An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative

Shloof, A. M.; Senu, Norazak; Ahmadian, Ali; Salahshour, Soheil

doi:10.1016/j.matcom.2021.04.019

Cited by 32 publications

(19 citation statements)

References 28 publications

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“…The fractional calculus theory is a study of the differential/integral operators of arbitrary order, as the fractional calculus was initially considered a purely mathematical idea with no applications; nevertheless, in recent decades, a noteworthy evolution has been achieved from both theoretical and applied points of views. Most of the fractional differential problems are difficult-sometimes impossible-to solve analytically; for this reason, there is a brawny need to handle them numerically [19][20][21].…”

Section: Gegenbauer (Ultraspherical) Cmentioning

confidence: 99%

“…Legendre operational matrix method [19]; • Variation of parameters method [27]; • Modified homotopy perturbation method [28]; • Trigonometric basic functions [22];…”

mentioning

confidence: 99%

“…Table 1. Comparison between our method [19,22], methods for φ(t) of Example 1, with n = 15, α = 0.9, β = 1 and ρ = 1.01. 1, it is worth mentioning here that we obtained the numerical approximate solutions in a few seconds, which emphasizes the effectiveness of the method.…”

mentioning

confidence: 99%

“…Comparison between our method[19,[27][28][29], methods for φ(t) of Example 1, with n = 15, α = β = ρ = 1.…”

mentioning

confidence: 99%

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Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Youssri

2021

Fractal Fract

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Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

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Section: Gegenbauer (Ultraspherical) Cmentioning

confidence: 99%

“…Legendre operational matrix method [19]; • Variation of parameters method [27]; • Modified homotopy perturbation method [28]; • Trigonometric basic functions [22];…”

mentioning

confidence: 99%

mentioning

confidence: 99%

“…Comparison between our method[19,[27][28][29], methods for φ(t) of Example 1, with n = 15, α = β = ρ = 1.…”

mentioning

confidence: 99%

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Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Youssri

2021

Fractal Fract

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“…Initially, fractional calculus was treated as an abstract mathematical idea with almost no application. But within the last few decades, a significant development has been observed in the fields of FC, such as Geo-Hydrology, chaotic processes [1,2], wave propagation, rheology, finance system [3], groundwater flow, and fluid mechanics [4,5], fractional-order dynamical systems in control theory [6], fractional order controller [7], fractional Brownian motion [8], generalized Mittag-Leffler function [9] etc. Until now, various operators of arbitrary order have been introduced in which the two most commonly used with singular kernels are Riemann-Liouville (RL) and Caputo.…”

Section: Introductionmentioning

confidence: 99%

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Kamran¹,

Ahmad²,

Shah³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects. Using the Laplace transform for solving differential equations, however, sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analytical means. Thus, we need numerical inversion methods to convert the obtained solution from Laplace domain to a real domain. In this paper, we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with order α, β. Our proposed numerical scheme is based on three main steps. First, we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense, and then into Caputo sense. Secondly, we transform the fractional differential equation in Caputo sense to an equivalent equation in Laplace space. Then the solution of the transformed equation is obtained in Laplace domain. Finally, the solution is converted into the real domain using numerical inversion of Laplace transform. Three inversion methods are evaluated in this paper, and their convergence is also discussed. Three test problems are used to validate the inversion methods. We demonstrate our results with the help of tables and figures. The obtained results show that Euler's and Talbot's methods performed better than Stehfest's method.

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A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Shloof

Ahmadian

Senu

et al. 2023

Numerical Meth Engineering

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Artificial neural networks have great potential for learning and stability in the face of tiny input data changes. As a result, artificial intelligence techniques and modeling tools have a growing variety of applications. To estimate a solution for fractal‐fractional differential equations (FFDEs) of high‐order linear (HOL) with variable coefficients, an iterative methodology based on a mix of a power series method and a neural network approach was applied in this study. In the algorithm's equation, an appropriate truncated series of the solution functions was replaced. To tackle the issue, this study uses a series expansion of an unidentified function, where this function is approximated using a neural architecture. Some examples were presented to illustrate the efficiency and usefulness of this technique to prove the concept's applicability. The proposed methodology was found to be very accurate when compared to other available traditional procedures. To determine the approximate solution to FFDEs‐HOL, the suggested technique is simple, highly efficient, and resilient.

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An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative

Cited by 32 publications

References 28 publications

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

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