Clemente Cesarano

UniNettuno University · Section of Mathematics

This article investigates the properties and monomiality principle within Bell-based Apostol-Bernoulli-type polynomials. Beginning with the establishment of a generating function, the study proceeds to derive explicit expressions for these polynomials, providing insight into their structural characteristics. Summation formulae are then derived, fac...

In this paper, we introduce degenerate versions of the hypergeometric Bernoulli and Euler polynomials. We demonstrate that they form $\Delta_\lambda$-Appell sets and provide some of their algebraic properties, including inversion formulas, as well as the associated matrix formulation. Additionally , we focus our attention on the monomiality princip...

This article comes up with criteria to make sure that the solutions to superlinear, half-linear, and noncanonical dynamic equations oscillate in both advanced and delayed cases; these criteria are comparable to the Hille-type and Ohriska-type criteria for the canonical nonlinear dynamic equations; and also these results solve an open problem in man...

This work presents a general framework that innovates and explores different mathematical aspects associated with special functions by utilizing the mathematical physics-based idea of monomiality. This study presents a unique family of multivariable Hermite polynomials that are closely related to Frobenius–Genocchi polynomials of Apostol type. The...

In this article, we introduce a new class of polynomials, known as Apostol Hermite Bernoulli-type polynomials, and explore some of their algebraic properties, including summation formulas and their determinant form. The majority of our results are proven using generating function methods. Additionally, we investigate the monomiality principle relat...

We investigate the numerical calculation of the general Heun equation using Wolfram Mathematica’s functions, comparing the numerical solutions with hypergeometric and explicit solutions. This exploration sheds light on the efficacy and accuracy of the numerical algorithm implemented in Mathematica for computing Heun functions.

This paper is devoted to building a general framework for constructing a solution to fractional Phi-4 differential equations using a Caputo definition with two parameters. We briefly introduce some definitions and properties of fractional calculus in two parameters and the Phi-4 equation. By investigating the homotopy analysis method, we built the...

This survey highlights the significant role of exponential operators and the monomiality principle in the theory of special polynomials. Using operational calculus formalism, we revisited classical and current results corresponding to a broad class of special polynomials. For instance, we explore the 2D Hermite polynomials and their generalizations...

In this research, we leverage various $ q $-calculus identities to introduce the notion of $ q $-Hermite-Appell polynomials involving three variables, elucidating their formalism. We delve into numerous properties and unveil novel findings regarding these $ q $-Hermite-Appell polynomials, encompassing their generating function, series representatio...

In this paper, the fractional Riemann wave equation with M-truncated derivative (FRWE-MTD) is considered. The Jacobi elliptic function method and the modified extended tanh function method are applied to acquire new elliptic, rational, hyperbolic, and trigonometric functions solutions. Moreover, we expand some earlier studies. The obtained solution...

This paper introduces new families of Fubini-Euler type and Apostol Fubini-Euler type polynomials, providing expressions, recurrence relations, and identities. We also derive Fourier series, and integral representations, and present their rational argument representation.

In this paper, we introduce the q-truncated exponential polynomials by means of the integral form. Certain properties of the q-truncated exponential polynomials like series definition, recurrence relations, q-differential equations and integral representations are obtained. Also, we introduce the associated q-truncated exponential polynomials, high...

The aim of this study is to refine the known Riccati transformation technique to provide new oscillation criteria for solutions to second-order dynamic equations over time. It is important to note that the convergence or divergence of some improper integrals on time scales depends not only on the integration function but also on the integration tim...

УДК 517.5 За різних умов на параметри гіпергеометричної функції Горна H 4 досліджено різні області збіжності гіллястих ланцюгових дробових розвинень відношень цих функцій.

In this paper, we study the jerk vector that is the rate of change of the acceleration vector over time. In three‐dimensional space, the decomposition of the jerk vector is a new concept in the field. This decomposition expresses the jerk vector as the sum of three unique components in specific directions: the tangential direction, the radial direc...

