Browsing by Author "Martinovic, Jan"

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Analysis of the Integro-Differential Jaulent-Miodek Evolution Equation by Nonlinear Self-Adjointness and Lie Theory
(Springer, 2025) Junaid-U-Rehman, Muhammad; Riaz, Muhammad Bilal; Tunc, Osman; Martinovic, Jan
The current exploration is related to the extraction of the exact explicit solutions of the Integrodifferential Jaulent-Miodek evolution (IDJME) equation. In general, the Jaulent-Miodek equation has many applications in many divisions of physics, for example optics and fluid dynamics. The symmetries of this (2 + 1)-dimensional (IDJME) equation are derived, and it admits the 8th Lie algebra. The similarity transformation method is considered to convert the NLPDE to the nonlinear ODE by using the translational symmetries. Then, we calculated the traveling-wave solutions by an analytical method, namely the extended direct algebraic method. Some obtained solutions are represented by giving suitable parameter values to understand their physical interpretation. The results obtained from this method involve various types of functions, for example, exponential, logarithmic, hyperbolic, and trigonometric. The self-adjointness theory is employed to classify and help us to compute the conserved quantities of the assumed model. A similar work related to this does not exist in the literature.
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, Osman
In 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.
Exploring Analytical Solutions and Modulation Instability for the Nonlinear Fractional Gilson-Pickering Equation
(Elsevier, 2024) Rahman, Riaz Ur; Riaz, Muhammad Bilal; Martinovic, Jan; Tunc, Osman
The 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.
Modeling and Simulations for the Mitigation of Atmospheric Carbon Dioxide Through Forest Management Programs
(Amer inst Mathematical Sciences-aims, 2024) Riaz, Muhammad Bilal; Raza, Nauman; Martinovic, Jan; Bakar, Abu; Tunc, Osman
The growing global population causes more anthropogenic carbon dioxide (CO2) 2 ) emissions and raises the need for forest products, which in turn causes deforestation and elevated CO2 2 levels. A rise in the concentration of carbon dioxide in the atmosphere is the major reason for global warming. Carbon dioxide concentrations must be reduced soon to achieve the mitigation of climate change. Forest management programs accommodate a way to manage atmospheric CO2 2 levels. For this purpose, we considered a nonlinear fractional model to analyze the impact of forest management policies on mitigating atmospheric CO2 2 concentration. In this investigation, fractional differential equations were solved by utilizing the Atangana Baleanu Caputo derivative operator. It captures memory effects and shows resilience and efficiency in collecting system dynamics with less processing power. This model consists of four compartments, the concentration of carbon dioxide C (t), human population N (t), forest biomass B (t), and forest management programs P (t) at any time t. The existence and uniqueness of the solution for the fractional model are shown. Physical properties of the solution, non-negativity, and boundedness are also proven. The equilibrium points of the model were computed and further analyzed for local and global asymptotic stability. For the numerical solution of the suggested model, the Atangana-Toufik numerical scheme was employed. The acquired results validate analytical results and show the significance of arbitrary order delta . The effect of deforestation activities and forest management strategies were also analyzed on the dynamics of atmospheric carbon dioxide and forest biomass under the suggested technique. The illustrated results describe that the concentration of CO2 2 can be minimized if deforestation activities are controlled and proper forest management policies are developed and implemented. Furthermore, it is determined that switching to low-carbon energy sources, and developing and implementing more effective mitigation measures will result in a decrease in the mitigation of CO 2 .