This study introduces novel numerical methods that employ spectral Galerkin and collocation techniques with shifted Schröder polynomials (SSPs) to solve linear and nonlinear second-order two-point boundary value problems (SOTBVPs). The proposed techniques are formulated through a reduced sequence of modified sets of SSPs. The unknown expansion coefficients are determined by using spectral Galerkin and collocation techniques. The resulting algebraic systems are efficiently solved using appropriate numerical solvers. Illustrative examples are provided to validate and demonstrate the accuracy, efficiency, and applicability of the proposed methodologies.
The main goal of this paper is to provide a logical advancement in the mathematical properties and representations related to Mittag-Leffler–Laguerre polynomials. Generating relations, finite summations, integral representations, and integral transforms for these polynomials are established. Some particular cases and consequences of the main results are also considered.
We present new sharp bounds for the function \((\sin x)/x, \) thus refining the well-known Jordan-type inequalities in the literature. A polynomial-trigonometric approach is used to establish the bounds. The main results are based on the series expansions, monotonicity rules, and the bounds of the ratio of even indexed Bernoulli numbers. We also generalize our main results using the concept of stratification.
In this work, we prove that the formal Stieltjes of q-Laguerre -hahn forms is a solution of many q-Ricatti equations. As a consequence , we show that the class of those forms depends on k ∈ ℕ. Some examples are highlighted.
This paper studies integral inequalities for a class of parameter-dependent weighted integral functionals involving two non-negative functions. We establish several inequalities describing the behavior of the associated integral functional under various structural assumptions on one of the functions, including monotonicity, convexity, log-convexity, and sub-multiplicativity. These results provide a unified framework that extends and generalizes inequalities obtained previously for certain special functions.
This paper proves a generalization of Hake’s Theorem for the Henstock‑Kurzweil‑Stieltjes (HKS) integral in the context of interval‑valued functions defined on time scales. The developed framework unifies the non‑absolute integration of Henstock‑Kurzweil type with Stieltjes integration on arbitrary time domains, thereby extending classical real analysis to settings that encompass both continuous and discrete dynamics. We provide a comprehensive theoretical extension with potential applications in uncertain dynamical systems modelled by set‑valued functions on hybrid time domains. The research covers fundamental theorems, properties and examples with suitable applications to interval-valued functions, demonstrating the Hake’s theorem significance in handling unbounded functions and infinite time scales.
We construct explicit strictly ascending chains of dense subalgebras of length 𝔠 in every separable infinite-dimensional complex Banach algebra. For large classes of commutative C*-algebras we also construct strictly descending chains of the same length. The constructions rely on algebraic independence, Stone–Weierstrass arguments, and transfinite recursion.
The multidimensional Fourier-Bessel transform is a generalization of Fourier-Bessel transform that obeys the same uncertainty principles as the classical Fourier transform. In this paper, we establish the following uncertainty principles; an \(L^p-L^q\)-version of Morgan’s theorem, the Donoho-Stark uncertainty principles and bandlimited principles of concentration type for the multidimensional Fourier-Bessel transform.
Let \(D\subset \mathbb{R}^3\) be a bounded domain. \(q\in C(D)\) be a real-valued compactly supported potential, \(A(\beta, \alpha,k)\) be its scattering amplitude, \(k>0\) be fixed, without loss of generality we assume \(k=1\), \(\beta\) be the unit vector in the direction of scattered field, \(\alpha\) be the unit vector in the direction of the incident field. Assume that the boundary of \(D\) is a smooth surface \(S\). Assume that \(D\subset Q_a:=\{x: |x|\le a\}\), and \(a>0\) is the minimal number such that \(q(x)=0\) for \(|x|>a\). Formula is derived for \(a\) in terms of the scattering amplitude.
We develop and analyze an adaptive spacetime finite element method for nonlinear parabolic equations of \(p\)–Laplace type. The model problem is governed by a strongly nonlinear diffusion operator that may be degenerate or singular depending on the exponent \(p\), which typically leads to limited regularity of weak solutions. To address these challenges, we formulate the problem in a unified spacetime variational framework and discretize it using conforming finite element spaces defined on adaptive spacetime meshes. We prove the well-posedness of both the continuous problem and the spacetime discrete formulation, and establish a discrete energy stability estimate that is uniform with respect to the mesh size. Based on residuals in the spacetime domain, we construct a posteriori error estimators and prove their reliability and local efficiency. These results form the foundation for an adaptive spacetime refinement strategy, for which we prove global convergence and quasi-optimal convergence rates without assuming additional regularity of the exact solution. Numerical experiments confirm the theoretical findings and demonstrate that the adaptive spacetime finite element method significantly outperforms uniform refinement and classical time-stepping finite element approaches, particularly for problems exhibiting localized spatial and temporal singularities.
