The College of Education for Pure Sciences, Department of Mathematics, has submitted a master's thesis on "Application of New Approximate Methods for Solving Boundary Initial Value Problems." The thesis, presented by Ali Faris Abdul-Ali, includes:
In this thesis, we propose a new method for finding approximate analytical solutions to various types of nonlinear flow problems. This technique relies on combining the q-Homotopy analysis method, the Kamal transform, and the Padé approximation (q-HAKPM). This method was used to address the problem of non-Newtonian elastic fluid flow, as well as the kinematically reduced equations of the local Navier-Stokes equations.
The effect of physical quantities on flow was studied in all flow problems, noting that this effect may vary from one problem to another depending on the nature of the problem. In the first section of the study, we conducted a comparative test between the approximate analytical solutions resulting from our proposed method and the solutions resulting from modern approximation methods proposed by several researchers to address the problem of heat transfer and flow of a non-Newtonian fluid over a turbine disc. The results showed that the proposed method is superior in terms of error measures. Furthermore, the resulting approximate analytical solutions exhibited higher accuracy and faster convergence compared to other methods.
The second contribution of this study was to analyze the effect of both low Reynolds number (Re) and Mach number (Ma) on the flow velocity of an incompressible fluid. The effects of physical quantities on the approximate analytical solutions resulting from our new method were presented in tabular and graphical form. The method was tested on two types of flow problems within a two-dimensional moving-cover cavity, and the results were compared with those published in the scientific literature. In all cases, the influence of physical parameters such as viscosity, elasticity, thermal conductivity, and dissipation function was thoroughly analyzed. The proposed method has demonstrated its high efficiency and ability to find solutions with high accuracy and high convergence speed when compared to solutions found in the literature. These results confirm the power and flexibility of the proposed mathematical tool, q-HAKPM, as well as its applicability to a wide range of complex nonlinear flow problems.
This study represents an important contribution to the development of analytical approximate solution methods and is in line with the main objective of this work, which is to provide effective techniques for solving complex systems of differential equations in the field of fluid mechanics, especially those representing initial and boundary value problems.