College of Computer Science and Mathematics

Discussion of a Doctorial Dissertation  in the College of Computer Science and Mathematics – Department of Mathematics Sciences entitled:

“ Enhanced Numerical Methods for Solving Fractional Partial Differential Equations“

 In continuation of the scientific research movement and with the follow-up and presence of the Dean of the College of Computer Science and Mathematics

   Prof. Dr. Duha Bashir Abdullah The esteemed

It was discussed in the discussion room at the Faculty of Computer Science and Mathematics at the University of Mosul on Thursday  28-3-2024

Doctorial Dissertation by student Almutasim  Abdulmuhsin  Hamed

, under the supervision of    Prof. Dr. Ekhlass Saadallah Ahmed

In this thesis, the use of finite difference methods has been expanded to find numerical solutions to nonlinear fractional partial differential equations in the three cases, which are temporal fractional derivative, spatial fractional derivative, and temporal and spatial fractional derivative. It included both the explicit finite difference method and the exponential finite difference method. The fractional derivative has been defined in two ways: first, the Grünwald-Letnikov formula in the case of the fractional derivative of time and space; second, the Caputo formula for the fractional derivative of time and the Grünwald-Letnikov formula for the fractional derivative of space.

For the first time, the fractional exponential finite difference method was used to solve non-linear fractional partial differential equations. New algorithms were written for both methods and based on examples of fractional derivation. Points that lie outside the solution region were also treated in the case of the temporal fractional derivative. We used four problems with different types of fractional derivatives in the practical application. The numerical results showed that the exponential finite difference method worked better and more accurately when the Caputo formula was used for the fractional derivative of time and the Grünwald-Letnikov formula was used for the fractional derivative of space.

New hybrid methods were also presented that link the finite difference methods with the Adomian polynomial. The goal is to treat nonlinear terms with the Adomian polynomial while writing an algorithm for each case of fractional derivative. These new techniques lead to improvements in the accuracy and effectiveness of the solutions to the four problems addressed in this study.

The scientific committee included the following members:

• Dr. Abdulghafoor Jasim Salim – Chairman .
• Dr. Saad Abdullah Manaa – Member.
• Prof. Dr. Ahmed Farooq Qasim – Member.
• Prof. Dr. Ahmed Entesar Githeeth – Member
• Prof. Dr. Amal Jasim Mohammed – Member .
• Dr. Ekhlass Saadallah Ahmed – Member and Supervisor .