Tamkang Journal of Mathematics

Convergence analysis of an efficient Chebyshev wavelet and its applications to differential equations via operational matrices of integration

2026-01-05T09:23:19+00:00

In this paper, the convergence analysis of the Chebyshev wavelet of the second kind is thoroughly carried out. Operational matrices for integration and product operations of the second kind Chebyshev wavelet are constructed, and these matrices are utilized to obtain solutions to the differential equations. A theorem related to the proposed operational matrix method is established. Solutions of the differential equations considered in this paper resemble their exact solutions. The characteristics of second kind Chebyshev wavelet are utilized to transform differential equations into systems of algebraic equations, which are solved very efficiently using a suitable method.

Sequence Spaces Constructed by using $\mathfrak{q}$-Pell-Lucas Matrix and its Geometric Properties

2026-01-05T09:40:33+00:00

In this paper, the $\mathfrak{q}$-Pell-Lucas matrix, denoted by
$\grave{\text{Q}}(\mathfrak{q}) = (\grave{\text{Q}}_{{\mathrm{n}} {{\mathrm{k}}}}^{(\mathfrak{q})})_{{\mathrm{n}},{{\mathrm{k}}} \in \mathbb{N}_0}$ is defined by
\[
\grave{\text{Q}}_{{\mathrm{n}} {{\mathrm{k}}}}^{(\mathfrak{q})} =
\begin{cases}
\dfrac{{(\mathfrak{q}+1)}\grave{\text{Q}}_{{{\mathrm{k}}}}(\mathfrak{q})}{{(2+\mathfrak{q})}\grave{\text{Q}}_{{\mathrm{n}}}(\mathfrak{q})+{\mathfrak{q}}\grave{\text{Q}}_{{\mathrm{n}}-1}(\mathfrak{q})}, & 0 \leq {{\mathrm{k}}} \leq {\mathrm{n}}, \\
0, & {{\mathrm{k}}} > {\mathrm{n}},
\end{cases}
\]
and $\{ \grave{\text{Q}}_{{\mathrm{k}}}{(\mathfrak{q})}\} $ corresponds to terms of the $ \mathfrak{q} $-Pell-Lucas sequence. The sequence $\{ \grave{\text{Q}}_{{\mathrm{k}}}{(\mathfrak{q} )}\}$ is defined by \[\grave{\text{Q}}_{{{\mathrm{k}}}}(\mathfrak{q})= \left\{2\grave{\text{Q}}_{{{\mathrm{k}}}-1}(\mathfrak{q} )+\mathfrak{q}^{{{}}}\grave{\text{Q}}_{{{\mathrm{k}}}-2}(\mathfrak{q} )\right\} \text{~for}~{{\mathrm{k}}} \leq 2,~ \grave{\text{Q}}_{0}(\mathfrak{q} )=2,\grave{\text{Q}}_{1}(\mathfrak{q} )=2.\]
The $\mathfrak{q}$-Pell-Lucas matrix serves as the foundation for the construction of matrix domains known as the $\mathfrak{q}$-Pell-Lucas sequence spaces. Within these spaces, we develop a Schauder basis, carry out a detailed investigation of operator ideals, and provide a comprehensive study of the geometric properties of $\ell_p(\grave{\text{Q}}(\mathfrak{q}))$ and $\ell_\infty(\grave{\text{Q}}(\mathfrak{q}))$, particularly addressing the Dunford--Pettis property, and the solidity property.}
\keywordstkjm{q-Pell-Lucas numbers, sequence space, Schauder basis, operator ideal, geometric property.

Approximation by diffusion of semiconductor Boltzmann equation

2025-12-15T05:38:29+00:00

In this paper, we deal with the asymptotic behaviour of a semiconductor Boltzmann equation by using the sigma convergence method. We first prove that the scaled model is well-posed in the usual Lebesgue space of square integrable functions, and we perform the a priori estimates. Then, assuming that the coefficients of the model are highly oscillating in space variable, we show that in the non-vanishing flux case, the homogenized problem is equivalent at first order, to a hyperbolic process modified by a perturbation of viscosity, and the diffusion term appears at second order. In the vanishing flux case, we obtain a diffusion model.

Strong convergence of Noor iteration in $L^p$ spaces via $p$-uniform convexity

2026-01-19T07:32:17+00:00

This paper investigates the strong convergence of Noor iteration for nonexpansive operators in $L^p$ spaces, where $1 < p < \infty$. By exploiting the $p$-uniform convexity of $L^p$, we derive explicit $p$-dependent constants in the asymptotic regularity inequalities and obtain quantitative convergence rates. We establish sufficient conditions for strong convergence, including the case of affine operators, demi-compact operators, and norm convergence. As an application, we analyze the $p$-Laplace equation for $p>1$ and validate our theoretical findings through comprehensive numerical experiments. The numerical results demonstrate the efficiency of the method and the importance of parameter selection across different nonlinearity regimes. Our work provides a quantitative refinement of known results in the literature and offers practical insights for solving nonlinear partial differential equations.

Finite-time blow-up of solutions for a Hartree type wave equation involving distributed delay, fractional conditions, and infinite memory effects

2026-03-14T13:15:40+00:00

This paper studies a Hartree-type wave equation featuring a distributed delay term and memory effects governed by a past history. The problem is formulated with coupling through fractional boundary conditions. Under suitable assumptions and for negative initial energy, we prove that solutions blow up in finite time.

Some Picture Fuzzy Information Inequalities with Applications in Market Behaviours and Pattern Recognition

2025-11-21T06:07:55+00:00

Quantifying uncertainty in robust datasets is essential in decision analysis. Merging such data with the concepts of information theory opens various aspects of uncertainty. The additional degree of uncertainty can be addressed by picture fuzzy divergences. This study precisely handles such datasets by introducing a novel picture fuzzy divergence measure (PFDM) using generalized f-divergence. The validity of the measure has been proved, and some properties are discussed. Furthermore, some information inequalities are established for classical dissimilarity measures. These inequalities offer critical mathematical boundaries for analyzing and comparing various fuzzy information measures. Additionally, the application of the measure is discussed for market behaviour trends and pattern recognition problems. The obtained results aims to minimize the divergence and maximize the result accuracy and the measures successfully quantify the variations within fuzzy sets.