Tamkang Journal of Mathematics

Approximating the fixed points of Suzuki's generalized non-expansive map via an efficient iterative scheme with an application

Pragati Gautam, Chanpreet Kaur — Sat, 01 Feb 2025 00:00:00 +0000

This paper is aimed at proving the efficiency of a faster iterative scheme called $PC^*$-iterative scheme to approximate the fixed points for the class of Suzuki's Generalized non-expansive mapping in a uniformly convex Banach space. We will prove some weak and strong convergence results. It is justified numerically that the $PC^*$-iterative scheme converges faster than many other remarkable iterative schemes. We will also provide numerical illustrations with graphical representations to prove the efficiency of $PC^*$ iterative scheme. As an application of the solution of a fractional differential equation is obtained by using $PC^*$ iterative scheme.

An orthogonal class of $p$-Legendre polynomials on variable interval

Nidhi R. joshi, B. I. Dave Dave — Sat, 01 Feb 2025 00:00:00 +0000

The work incorporates a generalization of the Legendre polynomial by introducing a parameter $p>0$ in its generating function. The coefficients thus generated, constitute a class of the polynomials which are termed as the $p$-Legendre polynomials. It is shown that this class turns out to be orthogonal with respect to the weight function: $(1-\sqrt{p}\ x)^{\frac{p+1}{2p}-1}(1+\sqrt{p}\ x)^{\frac{p+1}{2p}-1}$ over the interval $(-\frac{1}{\sqrt{p}}, \frac{1}{\sqrt{p}}).$ Among the other properties derived, include the Rodrigues formula, normalization, recurrence relation and zeros. A graphic depiction for $p=0.5, 1, 2,$ and $3$ is shown. The $p$-Legendre polynomials are used to estimate a function using the least squares approach. The approximations are graphically depicted for $p=0.7, 1, 2.$

Some inequalities for the numerical radius and spectral norm for operators in Hilbert $C^{\ast}$-modules space

Mohammad M.H Rashid — Sat, 01 Feb 2025 00:00:00 +0000

In this paper, a fresh approach to investigating the numerical radius of bounded operators on Hilbert $C^*$-modules is presented. By using our approach, we can produce some novel findings and extend certain established theorems for bounded adjointable operators on Hilbert $C^*$-module spaces. Moreover, we find an upper bound for power of the numerical radius of $t^{\alpha}ys^{1-\alpha}$
under assumption $0\leq \alpha\leq 1$. In fact, we prove
$$w_c\left(t^{\alpha}ys^{1-\alpha}\right)\leq{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert y \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}^r{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \alpha t^{r}+(1-\alpha)s^{r}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}$$
for all $0\leq \alpha\leq 1$ and $r \geq 2$.

Exploring helices in Minkowski $3$-space $\mathbb{E}_{1}^{3}$

Esra Betül Koç Öztürk, Ufuk Öztürk — Sat, 01 Feb 2025 00:00:00 +0000

This study focuses on the geometric exploration of helices in Minkowski $3-$ space. For this purpose, we study the problem of constructing a spacelike general helix, slant helix, or Darboux helix with a timelike principal from a given plane curve, fixed vector, and constant angle. We obtain a parametric representation for the helices whose projection is onto the plane $P$ perpendicular to the fixed vector $U$ share the same fixed vector. In addition, we furnish various illustrative examples showcasing the geometric characteristics of these helices.

A fitted parameter convergent finite difference scheme for two-parameter singularly perturbed parabolic differential equations

Mekashaw Ali, Justin B. Munyakazi, Tekle Gemechu Dinka — Sat, 01 Feb 2025 00:00:00 +0000

The objective of this paper is to develop a numerical scheme that is uniform in its parameters for a specific type of time-dependent parabolic problem with two perturbation parameters. The existence of these two parameters in the terms with the highest-order derivatives results in the formation of boundary layer(s) in the solution of such problems. Solving these model problems using classical methods does not yield satisfactory results due to the layer behavior. Therefore, nonstandard finite difference schemes have been developed as a means to obtain numerical solutions for these problems. To develop the scheme, we employ the Crank-Nicolson
discretization on a uniform time mesh and apply a fitted operator method with a uniform spatial mesh. We have established the stability and convergence of the proposed scheme. The proposed scheme exhibits uniform convergence of second order in the temporal direction and first order in the spatial direction. However, temporal mesh refinements is employed to enhance the order to two in both directions. Model examples are provided to validate the practicality of the proposed numerical scheme.

Homogenization of partial differential equations with Preisach operators

Achille Landri Pokam Kakeu — Sat, 01 Feb 2025 00:00:00 +0000

The current work deals with initial boundary value parabolic problems with Preisach hysteresis whose the density functions are allowed to depend on the variable of space. The model contains nonlinear monotone operators in the diffusion term, arising from an energy. Thanks to the properties of Preisach hysteresis operators and to the sigma-convergence method, we obtain the convergence of the
microscopic solutions to the solution of the homogenized problem. The effective operator is obtained in terms of a solution of a nonlinear corrector equation addressed in the usual sense of distributions, leading in an approximate scheme for the homogenized
coefficient which is an important step towards the numerical implementation of the results from the homogenization theory beyond the periodic setting.