A space–time spectral approximation for solving nonlinear variable-order fractional sine and Klein–Gordon differential equations

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

Section: Illustrative Examplesmentioning

confidence: 99%

“…In [12], Lin et al investigated the stability and convergence of an explicit finite-difference approximation for the VO nonlinear fractional diffusion equation. For more relevant references, the readers can refer to [13][14][15][16][17][18].…”

Section: Introductionmentioning

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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“…Some applications of the Prabhakar function can be seen in mathematics and physics as a fractional Poisson process [16], Havriliak-Negami relaxation functions [18,19], irregular case of the dielectric relaxation responses [20], a model of anomalous relaxation in dielectrics of fractional order [21], fractional thermoelasticity [10], telegraph equations [22], thermodynamics [23], and fractal time random [24]. By placing α 1 ðp, qÞ = 2, α 2 ðp, qÞ = 2 in Equation 3, the coupled nonlinear sine-Gordon equations of fractional variable orders given in (3) change into the classical coupled nonlinear sine-Gordon equations which are defined by Equation (2), and the classical coupled nonlinear sine-Gordon equations have many applications in physics as nonlinear models [25,26], plasma [27], quantum [28], optics [29], and mathematics [13,30,31]. Getting analytic solutions to fractional differential equations in general are not easy; therefore, numerical methods are used to obtain the solutions of this type of equations.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

Derakhshan

2021

Abstract and Applied Analysis

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In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

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“…A linearized second-order scheme was introduced in Lyu and Vong [44] to solve non-linear time-fractional Klein-Gordon-type equations. Later on, in Doha et al [45], a space-time spectral approximation was proposed for solving non-linear variable-order fractional Klein/sine-Gordon differential equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

et al. 2020

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In this article we develop a numerical algorithm based on redefined extended cubic B-spline functions to explore the approximate solution of the time-fractional Klein-Gordon equation. The proposed technique employs the finite difference formulation to discretize the Caputo fractional time derivative of order α ∈ (1, 2] and uses redefined extended cubic B-spline functions to interpolate the solution curve over a spatial grid. A stability analysis of the scheme is conducted, which confirms that the errors do not amplify during execution of the numerical procedure. The derivation of a uniform convergence result reveals that the scheme is O(h 2 + t 2−α) accurate. Some computational experiments are carried out to verify the theoretical results. Numerical simulations comparing the proposed method with existing techniques demonstrate that our scheme yields superior outcomes.

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A space–time spectral approximation for solving nonlinear variable-order fractional sine and Klein–Gordon differential equations

Cited by 14 publications

References 33 publications

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

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

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

Numerical Treatment of Time-Fractional Klein–Gordon Equation Using Redefined Extended Cubic B-Spline Functions

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