College of Computer Science and Mathematics

Discussion of Ph.D. Dissertation  in the College of Computer Science and Mathematics – Department of Mathematics Sciences entitled:

“ Weighted Residual Spectral Methods for Solving Generalized Fractional Differential Equations”  

It was discussed in the discussion room at the Faculty of Computer Science and Mathematics at the University of Mosul on Sunday, 18 -5-2025.

Ph.D. Dissertation by Student Abdulrazzaq Talal Abed

under the supervision of Prof. Dr. Ekhlass Saadallah Ahmed

In real-world problems, complex systems often lead to complex mathematical models, and researchers strive to provide the best mathematical models to adequately describe these phenomena. The advent of the fractional derivative has led to the solution of many applied problems. However, assigning a single value to the order in systems whose physical properties change over time, or whose internal interactions and multiple factors affect the system, makes it difficult to describe these situations using fixed orders. As a result, these phenomena have led to the emergence of two independent concepts of the fractional derivative: distributed order operators and variable order operators. This thesis studies the numerical solutions of linear and nonlinear differential equations with fixed, variable, and distributed fractional orders and delay problems. Three models in viscoelasticity are highlighted: the Bagley-Torvik equation, the Atanakovic dynamic equation, and Pantograph differential equation. Spectral methods based on several orthogonal and non-orthogonal polynomials were presented and combined with weighted residual methods to obtain the best numerical solutions. The results of each method were compared, the convergence of the proposed combined method was studied, and the error resulting from approximation was analysed. High-accuracy results were obtained in several tests.

The scientific committee included the following members:

• Dr. Saad Fawzi Jasim – Chairman
• Dr. Abdulghafor Mohammed Ameen Khudhur- Member.
• Prof. Dr. Ahmed Farooq Qasim – Member.
• Prof. Dr. Ammar Isam Edress- Member.
• Prof. Dr. Badran Jasim Salim – Member.
• Dr. Ekhlass Saadallah Ahmed – Member and supervisor.