Türkiye Mathematical Sciences

Adaptive Physics-Informed Neural Networks for Singularly Perturbed Convection-Diffusion Problems

Suayip Toprakseven — 2025-12-31

\begin{abstract}
This paper presents an adaptive physics-informed neural network (PINN) framework for the numerical solution of one-dimensional singularly perturbed differential equations. Such problems are characterized by the presence of small perturbation parameters multiplying the highest-order derivatives, which typically generate sharp boundary or interior layers and lead to severe numerical difficulties for standard discretization methods. The proposed approach integrates a residual-based adaptive sampling strategy with a dynamically refined neural network training process, allowing the method to focus computational effort in regions of rapid solution variation.

The governing differential equation and associated boundary conditions are incorporated directly into the loss function, ensuring consistency with the underlying physics. To enhance stability and accuracy in the layer regions, the training data are progressively enriched using an error indicator derived from the local PDE residual. Numerical experiments on representative singularly perturbed convection--diffusion and reaction--diffusion problems demonstrate that the adaptive PINN significantly improves pointwise accuracy compared to standard PINNs, particularly in boundary-layer regions, while maintaining computational efficiency.

The results confirm that adaptive sampling combined with physics-informed learning provides a robust and flexible tool for solving one-dimensional singularly perturbed problems without requiring a priori knowledge of layer locations or specialized meshes.
\end{abstract}

A new decision-making approach for the green finance investment strategies with interval-valued Pythagorean fuzzy sets

Murat Kirişci — 2025-12-31

Based on criteria derived from environmental, social, and governance (ESG) sub-criteria, this study presents a decision support system to aid in selecting the optimal green finance investment plan. For interval-valued Pythagorean fuzzy sets, a scoring function, distance measure, similarity measure, and entropy measure are introduced as a set of new mathematical tools for decision-making under uncertainty. The Interval-Valued Pythagorean Fuzzy Sets framework is employed to evaluate seven popular sustainable investment strategies: Impact Investing, Environmental, Social, and Governance (ESG) Integration, Green Bonds, Sustainable Agriculture Funds, Shareholder Engagement, Renewable Energy Funds, and Thematic Investing. This work primarily utilizes a score function and distance metric for interval-valued Pythagorean fuzzy numbers to address specific comparative challenges. We used an entropy measure based on an interval-valued Pythagorean fuzzy set to calculate the objective weights. We then used the weighted distance-based approximation approach. The best option may be close to the negative-ideal solution (AIP-worst plan) and far from the positive-ideal solution (PIS-best plan), according to the weighted distance-based approximation technique.

Finite Time Blow up in a Triharmonic Nonlinear Wave Model with Variable Damping

Erhan Pişkin — 2025-12-31

In this work, we deal with the triharmonic wave equations. We established blow up solution with negative initial energy under suitable conditions on variable exponents.

ON DYNAMIC INEQUALITIES OF HILBERT-TYPE

zeynep alpsoy — 2025-12-31

In this paper, we present new weighted diamond alpha dynamic inequalities of Hilbert type on time scales. Our approach is based on the application of the reversed Hölder’s inequality, the chain rule, and the mean inequality within the framework of diamond alpha calculus. Furthermore, we demonstrate that, as particular cases of our general results (for T=N and T=R ), one can obtain the reversed discrete and continuous forms of Hilbert-type inequalities.

Optimal Scaling in Double-Contact Regular Polygon Containment

Cemil Karaçam — 2025-12-31

This study presents a detailed derivation of a trigonometric identity governing the optimal scaling of a regular m-gon inscribed within a regular n-gon underdouble-contact constraints. Building on prior work that established containment inequalities for nested polygons in the complex plane, we focus on the symmetric configuration where rotational and vertical translation components vanish (b = 0, d = 0). In this setting, we derive a closed-form expression for the scaling factor c by equating two distinct contact conditions involving edge-vertex interactions. The resulting identity incorporates cosine and cotangent terms and reveals how geometric symmetry leads to algebraic simplification. We also provide a long-form factorization and numerical examples to illustrate the identity’s behavior across different polygon pairs. This work contributes to the broader theory of polygonal optimization and symbolic encoding in geometric configurations.