College of Computer Science and Mathematics

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

Using the developed approximated methods to solve fractional Hirota-Satsuma coupled KdV equations “

It was discussed in the discussion room at the Faculty of Computer Science and Mathematics at the University of Mosul on Sunday, 27 -7-2025.

Ph.D. Dissertation by Student Mohammed Ahmed Khader Ibrahim

under the supervision of Prof. Dr. Abdulghafoor  Muhammad Amin Khader – Member and supervisor.

The chief objective of this thesis is to use four developed approximation methods to solve the fractional-order coupled Hirota-Satsuma HSC-KdV equations with the lowest error rate. These equations are used to model longitudinal waves propagating in shallow water and are used in many applications in fluid mechanics. This is accomplished in two stages:

The first: Transforming a fractional partial differential equation into an ordinary differential equation using transformations.

The second: Using semi-analytical methods to solve nonlinear ordinary differential equations. This is represented by introducing a new tool for solving any mathematical system in various fields, which is the hybridization of the Kalla method with the Sumudu transform (ST-KDM), provided it is represented by fractional partial differential equations. This hybrid method combines the Sumudu transform with the Kalla decomposition method. This method reduces computational effort compare to ADM. The Sumudu transform eliminates the fractional order in the differential equations and converts them to ordinary differential equations. The Kalla polynomial then eliminates the nonlinear terms. ST-KDM provides solutions in a series converges to the exact solution. The steps of this method are explained in a simple manner, its convergence is studied, its algorithm is created, and it is applied to three different examples of fractional-order differential equations to demonstrate its efficiency and accuracy in finding analytical solutions. One of these examples is the HSC-KdV equation. Tables are created to compare the analytical solutions with the exact solutions, study the LSWF error, and plot both solutions in two-dimensional and three-dimensional forms. The effect of the fractional order 𝛼 on the approximate solution is also plotted.

The scientific committee included the following members:

• Dr. Ekhlass Saadallah Ahmed – Chairman
• Prof. Dr. Firas Adel Fawzi (Member).
• Prof. Dr. Ahmed Farouk Qasim – Member.
• Prof. Dr. Badran Jasim Salim – Member.
• Prof. Dr. Ahmed Intisar Ghatheeth / Member
• Dr. Abdulghafoor Muhammad Amin Khader – Member and supervisor.