About Some Possible Implementations of the Fractional Calculus

Velasco, M. Pilar; Usero, D.; Jiménez, Salvador; Vázquez, Luis; Vázquez-Poletti, José Luis; Mortazavi, Masood

doi:10.3390/math8060893

Cited by 17 publications

(8 citation statements)

References 34 publications

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“…Fractional calculus in ref. [4], is a highly complicated branch of mathematics, that resides under the umbrella of mathematical analysis. It has a plethora of applications across the scientific fields, including, but not limited to chemistry, engineering, and physics.…”

Section: The Mathematical Toolkit 21 Fractional Calculusmentioning

confidence: 99%

Free Fractional Gauge Fields: An Elementary Model Based on the Standard Model (SM)

Eastmond

2024

Preprint

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We study the Lagrangian density for freely propagating gauge fields on the basis of the Stan- dard Model (SM); namely its fractional counterpart. From this, we attain some free fractional Klein- Gordon equations, associated with the electromagnetic, weak, and strong forces. In addition to that, we verify that the Lagrange density is gauge invariant by virtue of the fractional Hessian condition. Also, using the fractional energy density we establish the fractional continuity equation for the freely propa- gating gauge bosons. And finally, a fractional analog of the Riemann-Silberstein vector and its equation is presented. Hence, we formulate fractional analogs of the Riemann-Silberstein and gravitoelectro- magnetic equations, which are compared to the fractional Maxwell’s equations.

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Section: The Mathematical Toolkit 21 Fractional Calculusmentioning

confidence: 99%

Free Fractional Gauge Fields: An Elementary Model Based on the Standard Model (SM)

Eastmond

2024

Preprint

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“…", the insufficient but short answer of Leibniz was: "It leads to a paradox, from which one day useful consequences will be drawn". In the same year, the derivative of general order was mentioned in the letter from Leibniz to J. Bernoulli [3,10,15,29,45].…”

Section: Introductionmentioning

confidence: 99%

A Comprehensive Study of Fractional-Order Derivative and Their Interplay with Basic Functions

Pandey,

Phaijoo,

Gurung

2024

J. Nep. Math. Soc.

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Fractional calculus from the nineteenth century to date has gained considerable attention due to its versatile applications in various scientific and engineering domains. This work examines the complex relationship between fractional-order derivative and basic functions, unraveling the profound interplay between mathematics and simulation. In this study, we illustrate the Mittag-Leffler function, Grunwald-Letnikov’s, Riemann-Liouville’s, and Caputo’s fractional derivative and integral are presented with examples of basic functions and their graphical presentations. The purpose of this study is to examine the features of fractional derivatives from the perspective of researchers’ motivations and interests.

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“…Differential equations containing derivatives of fractional (i.e., of any real positive) order are useful in various fields of science and engineering, especially when modelling real-life processes with memory properties [3,23,26]. For the fundamental theory of fractional derivatives and equations containing them we refer the reader to [8,11,21,22], see also [25].…”

Section: Introductionmentioning

confidence: 99%

Collocation Based Approximations for a Class of Fractional Boundary Value Problems

Soots

Lätt

Pedas

2023

Mathematical Modelling and Analysis

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A boundary value problem for fractional integro-differential equations with weakly singular kernels is considered. The problem is reformulated as an integral equation of the second kind with respect to, the Caputo fractional derivative of y of order α, with 1 < α < 2, where y is the solution of the original problem. Using this reformulation, the regularity properties of both y and its Caputo derivative z are studied. Based on this information a piecewise polynomial collocation method is developed for finding an approximate solution zN of the reformulated problem. Using zN, an approximation yN for y is constructed and a detailed convergence analysis of the proposed method is given. In particular, the attainable order of convergence of the proposed method for appropriate values of grid and collocation parameters is established. To illustrate the performance of our approach, results of some numerical experiments are presented.

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About Some Possible Implementations of the Fractional Calculus

Cited by 17 publications

References 34 publications

Free Fractional Gauge Fields: An Elementary Model Based on the Standard Model (SM)

Free Fractional Gauge Fields: An Elementary Model Based on the Standard Model (SM)

A Comprehensive Study of Fractional-Order Derivative and Their Interplay with Basic Functions

Collocation Based Approximations for a Class of Fractional Boundary Value Problems

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