Search Results (152)

One of the main motivations of this paper is to construct generating functions for generalization of the Touchard polynomials (or generalization exponential functions) and certain special numbers. Many novel formulas and relations for these polynomials are found by using the Euler derivative operator and functional equations of these functions. Some novel relations among these polynomials, beta polynomials, Bernstein polynomials, related to Binomial distribution from discrete probability distribution classes, are given. Full article

This paper presents a novel generalization of the mth-order Laguerre and Laguerre-based Appell polynomials and examines their fundamental properties. By establishing quasi-monomiality, we derive key results, including recurrence relations, multiplicative and derivative operators, and the associated differential equation. Additionally, both series and determinant representations are provided for this new class of polynomials. Within this framework, several subpolynomial families are introduced and analyzed including the generalized mth-order Laguerre–Hermite Appell polynomials. Furthermore, the generalized mth-order Laguerre–Gould–Hopper-based Appell polynomials are defined using fractional operators and we investigate their structural characteristics. New families are also constructed, such as the mth-order Laguerre–Gould–Hopper–based Bernoulli, Laguerre–Gould–Hopper–based Euler, and Laguerre–Gould–Hopper–based Genocchi polynomials, exploring their operational and algebraic properties. The results contribute to the broader theory of special functions and have potential applications in mathematical physics and the theory of differential equations. Full article

This paper introduces fully Dowling polynomials of the first and second kinds, which are degenerate versions of the ordinary Dowling polynomials. Then, several important identities for these degenerate polynomials are derived. The relationship between fully degenerate Dowling polynomials and fully degenerate Bell polynomials, degenerate Bernoulli polynomials, degenerate Euler polynomials, and so on is obtained using umbral calculus. Full article

The viscoelastic behavior of functionally graded (FG) materials significantly affects their vibration performance, making it necessary to establish theoretical analysis methods. Although fractional-order methods can be used to set up the vibration differential equations for viscoelastic, functionally graded beams, solving these fractional differential equations typically involves complex iterative processes, which makes the vibration performance analysis of viscoelastic FG materials challenging. To address this issue, this paper proposes a simple method to predict the vibration behavior of viscoelastic FG beams. The fractional viscoelastic, functionally graded porous (FGP) beam is modeled based on the Euler–Bernoulli theory and the Kelvin–Voigt fractional derivative stress-strain relation. Employing the variational principle and the Hamilton principle, the partial fractional differential equation is derived. A method based on Bernstein polynomials is proposed to directly solve fractional vibration differential equations in the time domain, thereby avoiding the complex iterative procedures of traditional methods. The viscoelastic, functionally graded porous beams with four porosity distributions and four boundary conditions are investigated. The effects of the porosity coefficient, pore distribution, boundary conditions, fractional order, and viscoelastic coefficient are analyzed. The results show that this is a feasible method for analyzing the viscoelastic behavior of FGP materials. Full article

This manuscript associates with a study of Frobenius–Euler–Simsek-type Polynomials. In this research work, we construct a new sequence of Szász–Beta type operators via Frobenius–Euler–Simsek-type Polynomials to discuss approximation properties for the Lebesgue integrable functions, i.e., $L^p[0,\infty )$, $1\le p<\infty$. Furthermore, estimates in view of test functions and central moments are studied. Next, rate of convergence is discussed with the aid of the Korovkin theorem and the Voronovskaja type theorem. Moreover, direct approximation results in terms of modulus of continuity of first- and second-order, Peetre’s K-functional, Lipschitz type space, and the $r^{th}$-order Lipschitz type maximal functions are investigated. In the subsequent section, we present weighted approximation results, and statistical approximation theorems are discussed. To demonstrate the effectiveness and applicability of the proposed operators, we present several illustrative examples and visualize the results graphically. Full article

The method of particular solutions using polynomial basis functions (MPS-PBF) has been extensively used to solve various types of partial differential equations. Traditional methods employing radial basis functions (RBFs)—such as Gaussian, multiquadric, and Matérn functions—often suffer from accuracy issues due to their dependence on a shape parameter, which is very difficult to select optimally. In this study, we adopt the MPS-PBF to solve the time-dependent Schrödinger equation in two dimensions. By utilizing polynomial basis functions, our approach eliminates the need to determine a shape parameter, thereby simplifying the solution process. Spatial discretization is performed using the MPS-PBF, while temporal discretization is handled via the backward Euler and Crank–Nicolson methods. To address the ill conditioning of the resulting system matrix, we incorporate a multi-scale technique. To validate the efficacy of the proposed scheme, we present four numerical examples and compare the results with known analytical solutions, demonstrating the accuracy and robustness of the scheme. Full article

The grey prediction model is designed to characterize systems comprising both partially known information (referred to as white) and partially unknown dynamics (referred to as black). However, traditional GM(1,1) models are based on linear differential equations, which limits their capacity to capture nonlinear and non-stationary behaviors. To address this issue, this paper develops a generalized grey differential prediction approach based on the Bernoulli equation framework. We incorporate the Bernoulli mechanism with a nonlinear exponent n and a dynamic polynomial-driven term. In this work, we propose a new model designated as BPGM(1,1). Another key innovation of this work is the adoption of a nonlinear least squares direct parameter identification strategy to calculate the exponent and polynomial parameters in the Bernoulli equation, which achieves a higher degree of freedom in parameter selection and effectively circumvents the model distortion issues caused by traditional background value estimation. Furthermore, the Euler discretization method is utilized for numerical solving, reducing the reliance on traditional analytical solutions for linear structures. Numerical experiments indicate that BPGM(1,1) surpasses GM(1,1), NFBM(1,1), and their improved versions. By leveraging the synergistic mechanism between Bernoulli-type nonlinear regulation and polynomial-driven external excitation, this framework significantly enhances prediction accuracy for systems characterized by non-stationary behaviors and multi-scale trends. Full article