The generalized nonlinear Schrödinger equation with M-truncated derivatives (GNLSE-MTD) is studied here. By using generalized Riccati equation and mapping methods, new elliptic, hyperbolic, trigonometric, and rational solutions are discovered. Because the GNLSE is widely employed in communication, heat pulse propagation in materials, optical fiber...

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various anal...

Appell’s functions F1–F4 turned out to be particularly useful in solving a variety of problems in both pure and applied mathematics. In literature, there have been published a significant number of interesting and useful results on these functions. In this paper, we prove that the branched continued fraction, which is an expansion of ratio of hyper...

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched conti...

In this study, the stochastic fractional Fokas system (SFFS) with M-truncated derivatives is considered. A certain wave transformation is applied to convert this system to a one-dimensional conservative Hamiltonian system. Based on the qualitative theory of dynamical systems, the bifurcation and phase portrait are examined. Utilizing the conserved...

This article comes up with criteria to make sure that the solutions to superlinear, half-linear, and noncanonical dynamic equations oscillate in both advanced and delayed cases; these criteria are comparable to the Hille-type and Ohriska-type criteria for the canonical nonlinear dynamic equations; and also these results solve an open problem in man...

In this paper, we establish iterative Hille‐type criteria for advanced functional half‐linear dynamic equations of the second order. These results extend and improve recent criteria established by multiple authors for the same equation and encompass classical criteria. We provide an example to demonstrate the significance of the results obtained.

In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional deriva...

This paper investigates the oscillatory behavior of nonlinear third-order dynamic equations on time scales. Our main approach is to transform the equation from its semi-canonical form into a more tractable canonical form. This transformation simplifies the analysis of oscillation behavior and allows us to derive new oscillation criteria. These crit...

For fourth-order neutral differential equations (NDE) in the canonical case, we present new relationships between the solution and its corresponding function in two casses: $ p < 1 $ and $ p > 1 $. Through these relationships, we discover new monotonic properties for this equation of fourth order. Using the new relationships and properties, we deri...

The main objective of this paper is to introduce and explore two novel classes of degenerate biparametric Apostol-type polynomials, which are based on a definition of degenerate Apostol-type polynomials provided by Subuhi Khan et al. We derive various algebraic and differential properties associated with these polynomials. Additionally, we provide...

In this lecture, we will study families of Appell-type polynomials, particularly Bernoulli polynomials and Euler polynomials, together with their generalizations, as well as their degenerate version, which have been studied over time. In addition, we will explore the degenerate sine and cosine functions. On this basis, we will introduce the degener...

In this paper, we introduce the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomial Pas...

In this article, the stochastic Riemann wave equation (SRWE) forced by white noise in the Itô sense is considered. The extended tanh function and mapping methods are applied to obtain new elliptic, rational, hyperbolic, and trigonometric stochastic solutions. Furthermore, we generalize some previous studies. The obtained solutions are important in...

This article focuses on developing and applying approximation techniques to derive conservation laws for the Timoshenko-Prescott mixed derivatives perturbed partial differential equations (PDEs). Central to our approach is employing approximate Noether-type symmetry operators linked to a conventional Lagrangian one. Within this framework, this pape...

In this paper, we study the equiform Bishop formulae for the equiform timelike curves in 3-dimensional Minkowski space where the equiform timelike spherical curves are defined according to the equiform Bishop frame. We establish a necessary and sufficient condition for an equiform timelike curve to be an equiform timelike spherical curve. Furthermo...

This paper delves into the analysis of oscillation characteristics within third-order quasi-linear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions-whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches th...

congress of the Italian Society of Applied and Industrial Mathematics (SIMAI)

In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that ther...

In this paper, we take into account the coupled stochastic Korteweg–De Vries (CSKdV) equations in the Itô sense. Using the mapping method, new trigonometric, rational, hyperbolic, and elliptic stochastic solutions are obtained. These obtained solutions can be applied to the analysis of a wide variety of crucial physical phenomena because the couple...

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for explaining the development o...

