Browsing by Author "Rahman, Riaz Ur"
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Article Analytical Study of Fractional Dna Dynamics in the Peyrard-Bishop Oscillator-Chain Model(Elsevier, 2024) Riaz, Muhammad Bilal; Fayyaz, Marriam; Rahman, Riaz Ur; Martinovic, Jan; Tunc, OsmanIn this research, we present a new auxiliary equation approach, which uses two distinct fractional derivatives: /3- and M-truncated fractional derivatives to explore the space-time fractional Peyrard-Bishop DNA dynamic model equation. This examines the nonlinear interplay between neighboring displacements and hydrogen bonds through mathematical modeling of DNA vibration dynamics. The solutions are tasked with examining the nonlinear interaction among neighboring displacements of the DNA strand. The generated solutions exhibit various wave patterns under varying fractional values and parametric conditions: w-shape, bright, combined periodic wave solutions, dark-bright, bell shaped, m-shaped, w-shaped with two bright solutions, and m-shape with two dark solutions. Graphical representations provide a complete analysis of these physical features. The results demonstrate the successful implementation of the proposed approach, which will be advantageous for locating analytical remedies to more nonlinear challenges.Article Exploring Analytical Solutions and Modulation Instability for the Nonlinear Fractional Gilson-Pickering Equation(Elsevier, 2024) Rahman, Riaz Ur; Riaz, Muhammad Bilal; Martinovic, Jan; Tunc, OsmanThe primary goal of this research is to explore the complex dynamics of wave propagation as described by the nonlinear fractional Gilson-Pickering equation (fGPE), a pivotal model in plasma physics and crystal lattice theory. Two alternative fractional derivatives, termed fi and M -truncated, are employed in the analysis. The new auxiliary equation method (NAEM) is applied to create diverse explicit solutions for surface waves in the given equation. This study includes a comparative evaluation of these solutions using different types of fractional derivatives. The derived solutions of the nonlinear fGPE, which include unique forms like dark, bright, and periodic solitary waves, are visually represented through 3D and 2D graphs. These visualizations highlight the shapes and behaviors of the solutions, indicating significant implications for industry and innovation. The proposed method's ability to provide analytical solutions demonstrates its effectiveness and reliability in analyzing nonlinear models across various scientific and technical domains. A comprehensive sensitivity analysis is conducted on the dynamical system of the f GPE. Additionally, modulation instability analysis is used to assess the model's stability, confirming its robustness. This analysis verifies the stability and accuracy of all derived solutions.