The College of Education for Pure Sciences at the University of Basra has discussed a master’s thesis on (The Perspective Power of Bernstein Fractional Polynomials)
The thesis presented by a master's student (Iman Aziz Abdel-Samad) included a definition of the visible power of Bernstein's fractal polynomials, which is a generalization of the definition of quadratic Bernstein fractal polynomials presented by Gavrea and Ivan in 2017. She studied the convergence theorem, the definition of moment and Vronovsky's formula for these polynomials in ordinary approximation. Some numerical examples were given by choosing the test functions and in the continuous function space. The numerical results from the application of these examples showed that Bernstein's fractional polynomials from the visible power give better results than the quadratic Bernstein fractional polynomials and from the ordinary Bernstein polynomials when the power is greater than 2. A numerical comparison was made using graphs of functions and their approximations, and the rate of absolute errors that occurred between these approximations.
Also, another series of fractal Bernstein-Kantrovich polynomials is presented from the see power which is a generalization of ordinary Bernstein-Kantrovich polynomials and studied theoretically in ordinary approximation and in the same way as previous polynomials. It turns out that it converges numerically better than the usual Bernstein-Kantrovich polynomials.
Finally, fractional Bernstein polynomials of visual power were studied in multiple (simultaneous) approximation. Prove the convergence theorem and Vronovsky's formula and apply some numerical examples to show the convergence of the derivatives of these manifolds to the derivatives of the two test functions
Objectives of the study
Generalization of the study presented by Gavrea and Ivan in 2017 with the definition of Bernstein's fractal quadratic sequence in normal approximation, as the researchers did not know the visible power sequence and were satisfied with the quadratic sequence and the study was limited to the normal approximation only. Here the definition of any visible force has been generalized and the theoretically defined sequence is studied and applied numerically in ordinary approximation. The idea of generalization was also applied to obtain generalization of Bernstein-Kantroweg fractional polynomials from the see power and studied them theoretically and numerically in ordinary approximation. Finally, the Rattan Bernstein polynomials were studied in (simultaneous) multiple approximation.
The study concluded that the sequences of Bernstein's limits seen are better than their quadratic and ordinary counterparts, and this thing was evident to us in the numerical examples given, as it showed the accuracy of the approximation of the sequence that we knew compared with other sequences. That is why we recommend using these sequences in mathematical applications