In many contexts a number of interesting and useful identities involving special polynomials (e.g., Bernoulli, Euler, Bell, Apostol, Jacobi, Laguerre polynomials and their generalizations and q-analogues) can be obtained from a matrix representation. Particularly interesting are those contexts in which such identities are provided via Pascal matric...

The Fokas system with M-truncated derivative (FS-MTD) was considered in this study. To get analytical solutions of FS-MTD in the forms of elliptic, rational, hyperbolic, and trigonometric functions, we employed the extend F-expansion approach and the Jacobi elliptic function method. Since nonlinear pulse transmission in monomode optical fibers is e...

The stochastic Fokas system (SFS), driven by multiplicative noise in the Itô sense, was investigated in this study. Novel trigonometric, rational, hyperbolic, and elliptic stochastic solutions are found using a modified mapping method. Because the Fokas system is used to explain nonlinear pulse propagation in monomode optical fibers, the solutions...

In this article, we examine the Kraenkel–Manna–Merle system (KMMS) with an M-truncated derivative (MTD). Our goal is to obtain rational, hyperbolic, and trigonometric solutions by using the F-expansion technique with the Riccati equation. To our knowledge, no one has studied the exact solutions to the KMMS in the presence/absence of a damping effec...

The objective of this paper is to explore novel unified continuous and discrete versions of the Trapezium-Jensen-Mercer (TJM) inequality, incorporating the concept of convex mapping within the framework of q-calculus, and utilizing majorized tuples as a tool. To accomplish this goal, we establish two fundamental lemmas that utilize the ς 1 q and ς...

In this study, we extended and improved the oscillation criteria previously established for second-order differential equations to even-order differential equations. Some examples are given to demonstrate the significance of the results accomplished.

We study the (3+1)-dimensional stochastic Jimbo–Miwa (SJM) equation induced by multiplicative white noise in the Itô sense. We employ the Riccati equation mapping and He’s semi-inverse techniques to provide trigonometric, hyperbolic, and rational function solutions of SJME. Due to the applications of the Jimbo–Miwa equation in ocean studies and oth...

We take into account the stochastic Boiti–Leon–Manna–Pempinelli equation (SBLMPE), which is perturbed by a multiplicative Brownian motion. By applying He’s semi-inverse method and the Riccati equation mapping method, we can acquire the solutions of the SBLMPE. Since the Boiti–Leon–Manna–Pempinelli equation is utilized to explain incompressible liqu...

Quantum calculus provides a significant generalization of classical concepts and overcomes the limitations of classical calculus in tackling non-differentiable functions. Implementing the q-concepts to obtain fresh variants of classical outcomes is a very intriguing aspect of research in mathematical analysis. The objective of this article is to es...

In this work, we consider the Boiti–Leon–Manna–Pempinelli equation with the M-truncated derivative (BLMPE-MTD). Our aim here is to obtain trigonometric, rational and hyperbolic solutions of BLMPE-MTD by employing two diverse methods, namely, He’s semi-inverse method and the extended tanh function method. In addition, we generalize some previous res...

In the current study, we investigate the stochastic Benjamin–Bona–Mahony equation with beta derivative (SBBME-BD). The considered stochastic term is the multiplicative noise in the Itô sense. By combining the F-expansion approach with two separate equations, such as the Riccati and elliptic equations, new hyperbolic, trigonometric, rational, and Ja...

In this paper, we construct an analytical solution of the coupled Burgers’ equation, using the homotopy analysis method, which is a semi-analytical method, the approximate solution obtained by this method is convergent for different values of the convergence control parameter ℏ, the optimal value of ℏ corresponding with the minimum error to be dete...

The Grad–Shafranov plasma equilibrium equation was originally solved analytically in toroidal geometry, which fitted the geometric shape of the first Tokamaks. The poloidal surface of the Tokamak has evolved over the years from a circular to a D-shaped ellipse. The natural geometry that describes such a shape is the prolate elliptical one, i.e., th...

This article presents a generalization of new classes of degenerated Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi Hermite polynomials of level m. We establish some algebraic and differential properties for generalizations of new classes of degenerated Apostol-Bernoulli polynomials. These results are shown using generating function methods...

