On (2 + 1)-dimensional physical models endowed with decoupled spatial and temporal memory indices⋆

Jaradat, Imad; Alquran, Marwan; Yousef, Feras; Momani, Shaher; Băleanu, Dumitru

doi:10.1140/epjp/i2019-12769-8

Cited by 15 publications

(4 citation statements)

References 42 publications

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“…How to simplify this computation is an important research topic in the fractional field. Many approximate approaches have been proposed for this issue [7][8][9][10][11][12]. Without exception, only an approximation solution, but not an exact solution, can be obtained by any approximate method.…”

Section: Introductionmentioning

confidence: 99%

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Analyzing the Approximate Error and the Applicable Condition of Fractional Reduced Differential Transform Method

2024

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The fractional reduced differential transform method is a finite iterative method based on infinite fractional expansions. The result obtained is the approximation of the real value. There are few reports on the approximate error and the applicable condition. In this paper, according to the fractional expansions, we study the factors related to the approximate errors. Our research shows that the approximate errors relate not only to fractional order but also to time $t$ and increase rapidly with time $t$. This method can only be applied within a certain time range and the time range is relevant to fractional order and fractional expansions. Then, many obtained achievements may be incorrect if the applicable conditions are not satisfied. Some examples presented in this paper verify our analysis.

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Section: Introductionmentioning

confidence: 99%

“…Or else, the achievements obtained may be incorrect [13][14][15]. For example, Ahmadian A. obtained the approximate solution in the time domain by its approximate value in Laplace domain [12], but Zhao L. etc [16] analyzed the approximate error and pointed out that this approach may mislead.…”

Section: Introductionmentioning

confidence: 99%

Analyzing the Approximate Error and the Applicable Condition of Fractional Reduced Differential Transform Method

2024

Preprint

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“…However, several studies have revealed that the power-law memory instilled in process and materials could also be in the space coordinate [46,47]. Motivated by this lack, some analytical methods were recently developed to handle and study mathematical models embedded entirely in a fractional space [48][49][50][51]. Continuing in this direction, this research examines the joint influence for the existence of both time and space fractional derivatives in higher-dimensional PDEs.…”

Section: Introductionmentioning

confidence: 99%

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

et al. 2020

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The purpose of this work is to analytically simulate the mutual impact for the existence of both temporal and spatial Caputo fractional derivative parameters in higher-dimensional physical models. For this purpose, we employ the γ γ γ-Maclaurin series along with an amendment of the power series technique. To supplement our idea, we present the necessary convergence analysis regarding the γ γ γ-Maclaurin series. As for the application side, we solved versions of the higher-dimensional heat and wave models with spatial and temporal Caputo fractional derivatives in terms of a rapidly convergent γ γ γ-Maclaurin series. The method performed extremely well, and the projections of the obtained solutions into the integer space are compatible with solutions available in the literature. Finally, the graphical analysis showed a possibility that the Caputo fractional derivatives reflect some memory characteristics.

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“…Arikoglu and Ozkol [17] proposed an analytical technique, called the fractional differential transform method (FDTM), for solving (non)linear differential equations endowed with one memory index α. Very recently, Jaradat et al [18,19] developed this method to address (non)linear differential equations endowed with two memory indices α 1 and α 2 . It is worth mentioning here that some recent advancements in analytical methods can be also found in [20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Ternary-fractional differential transform schema: theory and application

et al. 2019

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In this article, we propose a novel fractional generalization of the three-dimensional differential transform method, namely the ternary-fractional differential transform method, that extends its applicability to encompass initial value problems in the fractal 3D space. Several illustrative applications, including the Schrödinger, wave, Klein-Gordon, telegraph, and Burgers' models that are fully embedded in the fractal 3D space, are considered to demonstrate the superiority of the proposed method compared with other generalized methods in the literature. The obtained solution is expressed in a form of an α α α-fractional power series, with easily computed coefficients, that converges rapidly to its closed-form solution. Moreover, the projection of the solutions into the integer 3D space corresponds with the solutions of the classical copies for these models. This reveals that the suggested technique is effective and accurate for handling many other linear and nonlinear models in the fractal 3D space. Thus, research on this trend is worth tracking.

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On (2 + 1)-dimensional physical models endowed with decoupled spatial and temporal memory indices⋆

Cited by 15 publications

References 42 publications

Analyzing the Approximate Error and the Applicable Condition of Fractional Reduced Differential Transform Method

Analyzing the Approximate Error and the Applicable Condition of Fractional Reduced Differential Transform Method

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

Ternary-fractional differential transform schema: theory and application

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