The College of Education for Pure Sciences, Department of Mathematics, reviewed a doctoral dissertation on the Petrov-Kalerkin Least Squares method for solving fractional diffusion models.

The dissertation, submitted by researcher Sarah Qasim Juma, presented a new and efficient numerical framework for solving fractional diffusion problems formulated according to the Caputo-Fabrizo concept. This framework addresses the limitations of traditional numerical methods, particularly their low accuracy and high computational cost.

The proposed method combines the least squares method with the Petrov-Kalerkin method, utilizing Laqueur polynomials and Shpishev functions as test and approximation functions, respectively. This combination enabled the construction of an accurate numerical approximation for fractional diffusion problems in one and two dimensions, while maintaining adequate computational efficiency. The application of the fractional derivative in the Caputo-Fabrizo sense to time, space, and space-time formulas enabled a comprehensive theoretical study of the proposed method's properties.

A thorough theoretical analysis of the convergence and error estimation properties provided a solid mathematical foundation for the method's reliability. Numerical experiments performed on several test problems demonstrated a significant decrease in absolute error values ​​and a marked improvement in performance efficiency compared to methods published in the relevant literature. These results confirm that the proposed method is an accurate and efficient numerical tool for addressing fractional diffusion problems.

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