Higher-order fractional linear multi-step methods

Marasi, Hamidreza; Derakhshan, Mohammadhossein; Joujehi, A Soltani; Kumar, Pushpendra

doi:10.1088/1402-4896/acad42

Cited by 7 publications

(3 citation statements)

References 21 publications

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“…Fractional differential inequalities are also helpful to establish the properties of solutions of different problems in flow phenomena and biomathematics. Correspondingly, another study proposes two arrays that contain the coefficients of fractional Adams-Bashforth and Adams-Moulton methods along with the recursive relations to generate the elements of the related arrays [24]. As contribution, the authors demonstrate the efficiency of the new method compared to some existing earlier brings about an explanatory test problem.…”

Section: Work In Progressmentioning

confidence: 94%

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: 94%

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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“…Along with the derivation of the algorithm of the method, error and stability were analyzed, and the validity and effectiveness of the method were briefly explored. Marasi et al [31] provided two arrays that contained the coefficients of the fractional Adams-Bashforth and Adams-Moulton techniques, as well as recursive relations to generate the members of these arrays. Kumar et al [32] were to propose generalized forms of three well-known fractional numerical methods, namely Euler, Runge-Kutta 2-step, and Runge-Kutta 4-step, respectively.…”

Section: Introductionmentioning

confidence: 99%

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

Teppawar,

Ingle,

Muneshwar

2024

Contemp. Math.

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In this article, we use a new technique called conformable fractional reduced differential transform (CFRDT) with Adomian decomposition to estimate the solution of one and two-dimensional time-fractional partial linear and nonlinear differential equations with initial values. We explain the convergence analysis of this technique. The obtained results illustrate that the novel method is efficient and easy to use to find approximate solutions for the time-fractional partial differential equations (PDEs). Thus, the suggested method has a significant impact on how engineering, physics, and other disciplines solve fractional PDEs. Furthermore, we analyze the solution of problems with a 2D or 3D graphical representation by using Mathematica software.

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“…Spectral methods employ orthogonally based functions such as the Jacobi and Chebyshev polynomials to approximate solutions of fractional differential equations [31,32]. Fractional multistep methods, which can be viewed as an extension of classical multistep methods, were studied in [33,34]. Methods using spline interpolation were presented in [35,36].…”

Section: Introductionmentioning

confidence: 99%

Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

Dimitrov,

Georgiev,

Todorov

2023

Fractal Fract

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In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments.

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Higher-order fractional linear multi-step methods

Cited by 7 publications

References 21 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

Solving Nonlinear Time-Fractional Partial Differential Equations Using Conformable Fractional Reduced Differential Transform with Adomian Decomposition Method

Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations

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