In this study, we take into account the fractional stochastic Kraenkel–Manna–Merle system (FSKMMS). The mapping approach may be used to produce various type of stochastic fractional solutions, such as elliptic, hyperbolic, and trigonometric functions. Solutions to the Kraenkel–Manna–Merle system equation, which explains the propagation of a magneti...

This work aims to derive new inequalities that improve the asymptotic and oscillatory properties of solutions to fourth-order neutral differential equations. The relationships between the solution and its corresponding function play an important role in the oscillation theory of neutral differential equations. Therefore, we improve these relationsh...

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained by using the F-expansion and the Jacobi ellipt...

The stochastic shallow water wave equation (SSWWE) in the sense of the beta-derivative is considered in this study. The solutions of the SSWWE are obtained using the F-expansion technique with the Riccati equation and He’s semi-inverse method. Since the shallow water equation has many uses in ocean engineering, including river irrigation flows, tid...

In this paper, the oscillatory properties of certain second-order differential equations of neutral type are investigated. We obtain new oscillation criteria, which guarantee that every solution of these equations oscillates. Further, we get conditions of an iterative nature. These results complement and extend some beforehand results obtained in t...

In this work, we study the oscillatory properties of a higher-order neutral delay differential equation, and introduce a set of new oscillation criteria for this equation. The study of the qualitative behavior of delay differential equations, especially the neutral ones, is of great importance from the practical side as well as the theoretical side...

In this paper, we consider the (3 + 1)-dimensional fractional-stochastic quantum Zakharov–Kuznetsov equation (FSQZKE) with M-truncated derivative. To find novel trigonometric, hyperbolic, elliptic, and rational fractional solutions, two techniques are used: the Jacobi elliptic function approach and the modified F-expansion method. We also expand on...

Investigating the jounce vector in planar and space motion is the primary objective of this paper. For planar motion, the jounce vector is split into tangential‐normal and radial‐transverse components. Simple pendulum oscillation, a central force proportionate to distance, and Keplerian orbital motion are used as models for plane motion to show the...

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 solut...

In this paper, we look at the (4 + 1)‐dimensional stochastic Fokas equation (SFE) perturbed in the Itô sense by white noise. The tanh‐coth method and mapping method are used to acquire new trigonometric, hyperbolic, elliptic, and rational stochastic solutions. Also, we extend some earlier studies. Because the SFE equation is essential for describin...

In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some ex...

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the tanh–coth and Jacobi elliptic functi...

In this article, we establish several new generalized Hardy-type inequalities involving several functions on time-scale nabla calculus. Furthermore, we derive some new multidimensional Hardy-type inequalities on time scales nabla calculus. The main results are proved by applying Minkowski’s inequality, Jensen’s inequality and Arithmetic Mean–Geomet...

In this paper, the (2+1)-dimensional q-deformed Sinh-Gordon model has been investigated via (G′G,1G)-expansion and Sine-Gordon-expansion methods. These techniques successfully retrieve trigonometric as well as hyperbolic solutions, along necessary restricted conditions applied on parameters. In addition to these solutions, dark solitons and complex...

In this paper, we establish some new dynamic inequalities involving C-monotonic functions with C≥1, on time scales. As a special case of our results when C=1, we obtain the inequalities involving increasing or decreasing functions (where for C=1, the 1-decreasing function is decreasing and the 1-increasing function is increasing). The main results...

In the present article, we iteratively deduce new monotonic properties of a class from the positive solutions of fourth-order delay differential equations. We discuss the non-canonical case in which there are possible decreasing positive solutions. Then, we find iterative criteria that exclude the existence of these positive decreasing solutions. U...

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...

In this article, the stochastic Davey–Stewartson equations (SDSEs) forced by multiplicative noise are addressed. We use the mapping method to find new rational, elliptic, hyperbolic and trigonometric functions. In addition, we generalize some previously obtained results. Due to the significance of the Davey–Stewartson equations in plasma physics, n...