Pshtiwan Othman Mohammed

The dynamical behavior of nonlinear fractional convection-diffusion equations (CDEs) is very rich and extends beyond their applicability as transport models, but it has yet to be explored numerically. In this paper, it is demonstrated that a time- and space-fractional nonlinear convection-diffusion equation exhibits a transition from stable equilibrium to deterministic chaos its nonlinear convection parameter increases by employing an exponential cubic B-spline scheme for this said problem. The Caputo fractional derivative is discretized using the L1 formula, whereas the fractional spatial operators are approximated using the exponential spline method. Using phase portraits, bifurcation diagrams, and Lyapunov exponent spectra, we identify three different dynamical behaviors: a stable regime, periodic behavior, and finally chaos, arising from the increasing value of the nonlinear convection parameter. Therefore, this work establishes the nonlinear convection parameter as a true bifurcation parameter and demonstrates the possibility of complex nonlinear dynamics in fractional convection-diffusion systems.

We study the propagation of optical solitons in nonlinear media using a nonlinear Schrödinger equation with higher-order terms. We achieve accurate solutions as brilliant and dark solitons using the Extended Direct Algebraic Method (EDAM). The approach enables us to efficiently manage the dispersive and higher-order nonlinear factors. The impact of the coefficients on the soliton solutions is also investigated. Visual insights into the behaviour of the solitons are provided by the graphic analysis of the generated solutions using 2D and 3D plots. Furthermore, contour plots are created to show how the solutions vary depending on the different parameters, exposing complex structures and patterns. A greater comprehension of the physical phenomena that the equation describes is made possible by these graphical representations.

An extended direct algebraic method is used in this work to examine the soliton solutions of the fractional Hirota-Satsuma coupled Korteweg-de Vries equation. Understanding the dynamic behaviour of solitons in nonlinear systems using analytical solutions is our goal. To obtain precise soliton solutions, we utilise a logistic technique. These solutions are then shown graphically in three dimensions, two dimensions, and contours. Soliton interactions' complex dynamics and stability in fractional nonlinear systems are demonstrated by the results. This study clarifies the underlying dynamics and possible uses of soliton behaviour in complex systems, advancing our understanding of this phenomenon.

This study extends the discrete version of the Susceptible–Infected–Recovered (SIR) model to the non-autonomous case in which the rates of infection and cure are time-dependent. This generalization addresses a critical open problem in the literature. We develop a non-standard finite difference approach that maintains essential biological properties, such as population conservation and non-negativity, even under parameter variation. The main novelty here is the derivation of an exact semi-analytical solution to the non-autonomous nonlinear system. We also leverage the exact solution to develop an optimal control problem, which illustrates the potential for rigorous optimization of treatment strategies to minimize infection costs. The numerical simulations verify the theoretical results, showing not only the stability of the model when parameters change seasonally, as well as the efficiency of the optimal controls, but also the predictive power of the non-autonomous approach compared to classical autonomous models.