The College of Education for Pure Sciences in the Department of Mathematics at the University of Basrah  discussed a master's thesis on (constructing new numerical methods for solving initial value problems in ordinary differential equations based on quadratic formulas for integration using Bernstein polynomials and hybrid functions)
The thesis presented by the researcher (Zainab Jawad Kazem) included
In this study, the proposed methods are designed to provide an efficient and accurate solution to the initial value problem and are more suitable for problems with non-smooth solutions. The main idea behind the proposed methods is to combine the advantages of traditional numerical methods, such as Kutta-Runge and Taylor's series, with the strengths of modern quadratic integration formulas using Bernstein polynomials and hybrid functions. Furthermore, we have discussed Accuracy analysis, stability analysis, consistency property, and convergence of these methods The resulting methods can handle a wide range of problems, including those with singularities, discontinuities, and other non-smooth features.
Finally, to illustrate the performance of our proposed methods, we discussed two initial value problems, and compared them with other related methods. The results, which are documented in tables and figures, demonstrate the effectiveness and efficiency of our new methods in terms of accuracy, stability, consistency, and convergence. In addition, the good choice of approximate functions, represented by Bernstein polynomials of hybrid functions, plays a key role in constructing effective numerical methods for solving initial value problems in differential equations.