This study introduces a novel generalized class of special polynomials using a fractional operator approach. These polynomials are referred to as the generalized Gould–Hopper–Bell-based Appell polynomials. In view of the operational method, we first introduce the operational representation of the Gould–Hopper–Bell-based Appell polynomials; then, using a fractional operator, we establish a new generalized form of these polynomials. The associated generating function, series representations, and summation formulas are also obtained. Additionally, certain operational identities, as well as determinant representation, are derived. The investigation further explores specific members of this generalized family, including the generalized Gould–Hopper–Bell-based Bernoulli polynomials, the generalized Gould–Hopper–Bell-based Euler polynomials, and the generalized Gould–Hopper–Bell-based Genocchi polynomials, revealing analogous results for each. Finally, the study employs Mathematica to present computational outcomes, zero distributions, and graphical representations associated with the special member, generalized Gould–Hopper–Bell-based Bernoulli polynomials. Full article

This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. Next, we investigate approximation order uniform convergence via Korovkin result and the modulus of smoothness for functions in continuous functional spaces. A Voronovskaja theorem is also explored approximating functions which belongs to the class of function having first and second order continuous derivative. Further, we discuss numerical error and graphical analysis. In the last, two dimensional operators are constructed to discuss approximation for the class of two variable continuous functions. Full article

In this paper, a new and general form of truncated-exponential-based general-Appell polynomials is introduced using the two-variable general-Appell polynomials. For this new polynomial family, we present an explicit representation, recurrence relation, shift operators, differential equation, determinant representation, and some other properties. Finally, two special cases of this family, truncated-exponential-based Hermite-type and truncated-exponential-based Laguerre–Frobenius Euler polynomials, are introduced and their corresponding properties are obtained. Full article

This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence. Full article

This paper analyzes vibrations induced by a moving bogie passing through a single-layer railway track model. The emphasis is placed on the possibility of unstable behavior in the subcritical velocity range. All results are presented in dimensionless form to encompass a wide range of possible scenarios. The results are obtained semi-analytically, however, the only numerical step involves solving the roots of polynomial expressions. No numerical integration is used, allowing for the straightforward solution of completely undamped scenarios, as damping is not required for numerical stability. The vibration shapes are presented in the time domain in closed form. It is concluded that increased foundation damping worsens the situation. However, in general, the risk of instability in the subcritical velocity range for a moving bogie is lower than that of two moving masses, particularly for higher mass moments of inertia of the bogie bar and primary suspension damping. The study also examines how the results change when a Timoshenko-Rayleigh beam is considered instead of an Euler-Bernoulli beam. Although some cases may appear academic, it is demonstrated that instability in the supercritical velocity range cannot be assumed to be guaranteed. Full article

This paper aims to establish a new hybrid class of special polynomials, namely, the Fubini–Bell-based Appell polynomials. The monomiality principle is used to derive the generating function for these polynomials. Several related identities and properties, including symmetry identities, are explored. The determinant representation of the Fubini–Bell-based Appell polynomials is also established. Furthermore, some special members of the Fubini–Bell-based Appell family—such as the Fubini–Bell-based Bernoulli polynomials and the Fubini–Bell-based Euler polynomials—are derived, with analogous results presented for each. Finally, computational results and graphical representations of the zero distributions of these members are investigated. Full article

In this paper, we define a new generalization of three-variable q-Laguerre polynomials and derive some properties. By using these polynomials, we introduce a new generalization of three-variable q-Laguerre-based Appell polynomials (3VqLbAP) through a generating function approach involving zeroth-order q-Bessel–Tricomi functions. These polynomials are studied by means of generating function, series expansion, and determinant representation. Also, these polynomials are further examined within the framework of q-quasi-monomiality, leading to the establishment of essential operational identities. We then derive operational representations, as well as q-differential equations for the three-variable q-Laguerre-based Appell polynomials. Some examples are constructed in terms of q-Laguerre–Hermite-based Bernoulli, Euler, and Genocchi polynomials in order to illustrate the main results. Full article

Recently, several researchers have estimated the Maclaurin coefficients, namely $\left|q_2\right|$ and $\left|q_3\right|,$ and the Fekete–Szegö functional problem of functions belonging to some special subfamilies of analytic functions related to certain polynomials, such as Lucas polynomials, Legendrae polynomials, Chebyshev polynomials, and others. This study obtains the bounds of coefficients $\left|q_2\right|$ and $\left|q_3\right|$, and the Fekete–Szegö functional problem for functions belonging to the comprehensive subfamilies $T(\zeta ,ϵ,\delta )$ and $J(\phi ,\delta )$ of analytic functions in a symmetric domain $\mathbb{U}$, using the imaginary error function subordinate to Euler polynomials. After specializing the parameters used in our main results, a number of new special cases are also obtained. Full article