Department of Mathematics

Discussion of the doctoral thesis at the College of Computer Science and Mathematics, Department of Mathematics, entitled:

“Improving Wavelet Methods for Approximate Solutions of Nonlinear Partial Differential Equations”

supervised by Prof. Dr. Ekhlass Saadallah Ahmed.

The Thesis Addressed

This thesis aims to improve the numerical solutions of nonlinear ordinary and partial differential equations by developing hybrid methods combining wavelet analysis and Accelerated Adomian Analysis (ADM), using three wavelet families: Haar, CAS, and CAS-Haar.

The thesis included the derivation of new general formulas and algorithms for solving ordinary and partial differential equations, as well as systems of differential equations, while demonstrating the theoretical convergence of the proposed methods. Adomian polynomials were also employed to handle nonlinear terms, transforming complex problems into simpler and more efficient algebraic systems.

Finally, a new hybrid method using Accelerated Adomian Analysis (AADM) was developed. The results showed that this method achieved higher accuracy and better computational efficiency compared to the traditional hybrid method, with a significant reduction in absolute error and mean squared error (MSE).

Finally, the conclusions were presented based on theoretical analysis and numerical comparisons, along with a set of recommendations for expanding these methods and applying them to nonlinear systems and more complex medical models.

The Discussion Committee Consisted of

• Prof. Dr. Saad Fawzi Jasim — Chair
• Prof. Dr. Abdulghafor M. Al-Rozbayani — Member
• Asst. Prof. Dr. Ahmed Farooq Qasim— Member
• Asst. Prof. Dr. Qasim Abbas Hussein — Member
• Asst. Prof. Dr. Ahmed Entesar Githeeth— Member
• Prof. Dr. Ekhlass Saadallah Ahmed — Member and Supervisor