Genocchi Polynomials Research Articles

The operational matrices for Elliptic Partial Differential Equations with mixed boundary conditions

Abstract The purpose of this research is to implement the orthogonal polynomials associated with operational matrices to get the approximate solutions for solving two-dimensional elliptic partial differential equations (E-PDEs) with mixed boundary conditions. The orthogonal polynomials are based on the Standard polynomial (xi ), Legendre, Chebyshev, Bernoulli, Boubaker, and Genocchi polynomials. This study focuses on constructing quick and precise analytic approximations using a simple, elegant, and potent technique based on an orthogonal polynomial representation of the solution as a double power series. Consequently, a linear partial differential equation is transformed into a linear algebraic system which is solved by the Mathematica®12. Approximate solutions can be found if the answers are polynomials in and of itself. Three applications involving well-known linear problems Laplace, Poisson, and Helmholtz equations have been solved by using the proposed methods, and a comparison of the approaches has been provided. Furthermore, the computation of the error norm L∞ has been done to show the accuracy of the suggested approaches. The results clearly demonstrate how precise, efficient, and dependable the proposed methods are in obtaining rough solutions to the problem. Bernoulli was one of the best methods in most examples.

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations

Operational Matrices of Genocchi Polynomials for Solving High-Order Linear Fredholm Integro-Differential Equations

A highly accurate and efficient Genocchi‐based spectral technique applied to singular fractional order boundary value problems

This article focuses on an efficient and highly accurate approximate solver for a class of generalized singular boundary value problems (SBVPs) having nonlinearity and with two‐term fractional derivatives. The involved fractional derivative operators are given in the form of Liouville–Caputo. The developed algorithm for solving the generalized SBVPs consists of two main stages. The first stage is devoted to an iterative quasilinearization method (QLM) to conquer the (strong) nonlinearity of the governing SBVPs. Secondly, we employ the generalized Genocchi polynomials (GGPs) to treat the resulting sequence of linearized SBVPs numerically. An upper error estimate for the Genocchi series solution in the norm is obtained via a rigorous error analysis. The main benefit of the presented QLM‐GGPs procedure is that the required number of iteration in the first stage is within a few steps, and an accurate polynomial solution is obtained through computer implementations in the second stage. Three widely applicable test cases are investigated to observe the efficacy as well as the high‐order accuracy of the QLM‐GGPs algorithm. The comparable accuracy and robustness of the presented algorithm are validated by doing comparisons with the results of some well‐established available computational methods. It is apparently shown that the QLM‐GGPs algorithm provides a promising tool to solve strongly nonlinear SBVPs with two‐term fractional derivatives accurately and efficiently.

Exploring the Characteristics of Δh Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators

The objective of this article is to introduce the ∆h bivariate Appell polynomials ∆hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of ∆h bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the ∆h bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials ∆h are also proved by demonstrating that the ∆h bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of ∆h bivariate Appell polynomials is provided, and symmetric identities for the ∆h bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for ∆h bivariate Appell polynomials. Additionally, generating relations for the ∆h bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials.

Two-iterated degenerate Appell polynomials: properties and applications

In the development of hybrid special polynomials, it is essential to incorporate the monomiality principle, operational rules, and other related properties. This research study adopts this approach and introduces a novel polynomial type called the two-iterated degenerate Appell polynomials, utilizing the monomiality principle. This study yields new findings that align with previous research efforts. The study presents explicit formulas and key properties of these polynomials, while also establishing connections with other polynomial types such as the Bernoulli, Euler, and Genocchi polynomials. These connections facilitate the derivation of additional results. By employing the monomiality principle and considering the aforementioned polynomials as initial members of the Appell family, this study contributes to the expansion of knowledge in the field of special polynomials.

Approximate approach for solving pantograph type via Genocchi polynomials

Approximate approach for solving pantograph type via Genocchi polynomials

The forms of $ (q, h) $-difference equation and the roots structure of their solutions with degenerate quantum Genocchi polynomials

<p>We construct a new type of Genocchi polynomials using degenerate quantum exponential functions and find various forms of $ (q, h) $-difference equations with these polynomials as solutions. This paper includes properties of the symmetric structures of $ (q, h) $-difference equations and also presents $ (q, h) $-difference equations with other polynomials as coefficients. By understanding the approximate roots structure of degenerate quantum Genocchi polynomials (DQG), which are common solutions to various forms of $ (q, h) $-difference equations, we identify the properties of the solutions.</p>

Properties and applications of generalized 1-parameter 3-variable Hermite-based Appell polynomials

<p>We present a novel framework for introducing generalized 3-variable 1-parameter Hermite-based Appell polynomials. These polynomials are characterized by generating function, series definition, and determinant definition, elucidating their fundamental properties. Moreover, utilizing a factorization method, we established recurrence relations, shift operators, and various differential equations, including differential, integrodifferential, and partial differential equations. Special attention is given to exploring the specific cases of 3-variable 1-parameter generalized Hermite-based Bernoulli, Euler, and Genocchi polynomials, offering insights into their unique features and applications.</p>

Genocchi collocation method for accurate solution of nonlinear fractional differential equations with error analysis

In this study, we introduce an innovative fractional Genocchi collocation method for solving nonlinear fractional differential equations, which have significant applications in science and engineering. The fractional derivative is defined in the Caputo sense and by leveraging fractional-order Genocchi polynomials, we transform the nonlinear problem into a system of nonlinear algebraic equations. A novel technique is employed to solve this system, enabling the determination of unknown coefficients and ultimately the solution. We derive the error bound for our proposed method and validate its efficacy through several test problems. Our results demonstrate superior accuracy compared to existing techniques in the literature, suggesting the potential for extending this approach to tackle more complex problems of critical physical significance.

