In this work, we study the Chebyshev polynomials of the fourth kind as solutions of a singular Sturm–Liouville problem and establish their differential, orthogonality, and extremal properties. We derive a Rodrigues-type formula and a generating function, thereby establishing a direct connection with an extended Poisson kernel. Furthermore, we formulate an electrostatic model, in the spirit of Stieltjes’ classical interpretation, in which the zeros of these polynomials are characterized as the equilibrium positions of identical unit charges on (−1, 1) interacting through a logarithmic potential and subject to fixed external charges at the endpoints. We prove that the unique configuration minimizing the total energy coincides with the set of zeros of these polynomials. In addition, we introduce a principal SU(2)-bundle over R3 ${\mathbb{R}}^{3}$ to model internal spin degrees of freedom. Within this geometric framework, the polynomials arise naturally as components of eigenfunctions of the squared spin operator for spin-12 $\frac{1}{2}$ particles in the internal fiber. These results highlight the interplay between quantum spin geometry, orthogonal polynomials, and harmonic analysis on principal bundles.
This paper is devoted to establish existence and regularity theorems for the d- and ∂̄ $\bar{\partial }$ -equations in the domains C+n ${\mathbb{C}}_{+}^{n}$ and R+n+1 ${\mathbb{R}}_{+}^{n+1}$ , respectively, for differential forms in Hölder–Zygmund spaces. Consequently, the same regularity results are obtained for the ∂∂̄ $\partial \bar{\partial }$ -equation. Our analysis also extends to the case of differential forms admitting boundary values in the sense of currents.
This paper introduces a new family of multivariate Hermite-based Appell polynomials with a one-parameter extension, defined through an appropriate generating function. Explicit series representations, operational formulas, and a fractional integral representation are established, revealing their underlying algebraic structure. A determinant representation is derived, providing an effective characterization of the proposed polynomials. Moreover, several families of ordinary, integrodifferential, and partial differential equations satisfied by these polynomials are obtained via operational and factorization techniques. Various summation identities are also presented, enhancing their analytical and computational applicability. As notable special cases, the corresponding results for multivariate Hermite-based Bernoulli, Euler, and Genocchi polynomials are derived, demonstrating that the proposed framework unifies and generalizes several classical Appell-type polynomial families.
This paper proposes the definitions of general fractional operators with fixed memory length in several function spaces. Firstly, some properties of the general fractional integral with fixed memory length, such as boundedness, continuity, semigroup property, commutativity, and Leibniz rule, are discussed. Subsequently, the properties of the general fractional derivatives with fixed memory length, such as continuity and commutativity, are discussed. It is also proven that when the function μ satisfies certain conditions, the general fractional integral and derivatives with fixed memory length preserve the periodicity of periodic functions with same period T. Meanwhile, some illustrative examples are provided to enhance the comprehensibility of the results. Furthermore, fundamental theorem of calculus for general fractional operators with fixed memory length is explored.
This study investigates the solution of generalized second-order fractional Stokes equations through the application of a centroid interpolation collocation method. These equations serve as fundamental models in the field of complex fluid dynamics, with significant applications in areas such as viscoelastic fluid mechanics, porous media flow, and biomedical engineering. The proposed meshless barycentric interpolation collocation method (MBICM) eliminates the need for domain meshing, thereby enabling efficient and accurate numerical approximations on irregular domains. The governing equations are formulated based on the Caputo definition of time fractional derivatives, and the discrete matrix system is constructed using barycentric interpolation basis functions. The stability of the method is theoretically analyzed through Lebesgue constant estimation and error bound analysis. Numerical experiments, including linear biharmonic problems and two-dimensional fractional Stokes equations, are conducted to demonstrate the accuracy and effectiveness of the proposed approach.
We define a hyperbolic framed surface with a moving frame in hyperbolic 3-space. By the existence and uniqueness theorems for hyperbolic framed surfaces, we know that the hyperbolic framed surface is uniquely determined by basic invariants up to a Lorentz rotation. We then discuss both the results for regular surfaces and the properties of surfaces with singularities. Meanwhile, we consider the relationship between hyperbolic framed surfaces in hyperbolic 3-space and framed surfaces in Euclidean 3-space. As an application, we investigate hyperbolic tangent developable surfaces generated by hyperbolic framed curves and discuss the singularities of such surfaces.
In this paper, we introduce two new structured tensors, which are called generalized nonnegative tensors and GM $G\mathcal{M}$ -tensors. These concepts extend the concepts of nonnegative tensors and M $\mathcal{M}$ -tensors. An improved Ky Fan theorem for tensors is presented based on the properties of generalized nonnegative tensors.
