From Circular to Bessel Functions: A Transition through the Umbral Method

Dattoli, G.; Palma, E. Di; Licciardi, Silvia; Sabia, E.

doi:10.3390/fractalfract1010009

Cited by 8 publications

(5 citation statements)

References 13 publications

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“…The range of practical and theoretical issues whose mathematical description is based on functions of this type is very wide. An example is selected recent works [47][48][49][50][51][52][53], in which the theory presented in this manuscript can In this work, only examples related to fluid mechanics are presented, as this is the main research topic of the author of this work. The range of practical and theoretical issues whose mathematical description is based on functions of this type is very wide.…”

Section: Other Series and Their Possible Solutionsmentioning

confidence: 99%

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Infinite Series Based on Bessel Zeros

Urbanowicz

2023

Applied Sciences

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An interesting series based on Bessel function roots (zeros) is discussed and numerically analyzed. The novel-derived simplified general solutions are based on Lommel polynomials. This kind of series can have a large practical use in many scientific areas, such as solid mechanics, fluid mechanics, thermodynamics, electronics, physics, etc. Some practical examples connected with fluid mechanics are provided in this paper. The errors in Afanasiev solutions are corrected. In addition, the main solution for the series analyzed by Baricz and Angel is presented.

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Section: Other Series and Their Possible Solutionsmentioning

confidence: 99%

Section: Other Series and Their Possible Solutionsmentioning

confidence: 99%

Infinite Series Based on Bessel Zeros

Urbanowicz

2023

Applied Sciences

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“…Although it seems quite natural to extend classical operators and models to fractional order operators, there are several unsolved problems related to such a non-trivial generalization. For instance, in [1], the authors show that the Gaussian provides the umbral image of Bessel and circular functions. In particular, they are able to single out the role played by spherical Bessel functions, and by a family of associated auxiliary polynomials, as transition elements between these families of functions.…”

Section: Numerical Aspects In Fractional Dynamicsmentioning

confidence: 99%

Fractional Dynamics

Cattani

Spigler

2018

Fractal Fract

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Modelling, simulation, and applications of Fractional Calculus have recently become increasingly popular subjects, with impressive growth concerning applications. The founding and limited ideas on fractional derivatives have achieved an incredibly valuable status. The variety of applications in mathematics, physics, engineering, economics, biology, and medicine, have opened new, challenging fields of research. For instance, in soil mechanics, a suitable definition of the fractional operator has shed some light on viscoelasticity, explaining memory effects on materials. Needless to say, these applications require the development of practical mathematical tools in order to extract quantitative information from models, newly reformulated in terms of fractional differential equations. Even confining ourselves to the field of ordinary differential equations, the well-known Bagley-Torvik model showed that fractional derivatives may actually arise naturally within certain physical models, and are not merely fanciful mathematical generalizations. This Special Issue focuses on the most recent advances in fractional calculus, applied to dynamic problems, linear and nonlinear fractional ordinary and partial differential equations, integral fractional differential equations, and stochastic integral problems arising in all fields of science, engineering, and other applied fields. In this issue, we have collected several significant papers devoted to applications of fractional methods with a focus on dynamical aspects. The applications range from theoretical mathematical-numerical aspects [1,2] to bio-medical subjects [3-7]. Applications to complex materials are investigated in [8], aiming at proposing a generalized definition of fractional operators. Special diffusion models are studied in [9-11].

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“…They proposed a way to introduce a new family of polynomials that connects Laguerre polynomials with trigonometric functions. This family of polynomials is known as λ-polynomials [10]. They expanded on this concept by adding a parameter β and generalizing it to associated-λ polynomials.…”

Section: Introductionmentioning

confidence: 99%

Exploring Zeros of Hermite-λ Matrix Polynomials: A Numerical Approach

Alatawi,

Kumar,

Raza

et al. 2024

Mathematics

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This article aims to introduce a set of hybrid matrix polynomials associated with λ-polynomials and explore their properties using a symbolic approach. The main outcomes of this study include the derivation of generating functions, series definitions, and differential equations for the newly introduced two-variable Hermite λ-matrix polynomials. Furthermore, we establish the quasi-monomiality property of these polynomials, derive summation formulae and integral representations, and examine the graphical representation and symmetric structure of their approximate zeros using computer-aided programs. Finally, this article concludes by introducing the idea of 1-variable Hermite λ matrix polynomials and their structure of zeros using a computer-aided program.

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From Circular to Bessel Functions: A Transition through the Umbral Method

Cited by 8 publications

References 13 publications

Infinite Series Based on Bessel Zeros

Infinite Series Based on Bessel Zeros

Fractional Dynamics

Exploring Zeros of Hermite-λ Matrix Polynomials: A Numerical Approach

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