficiency of the proposed numerical algorithm in the dynamical analysis of the viscoelastic arch.

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Section: Introductionmentioning

confidence: 99%

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Cao

Chen

Wang

et al. 2021

Math Methods in App Sciences

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“…ese equations are highly nonlinear and describe wave structures in crystal lattice, plasma, water, and density stratified ocean waves, etc., that are explored by many researchers. Heydari et al observed fractional KdV-burger's equation by discrete Chebyshev polynomials [19], second order difference schemes for timefractional KdV-burger's is solved by Cen et al [20], fractional Kaup-Kupershmidt equation is analyzed by Shah et al [21], Iqbal et al [22] applied Atangana-Baleanu derivative on fractional Kersten-Krasil'shchik coupled KdV-mKdV system. KdV equations are also used in string theory with continuum limit.…”

Section: Introductionmentioning

confidence: 99%

Soliton Solutions of Generalized Third Order Time‐Fractional KdV Models Using Extended He‐Laplace Algorithm

et al. 2022

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In this research, the He-Laplace algorithm is extended to generalized third order, time-fractional, Korteweg-de Vries (KdV) models. In this algorithm, the Laplace transform is hybrid with homotopy perturbation and extended to highly nonlinear fractional KdVs, including potential and Burgers KdV models. Time-fractional derivatives are taken in Caputo sense throughout the manuscript. Convergence and error estimation are confirmed theoretically as well as numerically for the current model. Numerical convergence and error analysis is also performed by computing residual errors in the entire fractional domain. Graphical illustrations show the effect of fractional parameter on the solution as 2D and 3D plots. Analysis reveals that the He-Laplace algorithm is an efficient approach for time-fractional models and can be used for other families of equations.

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“…In recent years, discrete polynomials have been extensively applied for solving diverse problems. For instances, see [39][40][41][42][43][44][45][46][47].…”

Section: Introductionmentioning

confidence: 99%

An accurate approach based on the orthonormal shifted discrete Legendre polynomials for variable-order fractional Sobolev equation

Heydari

Atangana

2021

Adv Differ Equ

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This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach.

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Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

Cited by 13 publications

References 39 publications

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Numerical analysis of nonlinear variable fractional viscoelastic arch based on shifted Legendre polynomials

Soliton Solutions of Generalized Third Order Time‐Fractional KdV Models Using Extended He‐Laplace Algorithm

An accurate approach based on the orthonormal shifted discrete Legendre polynomials for variable-order fractional Sobolev equation

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