An Efficient Numerical Scheme for Solving Multiorder Tempered Fractional Differential Equations via Operational Matrix

Owoyemi, Abiodun Ezekiel; Phang, Chang; Toh, Yoke Teng

doi:10.1155/2022/7628592

Cited by 5 publications

(4 citation statements)

References 27 publications

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“…One of these is iteration procedures, which involve the homotopy analysis method of nonlinear equations arising in heat transfer [17] and the iteration perturbation method [18]. The iteration procedure is carried out via an operational matrix [19]. The non-perturbative variational iteration method (VIM) was developed in [20].…”

Section: Introductionmentioning

confidence: 99%

New Perturbation–Iteration Algorithm for Nonlinear Heat Transfer of Fractional Order

Abdel Aal

2024

Fractal Fract

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Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on extending the nonlinear heat equations to a fractional order in a Caputo order. A new perturbation iteration algorithm (PIA) of the fractional order is applied to solve the nonlinear heat equations. Solving numerical problems that involve fractional differential equations can be challenging due to their inherent complexity and high computational cost. To overcome these challenges, there is a need to develop numerical schemes such as the PIA method. This method can provide approximate solutions to problems that involve classical fractional derivatives. The results obtained from this algorithm are compared with those obtained from the perturbation iteration method (PIM), the variational iteration method (VIM), and the Bezier curve method (BCM). All solutions are tested with numerical simulations. The study found that the new PIA algorithm performs better than the PIM, VIM, and BCM, achieving high accuracy and low computational cost. One significant advantage of this algorithm is that the solutions obtained have established that the fractional values of alpha, specifically α, significantly influencing the accuracy of the outcome and the associated computational cost.

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Section: Introductionmentioning

confidence: 99%

New Perturbation–Iteration Algorithm for Nonlinear Heat Transfer of Fractional Order

Abdel Aal

2024

Fractal Fract

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“…Among the applications of the operational matrix method, we can mention the poly‐Bernoulli operational matrix, Jacobi wavelet operational matrix of fractional integration, etc., which are used for fractional delay differential equations and fractional integrodifferential equations, respectively. One of the main advantages of using operational matrix methods compared to other methods is its easy and applicable use in different subsidy systems 31,32 . In mathematics, there is a branch called graph theory that deals with the subject of vertices and edges.…”

Section: Introductionmentioning

confidence: 99%

“…One of the main advantages of using operational matrix methods compared to other methods is its easy and applicable use in different subsidy systems. 31,32 In mathematics, there is a branch called graph theory that deals with the subject of vertices and edges. Considering a set of e vertices called…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

Eidinejad,

Saadati,

Vahidi

et al. 2023

Int J Numerical Modelling

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We study the two‐dimensional stochastic time‐fractional Sine–Gordon equation (2D ST‐FS‐G) and provide a solution for it. We use the new clique polynomial method to obtain a numerical solution to the 2D TFS‐G equation. In this technique, the clique polynomial is considered as a basic function for operational matrices. By converting the 2D ST‐FS‐G equation to algebraic equations, a solution is obtained for the desired equation, which clearly shows that this approach is suitable and accurate for dealing with the equation. We further present the error boundary for the obtained approximation of the desired three‐variable function based on the clique polynomial. Finally, we compare some numerical results obtained with the exact ones by a few practical examples.

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“…Nowadays, differential equations of fractional order as extension of the differential equations of integer order play a fundamental role in the modeling of many scientific, practical, and important problems [1][2][3][4][5][6][7][8][9][10]. Recently, several numerical methods have been proposed to simulate various types of fractional-order systems [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

et al. 2023

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In this paper, we propose an effective numerical method, based on the use of two-dimensional Shifted fractional-order Gegenbauer Multi-wavelets, for finding the approximate solutions of the time-fractional distributed order non-linear partial differential equations. The method is applied to solve fractional distributed order non-linear Klein-Gordon equation, numerically. We derive an exact formula for the Riemann–Liouville fractionalintegral operator for the Shifted fractional Gegenbauer Multi-wavelets.Applying function approximations obtained by this method turns the considered equation into a system of algebraic equations. Error estimation and convergence analysis of the methodare also studied. Some numerical examples are included to show and check the effectiveness of the proposed method.

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An Efficient Numerical Scheme for Solving Multiorder Tempered Fractional Differential Equations via Operational Matrix

Cited by 5 publications

References 27 publications

New Perturbation–Iteration Algorithm for Nonlinear Heat Transfer of Fractional Order

New Perturbation–Iteration Algorithm for Nonlinear Heat Transfer of Fractional Order

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

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