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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 August 2023) | Viewed by 3022

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Dr. Clemente Cesarano

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Section of Mathematics, International Telematic University, Corso Vittorio Emanuele II, 39, 00186 Rome, Italy
Interests: special functions; orthogonal polynomials; differential equations; operator theory; multivariate approximation theory; lie algebra
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Dear Colleagues,

The theory of generalized orthogonal polynomials and special functions is applied in different branches of pure and applied mathematics, as well as in physics. A combination of techniques involving methods of an algebraic nature and numerical methods may offer a powerful tool to solve problems in pure and applied mathematics. In the last years, the combined use of operational methods, orthogonal polynomials, and special functions has provided solutions that are hardly achievable with conventional means. Furthermore, the structural properties of polynomials in the framework of standard L2 orthogonality with respect to a Borel measure (or a weight function) have been deeply studied for other patterns of orthogonality, like multiple orthogonal polynomials, orthogonal polynomials in several variables, or Sobolev orthogonal polynomials.

Dr. Clemente Cesarano
Guest Editor

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Published Papers (4 papers)

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Research

Open AccessArticle

q-Analogs of

H_{n, m}^{r} (σ)

and Their Applications

and

Mathematics 2023, 11(19), 4159; https://doi.org/10.3390/math11194159 - 03 Oct 2023

Viewed by 191

Abstract

In this paper, inspired by recent works, we define q-analogs of

H_{n, m} (σ)

and

H_{n, m}^{r} (σ) .

By implementing them, we obtain new interesting results by taking the derivative or using generating functions. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions-II)

Open AccessArticle

Fractal Divergences of Generalized Jacobi Polynomials

Răzvan-Cornel Sfetcu

and

Vasile Preda

Mathematics 2023, 11(16), 3500; https://doi.org/10.3390/math11163500 - 13 Aug 2023

Viewed by 301

Abstract

The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation for the study of the information measures of orthogonal polynomials is because these polynomials appear in the densities of many quantum mechanical systems with shape-invariant potentials (e.g., the harmonic oscillator and the hydrogenic systems). With the help of a sequence of some generalized Jacobi polynomials, we define a sequence of discrete probability distributions. We introduce fractal Kullback–Leibler divergence, fractal Tsallis divergence, and fractal Rényi divergence between every element of the sequence of probability distributions introduced above and the element of the equiprobability distribution corresponding to the same index. Practically, we obtain three sequences of fractal divergences and show that the first two are convergent and the last is divergent. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions-II)

Open AccessArticle

Solutions to the (4+1)-Dimensional Time-Fractional Fokas Equation with M-Truncated Derivative

Wael W. Mohammed

Clemente Cesarano

and

Farah M. Al-Askar

Mathematics 2023, 11(1), 194; https://doi.org/10.3390/math11010194 - 30 Dec 2022

Cited by 12 | Viewed by 1246

Abstract

In this paper, we consider the (4+1)-dimensional fractional Fokas equation (FFE) with an M-truncated derivative. The extended tanh–coth method and the Jacobi elliptic function method are utilized to attain new hyperbolic, trigonometric, elliptic, and rational fractional solutions. In addition, we generalize some previous results. The acquired solutions are beneficial in analyzing definite intriguing physical phenomena because the FFE equation is crucial for explaining various phenomena in optics, fluid mechanics and ocean engineering. To demonstrate how the M-truncated derivative affects the analytical solutions of the FFE, we simulate our figures in MATLAB and show several 2D and 3D graphs. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions-II)

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Open AccessArticle

The Analytical Solutions of the Stochastic mKdV Equation via the Mapping Method

Wael W. Mohammed

Farah M. Al-Askar

and

Clemente Cesarano

Mathematics 2022, 10(22), 4212; https://doi.org/10.3390/math10224212 - 11 Nov 2022

Cited by 15 | Viewed by 713

Abstract

Here, we analyze the (2+1)-dimensional stochastic modified Kordeweg–de Vries (SmKdV) equation perturbed by multiplicative white noise in the Stratonovich sense. We apply the mapping method to obtain new trigonometric, elliptic, and rational stochastic fractional solutions. Because of the importance of the KdV equation in characterizing the behavior of waves in shallow water, the obtained solutions are beneficial in interpreting certain fascinating physical phenomena. We plot our figures in MATLAB and show several 3D and 2D graphical representations to show how the multiplicative white noise affects the solutions of the SmKdV. We show that the white noise around zero stabilizes SmKdV solutions. Full article

(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions-II)

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Figure 1

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