Türkiye Mathematical Sciences

Approximate Solutions for Some Nonlinear Second-Order Ordinary Differential Equations Using Picard–Lindelof and Adomian Decomposition Methods

2026-06-27T12:55:52+03:00

This paper introduces a comparison between the Picard–Lindelof method (PIM) and the Adomian decomposition method (ADM) for solving second-order nonlinear differential equations. Both methods are applied to selected test problems to evaluate their accuracy and convergence rate. The Picard method is employed as an iterative scheme for successive approximations, while ADM provides a decomposition-based approach to handle nonlinearities. Numerical results indicate that both methods are effective, yet their performance varies depending on the form and complexity of the nonlinear terms. Through Examples 1–4, the paper presents a comparison between the two methods with respect to the exact solution.

Caputo Fractional Operator and Aboodh Transform Technique of WBK Equations: Some Comparisons of Analytical and Numerical Solutions with Other Techniques

2026-06-04T21:54:26+03:00

Among the most important equations in the theory of shallow water waves are the Whitham–Broer–Kaup equations (WBK equations), which form a coupled system of nonlinear partial differential equations describing the propagation of shallow water waves with different dispersion relations and diffusion effects. In this paper, we combine the Aboodh transform with the residual power series method to develop a new technique called the Aboodh Residual Power Series Technique (ARPST). This technique, together with the Caputo fractional operator, is used to obtain analytical and numerical solutions of the WBK equations by calculating the coefficients of the corresponding power series. The proposed ARPST is applied to solve the WBK equations in two cases: exact initial conditions and approximate initial conditions. To demonstrate the capability, efficiency, and reliability of the proposed technique, the numerical results of the analytical solutions are presented graphically and compared with results obtained by other established methods, including the Adomian decomposition method, the optimal homotopy asymptotic method, and the Sumudu decomposition method.

Entropy Extraction from Audio Signals Using Hilbert-Mod Transformation and Von Neumann Refinement

2025-11-11T21:48:21+03:00

This paper introduces a multi-stage framework for extracting high-entropy binary sequences from audio signals. The procedure consists of generating the an alytic signal via Hilbert transformation, applying thresholding and amplification, and performing modular reduction before binary encoding based on local ampli tude comparisons. Experiments were carried out using several prime moduli and amplification parameters, with optional post-processing through XOR aggregation and Von Neumann debiasing. Randomness quality was assessed using the NIST SP 800-22 test suite. The results indicate that configurations employing larger modulus values yield statistically robust bitstreams, while smaller moduli tend to introduce detectable structural patterns. Overall, the method provides effective entropy enhancement, particularly when dispersion and debiasing parameters are appropriately tuned.

Hub Number in Ring Fuzzy Graphs and Restricted Hyper-Totient Fuzzy Graphs

2026-06-04T20:11:58+03:00

In this paper, we introduce and study the hub number of ring fuzzy graphs. To make the construction self-contained, a ring fuzzy graph is treated as a weighted fuzzy structure RFΓ = (V, E, μ, σ) whose underlying vertices are ring-induced substructures and whose hub number is computed by the total vertex-membership value of a minimal hub set. We investigate this parameter for several important classes of ring fuzzy graphs, including complete, path, cycle, and star ring fuzzy graphs, and obtain explicit expressions in each case. In addition, we derive general bounds for the hub number in terms of the complement graph and several neighborhood-based parameters. Relationships between the hub number and other graph-theoretic quantities, such as domination-type and chromatic-type parameters, are also examined. Furthermore, we extend the study to restricted hyper-totient fuzzy graphs and present a characterization of their hub number. Finally, we analyze the behavior of the hub number under several operations on ring fuzzy graphs, including intersection, union, direct product, and direct sum, stating the required compatibility assumptions on the underlying vertex sets and membership functions. The results obtained in this work show that the hub number is a useful structural parameter for the analysis of ring fuzzy graphs and provide a foundation for further investigations in algebraic fuzzy graph theory.

Diamond-∝ Dynamic Type Inequalities

2025-10-20T14:23:10+03:00

In this work, we prove diamond inequalities of dynamic type by applying reversed Hölder’s inequality and the chain rule with diamond calculus on time scales. As particular cases of our results (when T = N and T = R), we get the reversed form of discrete and continuous inequalities.

A Predictor–Corrector Method for Conformable Fractional Differential Equations of Order α ∈ (1,2]

2026-06-09T02:47:51+03:00

In this paper, we develop a predictor–corrector numerical method for solving conformable fractional differential equations of order α ∈ (1,2. By transforming the conformable fractional problem into an equivalent system of first-order ordinary differential equations, we construct an Adams–Bashforth–Moulton type scheme adapted to the conformable framework. The proposed method is easy to implement and avoids memory effects typical of Caputo-type fractional models. A consistency and convergence analysis is provided, and the method achieves second-order accuracy under suitable smoothness assumptions