Approximation by Modified Bivariate Bernstein-Durrmeyer and GBS Bivariate Bernstein-Durrmeyer Operators on a Triangular Region

Year 2022, , 135 - 144, 01.06.2022

Harun Çiçek , Aydın İzgi

https://doi.org/10.33401/fujma.1009058

Cited By: 4

Abstract

In this paper, the approximation properties and the rate of convergence of modified bivariate Bernstein-Durrmeyer Operators on a triangular region are examined. Furthermore, definitions and some properties of modulus of continuity for functions of two variables are given. Voronovskaya and Gr\"{u}ss Voronovskaja type theorems are used to determine the order of approximation. The GBS (Generalized Boolean Sum) operator of Bivariate Bernstein-Durrmeyer type on a triangular region is studied. Lastly, some numerical examples are given and related graphs are plotted for comparison.

Keywords

Approximation properties, GBS operators, Modulus of continuity, Voronovskaja and Gr\"{u}ss Voronovskaja theorem

References

• [1] S. T. Yau, From Approximation Theory to Real World, Applications Workshop(TSIMF-PR China), 2017.
• [2] E. H. Kingsley, Bernstein polynomials for functions of two variables of class C(k), Proceedings of the AMS, 2(1) (1951), 64-71.
• [3] O. T. Pop, M. D. Farcas, About the bivariate operators of Kantorovich type, Acta Math. Univ. Comenianae, 1(78) (2009), 43-52.
• [4] D. D. Stancu, A method for obtaining polynomials of Bernstein type of two variables, The Amer. Math. Monthly, 3(70) (1963), 260-264.
• [5] O. T. Pop, The Generalızatıon of Voronovskaja’s Theorem for a Class of Bivariate Operators, Stud. Univ. Babe¸s-Bolyai Math., 2(53) (2008), 85-107.
• [6] T. Acar, A. Aral, Approximation properties of two dimensional Bernstein-Stancu-Chlodowsky operators, Le Matematiche, 13(68) (2013), 15-31.
• [7] S. P. Zhou, On comonotone approximation by polynomials in Lp space, Analysis, 4(13) (1993), 363-376.
• [8] M. Goyal, A. Kajla, P. N. Agrawal, S. Araci, Approximation by bivariate Bernstein-Durrmeyer operators on a triangle, Appl. Math. Inf. Sci., 3(13) (2017), 693-702.
• [9] R. Ruchi, B. Baxhaku, P. N. Agrawal, GBS operators of bivariate Bernstein-Durrmeyer-type on a triangle, Math. Meth. Appl. Sci., 7(41) (2018), 2673-2683.
• [10] S. Deshwal, N. Ispir, P. N. Agrawal, Blending type approximation by bivariate Bernstein-Kantorovich operators, Appl. Math., 2(11) (2017), 423-432.
• [11] A. Kajla, Generalized Bernstein-Kantorovich-type operators on a triangle, Math. Meth. App. Sci., 12(42) (2019), 4365-4377.
• [12] L. Aharouch, K. J. Ansari, M. Mursaleen, Approximation by B´ezier Variant of Baskakov-Durrmeyer-Type Hybrid Operators, J. Func. S., (2021), Article ID 6673663, 9 pages.
• [13] M. Mursaleen, M. Ahasan, K. J. Ansari, Bivariate Bernstein–Schurer-Stancu type GBS operators in (p, q) (p;q)-analogue, Adv. Diff. Eq., 1 (2020), 1-17.
• [14] M. Mursaleen, S. Rahman, K. J. Ansari, Approximation by Jakimovski-Leviatan-Stancu-Durrmeyer type operators, Filomat, 6(33) (2019), 1517-1530.
• [15] M. Mursaleen, A. Al-Abied, K. J. Ansari, Rate of convergence of Chlodowsky type Durrmeyer Jakimovski-Leviatan operators, Tbilisi Math. J., 2(10) (2017), 173-184.
• [16] M. Mursaleen, K. J. Ansari, On the stability of some positive linear operators from approximation theory, Bull. Math. Sci., 2(5) (2015), 147-157.
• [17] F. Usta, Approximation of functions by a new construction of Bernstein-Chlodowsky operators: Theory and applications, Num. Meth. Partial Diff. Eq., 37 (2021), 782-795.
• [18] F. Usta, Bernstein approximation technique for numerical solution of Volterra integral equations of the third kind, Comput. App. Math.,5(40) (2021), 1-11.
• [19] K. Bogel, Uber mehrdimensionale differentiation, integration und beschrankte variation.s, J. f¨ur die reine und angewandte Math., 1(173) (1935), 5-30.
• [20] K. Bogel, Mehrdimensionale differentiation von funktionen mehrerer reeller Ver¨anderlichen., J. f¨ur die reine und angewandte Math., 2(170) (1934), 197-217.
• [21] C. Badea, C. Cottin, Korovkin-type theorems for generalized boolean sum operators, C. Math. Soc. Janos Bolyai, 2(58) (1990), 51-67.
• [22] C. Badea,I. Badea, H. H. Gonska, A test function theorem and apporoximation by pseudopolynomials, C. Math. Soc. Janos Bolyai, 1(34) (1986), 53-64.
• [23] E. Dobrescu,I. Matei, The approximation by Bernstein type polynomials of bidimensionally continuous functions, Univ. Timisoara Ser. Sti. Mat.-Fiz., 1(4) (1961), 85-90.
• [24] P. N. Agrawal, N. Ispir, A. Kajla, GBS operators of Lupas¸-Durrmeyer type based on P´olya distribution, Results in Math., 3(69) (2016), 397-418.
• [25] P. N. Agrawal, D. Kumar, S. Araci, Linking of Bernstein-Chlodowsky and Sz´asz-Appell-Kantorovich type operators, J. Nonlinear Sci. Appl., 10 (2017), 3288-3302.
• [26] R. Ruchi, B.Baxhaku, P. N. Agrawal, GBS operators of bivariate Bernstein-Durrmeyer–type on a triangle, Math. Meth. App. Sci., 4(41) (2018), 2673-2683.
• [27] D. Barbosu, A. M. Acu, C. V. Muraru, On certain GBS-Durrmeyer operators based on q-integers, Turk. J. Math., 2(41) (2017), 368-380.