A new numerical method to solve fractional differential equations in terms of Caputo-Fabrizio derivatives

Mahatekar, Yogita; Scindia, Pallavi S; Kumar, Pushpendra

doi:10.1088/1402-4896/acaf1a

Cited by 15 publications

(10 citation statements)

References 28 publications

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“…Consequently, employing the homotopy analysis method to derive analytic solutions for the model [46,47] may enhance the sensitivity analysis, as these analytic solutions could potentially reduce computational expenses. Furthermore, it may be of interest to conduct a research study aimed at developing a computational tool for numerically solving the model, based on the methods proposed in the referenced papers [42,43]. In addition, addressing the positivity of the solutions of the model is an important consideration [48].…”

Section: Discussionmentioning

confidence: 99%

“…However, the convergence and accuracy of this method have not yet been evaluated. Recently, novel numerical methods have been developed to solve fractional-order ordinary differential equations efficiently, and their analyses have yielded several interesting results [42,43]. The employment of these methods in creating a new Python solver for fractional-order ordinary differential equations could lead to intriguing research opportunities.…”

Section: Numerical Methods For Solving the Fractional Differential Eq...mentioning

confidence: 99%

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Fractional Modelling of H2O2-Assisted Oxidation by Spanish broom peroxidase

Mai,

Nhan

2024

Mathematics

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The H2O2-assisted oxidation by a peroxidase enzyme takes place to help plants maintain the concentrations of organic compounds at physiological levels. Cells regulate the oxidation rate by inhibiting the action of this enzyme. The cells use two inhibitory processes to regulate the enzyme: a noncompetitive substrate inhibitory process and a competitive substrate inhibitory process. Numerous applications of peroxidase have been developed in clinical biochemistry, enzyme immunoassays, the treatment of waste water containing phenolic compounds, the synthesis of various aromatic chemicals, and the removal of peroxide from industrial wastes. The kinetic mechanism of the Spanish broom peroxidase enzyme is a Ping Pong Bi Bi mechanism with the presence of competitive inhibition by substrates. A mathematical model may help in identifying the key mechanism from amongst a set of competing mechanisms. In this study, we developed a fractional mathematical model to describe the H2O2-supported oxidation by the enzyme Spanish broom peroxidase. Numerical simulations of the model produced results that are consistent with the known behaviour of Spanish broom peroxidase. Finally, some future investigations of the study are briefly indicated as well.

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Section: Discussionmentioning

confidence: 99%

Section: Numerical Methods For Solving the Fractional Differential Eq...mentioning

confidence: 99%

Fractional Modelling of H2O2-Assisted Oxidation by Spanish broom peroxidase

Mai,

Nhan

2024

Mathematics

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“…The launching of a new fractional derivative by Caputo and Fabrizio has been followed by some related theoretical and applied results, and the interest in this approach is because of the requirement to describe material heterogeneities and structures with different scales. A new numerical method concerning the solution of fractional differential equations is presented in terms of Caputo-Fabrizio derivatives in the next study [25]. As a contribution, the authors of the study whose basic concepts including FC, numerical analysis as well as fixed point theory form the basis, derive a new numerical method to solve fractional differential equations with Caputo-Fabrizio derivatives.…”

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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“…The aforementioned problems were all buried by this derivative. In this regard, numerous numerical techniques have been broadly adopted and developed to solve a wide range of linear and nonlinear problems in FPDEs, such as the Adomian decomposition method [24], residual power series method [25], iterative Laplace transform method [26], Laplace homotopy analysis method [27], homotopy analysis method [28], variational iteration technique [29], homotopy perturbation technique [30], reduced differential transform method [31], modified variational iteration method [32], L1predictor-corrector method [33], and Daftardar-Gejji and Jafari's iterative method [34].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Technique to Study Time Fractional Extended Fisher-Kolmogorov Equation

Pavani,

Raghavendar,

Aruna

2024

Contemp. Math.

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Reaction-diffusion partial differential equations are among the most widely used equations in applied mathematical modelling. In this study, we examine the solutions of one such equation, namely, the time-fractional extended Fisher-Kolmogorov equation. This equation is widely used in the study of population growth and wave propagation dynamics. The technique we consider is a combination of the natural transform and the Adomian decomposition method. Fractional derivatives are considered with singular and non-singular kernels. The existence and uniqueness of the solutions are presented. We analyse two different cases of the proposed problem to determine the validity and efficacy of the proposed scheme. Additionally, numerical simulation is shown, and the nature of the achieved solution is captured in terms of plots for various fractional orders. The outcome demonstrates that the method is straightforward, efficient, and dependable. The proposed method does not require any predetermined assumptions, linearization, perturbation, or discretization, and it prevents rounding errors. Therefore, the technique is ready to be implemented for a variety of nonlinear time fractional partial differential equations.

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A new numerical method to solve fractional differential equations in terms of Caputo-Fabrizio derivatives

Cited by 15 publications

References 28 publications

Fractional Modelling of H2O2-Assisted Oxidation by Spanish broom peroxidase

Fractional Modelling of H2O2-Assisted Oxidation by Spanish broom peroxidase

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

An Efficient Technique to Study Time Fractional Extended Fisher-Kolmogorov Equation

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