College of Computer Science and Mathematics

Discussion of a master’s thesis in the College of Computer Science and Mathematics – Department of Mathematics Sciences entitled:

“Numerical and Analytical Methods for Solving Nonlinear Partial Differential Equations Arising in Mathematical Physics “

It was discussed in the discussion hall of Computer Science and Mathematics at the University of Mosul on sunday 2 2025/9/.

master’s thesis by Naglaa Sami Abed Marey

under the supervision of Asst. Prof. Dr .Mohammed Omar Shaaban Mohammed  

This thesis aims to conduct an in depth and detailed study of the analytical and approximate solutions of nonlinear partial differential equation systems in (2+1) dimensions, which describe the evolution of long waves affected by both dispersion and dissipation phenomena. To achieve this objective, two analytical methods are employed: the Simplest Equation Method (SEM), which aims to obtain exact closed-form solutions, and the Residual Power Series Method (RPSM), which provides highly accurate approximate solution

The SEM approach relies on transforming the partial differential equations into ordinary differential equations through the traveling wave transformation. This method yielded a diverse set of novel traveling wave solutions that deepen the understanding of the dynamic properties of the systems, including bell-shaped solitons, inverted bell-shaped solitons, and kink wave solutions. The physical behavior of these solitons was graphically demonstrated using two- and three-dimensional plots under appropriate parameter selections.

On the other hand the RPSM constructs the approximate solution as a power series expansion whose coefficients are determined by the residual error, without requiring additional assumptions such as linearization, perturbation techniques, or discretization, relying solely on the initial conditions of the problem. Numerical results showed that this method produces highly accurate solutions by comparing them with exact solutions, and the error can be reduced by adding further terms to the series.

These findings emphasize the scientific significance of both methods in analyzing important mathematical models in physics and engineering, and they open promising avenues for future research aimed at exploring more nonlinear phenomena using advanced analytical and approximate techniques.

The scientific committee included the following members:

• Prof. Dr. Ahmed Farouq Qasim-Chairman
• Prof. Dr. Badran  Jasim Salem. – Member
• Mahasen Thabet Younis- Member
• Prof. Dr. Mohammed Omar Shaaban Mohammed  – Member and supervisor.