Optimal Control of Non-linear Volterra Integral Equations with Weakly Singular Kernels Based on Genocchi Polynomials and Collocation Method

AbstractWe consider a problem of finding the best way to control a system, known as an optimal control problem (OCP), governed by non-linear Volterra Integral Equations with Weakly Singular kernels. The equations are based on Genocchi polynomials. Depending on the applicable properties of Genocchi polynomials, the considered OCP is converted to a non-linear programming problem (NLP). This method is speedy and provides a highly accurate solution with great precision using a small number of basis functions. The convergence analysis of the approach is also provided. The accuracy and flawless performance of the proposed technique and verification of the theory are examined with some examples.

New $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomials and their matrices

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 Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomial matrices, which involves the generalized Pascal matrix.

Properties of Multivariable Hermite Polynomials in Correlation with Frobenius–Genocchi Polynomials

The evolution of a physical system occurs through a set of variables, and the problems can be treated based on an approach employing multivariable Hermite polynomials. These polynomials possess beneficial properties exhibited in functional and differential equations, recurring and explicit relations as well as symmetric identities, and summation formulae, among other examples. In view of these points, comprehensive schemes have been developed to apply the principle of monomiality from mathematical physics to various mathematical concepts of special functions, the development of which has encompassed generalizations, extensions, and combinations of other functions. Accordingly, this paper presents research on a novel family of multivariable Hermite polynomials associated with Frobenius–Genocchi polynomials, deriving the generating expression, operational rule, differential equation, and other defining characteristics for these polynomials. Additionally, the monomiality principle for these polynomials is verified, as well as establishing the series representations, summation formulae, operational and symmetric identities, and recurrence relations satisfied by these polynomials. This proposed scheme aims to provide deeper insights into the behavior of these polynomials and to uncover new connections between these polynomials, to enhance understanding of their properties.

Genocchi polynomials for variable-order time fractional Fornberg–Whitham type equations

In this research, a kind of non-singular variable-order fractional derivative is utilized to define two types of variable-order fractional Fornberg–Whitham equations. The shifted Genocchi polynomials are employed to construct two operational matrix approaches for solving these equations. More precisely, a new variable-order fractional derivative matrix is obtained for these polynomials, and used with the collocation technique for solving these equations. In fact, using this approaches, solving such problems converts into solving simple systems of algebraic equations. Some examples are solved to illustrate the applicability and accuracy of the proposed algorithms.

A Note on Multi-Euler–Genocchi and Degenerate Multi-Euler–Genocchi Polynomials

Recently, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials are introduced. The aim of this note is to study the multi-Euler–Genocchi and degenerate multi-Euler–Genocchi polynomials which are defined by means of the multiple logarithm and generalize, respectively, the generalized Euler–Genocchi and generalized degenerate Euler–Genocchi polynomials. Especially, we express the former by the generalized Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind, and the latter by the generalized degenerate Euler–Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind.

Explicit relations on the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials and numbers

The main aim of this paper is to introduce and investigate the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials by using monomiality principle and operational methods. We give explicit relations and some identities for the degenerate type 2-unified Apostol–Bernoulli, Euler and Genocchi polynomials.

Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling

Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained.

Convergence Properties of a Kantorovich Type of Szász Operators Involving Negative Order Genocchi Polynomials

The goal of this research is to construct a generalization of a Kantorovich type of Szász operators involving negative-order Genocchi polynomials. With the aid of Korovkin’s theorem, modulus of continuity, Lipschitz class, and Peetre’s K-functional the approximation properties and convergence rate of these operators are established. To illustrate how operators converge to a certain function, we present some examples.

Several Characterizations of Δh-Doped Special Polynomials Associated with Appell Sequences

The study presented in this paper follows the line of research created by the fact that by employing the monomiality principle, new outcomes are produced. This article deals with the inducement of Δh tangent-based Appell polynomials and derivation of certain of its characterizations such as explicit form, determinant form, monomiality principle, etc. These polynomials are designed to exhibit certain symmetries themselves or to capture and describe symmetrical patterns in mathematical structures. Further, certain members of Δh Appell polynomials such as Δh Bernoulli, Euler, and Genocchi polynomials are taken, and their corresponding results are obtained.

A family of Apostol–Euler polynomials associated with Bell polynomials

Abstract Many authors investigated the characteristics of the Bell, Euler, Bernoulli, and Genocchi polynomials because of their numerous uses in statistics, number theory, and other branches of science. A generating function for mixed-type Apostol–Euler polynomials of order η related with Bell polynomials is presented in this study. We also construct essential identities of Apostol–Euler polynomials of order η connected with Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some association with Stirling numbers, and specific cases. Also we establish certain symmetric identities and their known consequences. In addition, we investigate several interesting relationships related to umbral calculus.

Numerical Solution for Arbitrary Domain of Fractional Integro-differential Equation via the General Shifted Genocchi Polynomials

The Genocchi polynomial has been increasingly used as a convenient tool to solve some fractional calculus problems, due to their nice properties. However, like some other members in the Appell polynomials, the nice properties are always limited to the interval defined in 0 , 1 . In this paper, we extend the Genocchi polynomials to the general shifted Genocchi polynomials, S n a , b x , which are defined for interval a , b . New properties for this general shifted Genocchi polynomials will be introduced, including the determinant form. This general shifted Genocchi polynomials can overcome the conventional formula of finding the Genocchi coefficients of a function f x that involves f n − 1 x which may not be defined at x = 0 , 1 . Hence, we use the general shifted Genocchi polynomials to derive the operational matrix and hence to solve the Fredholm-type fractional integro-differential equations with arbitrary domain a , b .