In this study, we investigate several well-established Erdös-Lax and Turán-type inequalities that connect the sup-norms of a univariate polynomial with complex coefficients and its ordinary derivative in the complex plane to those concerning the polar derivative. The results derived enhance some recently established Erdös-Lax and Turán-type inequalities for constrained polynomials and yield a range of inequalities that are more precise than those previously recognized in the literature, considering the location of the zeros. Furthermore, specific numerical examples are provided and illustrated graphically, demonstrating clearly that, in certain cases, the bounds obtained from our findings can be significantly sharper than those known earlier.
This paper proposes the use of interior penalty discontinuous Galerkin methods to solve a convection dominated singularly perturbed problem with an additional shift. Initially, the construction of Bakhvalov-type meshes and their associated properties are detailed. Following this, in the energy norm, an optimal parameter-uniform convergence rate k is rigorously demonstrated, where k denotes the degree of the piecewise polynomial space. Finally, a series of numerical results are performed to substantiate the theoretical claims presented in this study.
In this paper, we investigate the β −Sturm-Liouville problem within the framework of a generalized quantum difference calculus, which extends Hahn’s difference operator to the half-line. Our primary focus is on establishing the existence of a spectral function associated with the β −Sturm-Liouville operator. In addition to this, we derive the Parseval identity corresponding to the problem. Furthermore, we obtain an explicit expansion formula in terms of these eigenfunctions.
In this paper, we consider a two-point boundary value problem for a class of coupled systems of nonlinear fractional differential equations. The existence of positive solutions is obtained via the nonlinear alternative of Leray–Schauder type. Especially, an approximate solution is given by Adomian decomposition method. In addition, numerical examples are presented to demonstrate the application of our main results.
Abstract Three accelerating transformation formulae for hypergeometric series are systematically analyzed by the “coefficient extraction method”. Numerous harmonic series of convergence ratios “ ± 4 27 $\frac{{\pm}4}{27}$ ” are evaluated in closed form, which fills a gap of series calculus since similar ones have not appeared often in the literature.
This paper presents a constructive approach to a class of optimal control problems governed by polyhedral discrete and differential inclusions subject to multiple time delays and state constraints. While the theoretical exposition initially focuses on systems with two distinct delays to maintain notational clarity, the proposed framework provides a rigorous basis for generalizing the results to systems governed by multiple delays. By exploiting the geometric structure of polyhedral set-valued mappings, we rigorously employ the method of discrete approximations, utilizing polyhedral Euler–Lagrange type inclusions to bridge the gap between discrete and continuous-time formulations. Within a conjugate analysis framework, we establish that each delay parameter is associated with a distinct absolutely continuous function in the adjoint inclusion. Specifically, we derive necessary and sufficient optimality conditions for the discrete-approximate problem. Subsequently, by formally passing to the limit, we establish sufficient optimality criteria for the continuous-time problem. To demonstrate the computational applicability of the proposed method, numerical experiments are provided for both discrete and continuous settings. These examples explicitly verify the derived conditions and illustrate the behavior of conjugate variables under active state constraints, confirming the effectiveness of the discrete approximation scheme in solving complex constrained control problems.
In this paper, we discuss a class of implicit fractional differential equation with quasi-periodic boundary condition involving Caputo-Katugampola fractional derivative. Based on different parameter values, we utilize Mawhin’s continuation theorem, Schaefer’s fixed point theorem, and Banach’s contraction mapping principle, respectively, to provide the existence, uniqueness, and Ulam-type stability of the solutions to the considered problem, and provide several examples to illustrate the main results.
