Compact implicit difference approximation for time-fractional diffusion-wave equation

Ali, Umair; Iqbal, Azhar; Sohail, Muhammad; Abdullah, Farah Aini; Khan, Zohaib

doi:10.1016/j.aej.2021.09.005

Cited by 29 publications

(16 citation statements)

References 27 publications

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“…The integer order shows remembrance, whereas, the memory function represents the form of FO. The FO form is applied to the application of the real-world systems [26][27][28][29][30][31][32][33][34]. Few novel features of the ANNs-LMBM for the FO-NEE model are provided as:…”

Section: Introductionmentioning

confidence: 99%

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

Weera¹,

Zamart²,

Sabir³

et al. 2023

Computers, Materials &Amp; Continua

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The motive of these investigations is to provide the importance and significance of the fractional order (FO) derivatives in the nonlinear environmental and economic (NEE) model, i.e., FO-NEE model. The dynamics of the NEE model achieves more precise by using the form of the FO derivative. The investigations through the non-integer and nonlinear mathematical form to define the FO-NEE model are also provided in this study. The composition of the FO-NEE model is classified into three classes, execution cost of control, system competence of industrial elements and a new diagnostics technical exclusion cost. The mathematical FO-NEE system is numerically studied by using the artificial neural networks (ANNs) along with the Levenberg-Marquardt backpropagation method (ANNs-LMBM). Three different cases using the FO derivative have been examined to present the numerical performances of the FO-NEE model. The data is selected to solve the mathematical FO-NEE system is executed as 70% for training and 15% for both testing and certification. The exactness of the proposed ANNs-LMBM is observed through the comparison of the obtained and the Adams-Bashforth-Moulton database results. To ratify the aptitude, validity, constancy, exactness, and competence of the ANNs-LMBM, the numerical replications using the state transitions, regression, correlation, error histograms and mean square error are also described.

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Section: Introductionmentioning

confidence: 99%

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

Weera¹,

Zamart²,

Sabir³

et al. 2023

Computers, Materials &Amp; Continua

View full text Add to dashboard Cite

show abstract

“…As is known to all, the nonlinear partial differential equations (NLPDEs) play a key role in modeling the different complex phenomena in the fields of optics, 1–5 fluid mechanics, 6,7 plasma physics, 8–10 vibration, 11–13 diffusion, 14,15 magnetic field, 16 and so on 17–20 . It is of great significance to study the exact solution of the NLPDEs since it can enable us to better understand and make use of natural phenomena.…”

Section: Introductionmentioning

confidence: 99%

On the non‐differentiable exact solutions of the (2 + 1)‐dimensional local fractional breaking soliton equation on Cantor sets

Wang

2022

Math Methods in App Sciences

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In this article, a new (2 + 1)‐dimensional local fractional breaking soliton equation is derived with the local fractional derivative. Applying the traveling wave transform of the non‐differentiable type, the (2 + 1)‐dimensional local fractional breaking soliton equation is converted into a nonlinear local fractional ordinary differential equation. By defining a set of elementary functions on Cantor sets, a novel analytical technique namely the Mittag–Leffler function‐based method is employed for the first time ever to construct the exact solutions. The solutions on the Cantor sets are presented via the 3‐D contours. It reveals that the proposed method is effective and powerful and is expected to give some inspiration for the study of the local fractional PDEs.

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“…e study and application of arbitrary-order derivatives and integrals are associated with fractional calculus. e use of fractional-order calculus in a variety of elds of science and engineering, including geometric phenomena, has sparked a lot of interest in this area [1]. e rst discussion of fractional calculus took place between Leibniz and L'Hospital at the end of the seventeenth century [2].…”

Section: Introductionmentioning

confidence: 99%

The Rayleigh–Stokes Problem for a Heated Generalized Second-Grade Fluid with Fractional Derivative: An Implicit Scheme via Riemann–Liouville Integral

Ganie

Saeed

et al. 2022

Mathematical Problems in Engineering

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The goal of this study is to use the fast algorithm to solve the Rayleigh–Stokes problem for heated generalized second-grade fluid (RSP-HGSGF) with Riemann–Liouville time fractional derivative using the fast algorithm. The modified implicit scheme, which is formulated by the Riemann–Liouville integral formula and applied to the fractional RSP-HGSGF, is proposed. Numerical experiments will be carried out to demonstrate that the scheme is simple to implement, and the results will reveal the best way to implement the suggested technique. The proposed scheme’s stability and convergence will be examined using the Fourier series. The method is stable, and the approximation solution approaches the exact solution. A numerical demonstration will be provided to demonstrate the applicability and viability of the suggested strategy.

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Compact implicit difference approximation for time-fractional diffusion-wave equation

Cited by 29 publications

References 27 publications

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

Fractional Order Environmental and Economic Model Investigations Using Artificial Neural Network

On the non‐differentiable exact solutions of the (2 + 1)‐dimensional local fractional breaking soliton equation on Cantor sets

The Rayleigh–Stokes Problem for a Heated Generalized Second-Grade Fluid with Fractional Derivative: An Implicit Scheme via Riemann–Liouville Integral

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