A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method

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

doi:10.1088/1402-4896/acba5d

Cited by 4 publications

(4 citation statements)

References 34 publications

(60 reference statements)

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“…Most of the matrices used in this method have many zero entries, and this reduces the time required for computational processes. The subsequent paper aims to establish a numerical solution for the helium burning system (HBN) FFDEs [27]. As the other inputs, the authors model the nine FF gas models and discuss the impact of the fractal-fractional parameters on product abundances.…”

Section: Work In Progressmentioning

confidence: 99%

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Baleanu,

Karaca,

Vázquez

et al. 2023

Phys. Scr.

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Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems’ exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is ‘the study of every single phenomenon’ due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems’ actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.

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Section: Work In Progressmentioning

confidence: 99%

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Baleanu,

Karaca,

Vázquez

et al. 2023

Phys. Scr.

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“…Fractional calculus is one of the important topics in science, engineering, physics, and other disciplines due to its representation in the mathematical model of natural events [1]. The spectral method is considered a well-known numerical method to approximate the solution of various fractional differential equations (FDEs) [2][3][4] and fractional integral equations (FIEs) [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

An efficient approach for solving a class of fractional anomalous diffusion equation with convergence

Rashidinia,

Molavi-Arabshahi,

Yousefi

2024

Phys. Scr.

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This article presents a study on Fractional Anomalous Diﬀusion (FAD) and proposes a novel numerical algorithm for solving Cupoto’s type fractional sub-diﬀusion equations. to convert the fractional model into a set of nonlinear algebraic equations. These equations are eﬃciently solved using the Levenberg-Marquardt algorithm. The study provides the error analysis to validate the proposed method. The eﬀectiveness and accuracy of the method are demonstrated through several test problems, and its performance and reliability are compared with other existing methods in the literature. The results indicate that the proposed method is a reliable and eﬃcient technique for solving fractional sub-diﬀusion equations, with better accuracy and computational eﬃciency than other existing methods. The study’s ﬁndings have important implications for researchers working in the ﬁeld of fractional calculus. They could provide a valuable tool for solving sub-diﬀusion equations in various applications, including physics, chemistry, biology, and engineering.

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“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well-known algorithms can solve FDEs, such as operational matrix, [22][23][24][25][26] Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), [31][32][33] and finite difference. 34,35 In comparison to classic numerical approaches, the approximate calculation of ANN appears to be less sensitive to the spatial dimension.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well‐known algorithms can solve FDEs, such as operational matrix, 22‐26 Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), 31‐33 and finite difference 34,35 …”

Section: Introductionmentioning

confidence: 99%

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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A novel fractal-fractional analysis of the stellar helium burning network using extended operational matrix method

Cited by 4 publications

References 34 publications

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations

An efficient approach for solving a class of fractional anomalous diffusion equation with convergence

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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