This paper investigates a Dirichlet problem for a system of logarithmic double-phase equations with variable exponents. The system is given by−div|∇u|α1(τ)−2∇u+μ(τ)log(e+|∇u|)+|∇u|β1(τ)(e+|∇u|)|∇u|β1(τ)−2∇u=h1(u,v) in D,−div|∇v|α2(τ)−2∇v+μ(τ)log(e+|∇v|)+|∇v|β2(τ)(e+|∇v|)|∇v|β2(τ)−2∇v=h2(u,v) in D,u=0,v=0 on ∂D. $$\begin{cases}-\mathrm{d}\mathrm{i}\mathrm{v}\left(\vert \nabla u{\vert }^{{\alpha }_{1}\left(\tau \right)-2}\nabla u+\mu \left(\tau \right)\left[\mathrm{log}\left(e+\vert \nabla u\vert \right)+\frac{\vert \nabla u\vert }{{\beta }_{1}\left(\tau \right) \left(e+\vert \nabla u\vert \right)}\right]\vert \nabla u{\vert }^{{\beta }_{1}\left(\tau \right)-2}\nabla u\right)={h}_{1}\left(u,v\right)\hfill & \quad \text{in\,}\mathfrak{D},\hfill \\ -\mathrm{d}\mathrm{i}\mathrm{v}\left(\vert \nabla v{\vert }^{{\alpha }_{2}\left(\tau \right)-2}\nabla v+\mu \left(\tau \right)\left[\mathrm{log}\left(e+\vert \nabla v\vert \right)+\frac{\vert \nabla v\vert }{{\beta }_{2}\left(\tau \right) \left(e+\vert \nabla v\vert \right)}\right]\vert \nabla v{\vert }^{{\beta }_{2}\left(\tau \right)-2}\nabla v\right)={h}_{2}\left(u,v\right)\hfill & \quad \text{in\,}\mathfrak{D},\hfill \\ u=0,\quad v=0\hfill & \quad \text{on\,}\partial \mathfrak{D}.\hfill \end{cases}$$
In this paper, we consider Fermat-type functional equations of the formF2(z)+G2(z)=exp(c1z1+c2z2+c0), $${F}^{2}\left(z\right)+{G}^{2}\left(z\right)=\mathrm{exp}\left({c}_{1}{z}_{1}+{c}_{2}{z}_{2}+{c}_{0}\right),$$
This paper introduces a new subclass of bi-univalent functions and establishes sharp bounds for their second- and third-order Hankel determinants. These theoretical results extend earlier work in geometric function theory and contribute to the study of coefficient problems for analytic and bi-univalent functions. Building on these findings, we develop an image enhancement algorithm that incorporates Hankel determinants into a convolution-based framework. The proposed method is evaluated using standard quality metrics such as Peak Signal to Noise Ratio, Structural Similarity Index Measure, Mean Squared Error, and Pearson Correlation Coefficient, and further analyzed through histogram plots and mesh visualizations. Experimental results on natural images, flower datasets, and brain stroke Computed Tomography (CT) scans demonstrate that the algorithm produces clearer, more detailed, and structurally preserved images compared to existing coefficient-based methods. The study highlights the dual significance of Hankel determinants, both as a theoretical tool in function theory and as a practical mechanism for improving digital image quality, thereby bridging mathematical analysis with real-world applications in medical imaging and computer vision.
The Hermite–Hadamard inequality remains a central focus in mathematical research, with its profound significance continually motivating mathematicians to explore new areas for its enhancement and generalizations. The main objective of this paper is to introduce a novel approach to establish conticrete forms of the Hermite–Hadamard–Mercer type inequalities in fractional framework. A broader notion than synchronous and monotonic sequences known as separable sequences is employed in connection with convexity theory and Riemann–Liouville fractional operators to establish these inequalities. These inequalities are constructed by employing three ξ-sequences and dual bases in Rξ ${\mathbb{R}}^{\xi }$ . These results are further expanded by a number of corollaries utilizing various sequences and bases. The remarks presented at the end of the given corollaries exhibit applications of the primary result for synchronous sequences, monotonic sequences, nondecreasing sequences in P-mean, star-shaped sequences and convex sequences. Also, new and old inequalities are established as special cases of the major findings. Finally, the paper demonstrates applications of the main outcomes to special means.
This paper aims to introduce new significant forms of the Legendre matrix polynomials (LMPs) with two variables, and establish matrix differential recurrence relations and matrix partial differential recurrence relations for LMPs with two variables, and double generating functions. A bilinear double generating function, a special property, and expansion of LMPs with two variables as a series of Hermite matrix polynomials (HMPs) are discussed.
In this paper, we propose and study a new ternary three-dimensional fractional reaction-diffusion model. The model combines Caputo fractional derivatives in time with Riesz fractional derivatives in space. It is capable of describing multi-species anomalous transport processes with memory in time and nonlocal interactions in space. To approximate the system, we design a quadratic-weighted scheme for the Caputo derivative and a meshless scheme for the Riesz derivative. This leads to a fully discrete numerical method that is accurate, flexible, and easy to implement. We establish a rigorous stability analysis using the energy method. We also prove convergence, showing second-order accuracy in both time and space. Supporting the theory, we prove auxiliary results such as spatial coercivity, discrete energy monotonicity, and boundedness of the numerical solution. Two numerical examples are presented to validate the analysis. The computed errors and convergence rates are summarized in tables and illustrated with figures. The numerical results confirm the predicted second-order accuracy in both temporal and spatial directions.
