A novel variable‐order fractional nonlinear Klein Gordon model: A numerical approach

This article adopts a novel technique to numerical solution for fractional time-delay diffusion equation with variable-order derivative (VFDDEs). As a matter of fact, the variable-order fractional derivative (VFD) that has been used is in the Caputo sense. The first step of this technique is constructive on the construction of the solution using the shifted Legendre–Laguerre polynomials with unknown coefficients. The second step involves using a combination of the collocation method and the operational matrices (OMs) of the shifted Legendre–Laguerre polynomials, as well as the Newton–Cotes nodal points, to find the unknown coefficients. The final step focuses on solving the resulting algebraic equations by employing Newton’s iterative method. To illustrate and demonstrate the technique’s efficacy and applicability, two examples have been provided.

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“…1 is the pseudo-operational matrix of the integer order integral of the SLPs and 1 is defined by [10,20,33]…”

Section: Pseudo-operational Matrix Of Integer Order Integral Of the S...mentioning

confidence: 99%

Solving fractional time-delay diffusion equation with variable-order derivative based on shifted Legendre–Laguerre operational matrices

2023

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“…Example 2 Consider the nonlinear inhomogeneous VOF-KGE [13,38] where with the initial conditions and boundary conditions:…”

Section: Illustrative Examplesmentioning

confidence: 99%

Modified wavelet method for solving multitype variable-order fractional partial differential equations generated from the modeling of phenomena

2021

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The aim of this paper is to introduce a new wavelet method for presenting approximate solutions of multitype variable-order (VO) fractional partial differential equations arising from the modeling of phenomena. In specific, this paper focuses on the numerical solution of the VO-fractional mobile-immobile advection-dispersion equation, Klein Gordon equation and Burgers equation. These equations are converted into a system of algebraic equations with the assistance of the bivariate Genocchi wavelet functions, their operational matrices, and the variable-order fractional Caputo derivative operator. Also, we present a new technique to get the operational matrix of integration and VO-fractional derivative. The modified operational matrices for solving the proposed equations are powerful and effective. So that, the accuracy of these matrices directly affects the implementation process. Finally, we consider numerical examples to confirm the superiority of the scheme, and for each example, exhibit the results through graphs and tables.

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“…A Josephson junction, the motion of rigid pendula attached to a stretched wire can be described by sine Klein-Gordon equation and a non-local version of them are properly modeled by the fractional version of them [24]. In [25], Sweilam et al has constructed a new and e¤ective numerical scheme, namely weighted average nonstandard …nite di¤erence method, for analyzing the time variable-order fractional of nonlinear Klein-Gordon equation and so on.…”

Section: S • Inan Den • Izmentioning

confidence: 99%

On the stability analysis of the time-fractional variable order Klein-Gordon equation and a numerical simulation

Deni̇z¹

2020

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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In this paper, the Klein-Gordon equation is generalized using the concept of the variational order derivative. We try to construct the Crank-Nicholson scheme for numerical solutions of the modi…ed Klein-Gordon equation. Stability analysis of the Crank-Nicholson scheme is examined and analyzed to prove the proposed method is stable for solving the time-fractional variable order Klein-Gordon equation. Numerical examples are also given for illustration.

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A novel variable‐order fractional nonlinear Klein Gordon model: A numerical approach

Cited by 14 publications

References 47 publications

Solving fractional time-delay diffusion equation with variable-order derivative based on shifted Legendre–Laguerre operational matrices

Solving fractional time-delay diffusion equation with variable-order derivative based on shifted Legendre–Laguerre operational matrices

Modified wavelet method for solving multitype variable-order fractional partial differential equations generated from the modeling of phenomena

On the stability analysis of the time-fractional variable order Klein-Gordon equation and a numerical simulation

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