A Bernstein polynomial approach for solution of nonlinear integral equations

Volume 10, Issue 9, pp 4638--4647 http://dx.doi.org/10.22436/jnsa.010.09.07

Publication Date: September 08, 2017 Submission Date: June 23, 2017

Download PDF

Download XML

3358 Downloads
5998 Views

Authors

Nese Isler Acar - Department of Mathematics, Faculty of Arts and Sciences, Mehmet Akif Ersoy University, Burdur, Turkey. Aysegul Dascioglu - Department of Mathematics, Faculty of Arts and Sciences, Pamukkale University, Denizli, Turkey.

Abstract

In this study, a collocation method based on the generalized Bernstein polynomials is derivated for solving nonlinear Fredholm-Volterra integral equations (FVIEs) in the most general form via the quasilinearization technique. Moreover, quadratic convergence and error estimate of the proposed method is analyzed. Some examples are also presented to show the accuracy and applicability of the method. keywords

Share and Cite

ISRP Style

Nese Isler Acar, Aysegul Dascioglu, A Bernstein polynomial approach for solution of nonlinear integral equations, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4638--4647

AMA Style

Acar Nese Isler, Dascioglu Aysegul, A Bernstein polynomial approach for solution of nonlinear integral equations. J. Nonlinear Sci. Appl. (2017); 10(9):4638--4647

Chicago/Turabian Style

Acar, Nese Isler, Dascioglu, Aysegul. "A Bernstein polynomial approach for solution of nonlinear integral equations." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4638--4647

Keywords

Bernstein polynomial approach
nonlinear integral equations
quasilinearization technique
collocation method.

MSC

45A05
45G10
65L60

References

[1] R. P. Agarwal, Y. M. Chow, Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math., 10 (1984), 203–217.
- View Article
- Google Scholar

[2] B. Ahmad, R. Ali Khan, S. Sivasundaram , Generalized quasilinearization method for nonlinear functional differential equations, J. Appl. Math. Stochastic Anal., 16 (2003), 33–43.
- View Article
- MathSciNet

[3] A. Akyuz Dascioglu, N. Isler, Bernstein collocation method for solving nonlinear differential equations , Math. Comput. Appl., 18 (2013), 293–300.

[4] A. Akyüz-Daşcıoğlu, N. Işler Acar, C. Güler, Bernstein collocation method for solving nonlinear Fredholm-Volterra integrodifferential equations in the most general form, J. Appl. Math., 2014 (2014), 8 pages.

[5] A. C. Baird, Jr., Modified quasilinearization technique for the solution of boundary-value problems for ordinary differential equations, J. Optimization Theory Appl., 3 (1969), 227–242.

[6] R. E. Bellman, R. E. Kalaba, Quasilinearization and nonlinear boundary-value problems, Modern Analytic and Computional Methods in Science and Mathematics, American Elsevier Publishing Co., Inc., New York (1965)
- Google Scholar

[7] Z. Drici, F. A. McRae, J. V. Devi , Quasilinearization for functional differential equations with retardation and anticipation, Nonlinear Anal., 70 (2009), 1763–1775.

[8] R. T. Farouki, V. T. Rajan , Algorithms for polynomials in Bernstein form , Comput. Aided Geom. Design, 5 (1988), 1–26.
- View Article
- Google Scholar

[9] N. İşler Acar, A. Akyüz-Daşcıoğlu, A collocation method for Lane-Emden Type equations in terms of generalized Bernstein polynomials, Pioneer J. Math. Math. Sci., 12 (2014), 81–97.
- Google Scholar

[10] K. I. Joy, Bernstein polynomials, On-Line Geometric Modeling Notes , Visualization and Graphics Research Group, Department of Computer Science, University of California, Davis, (2000), 1–13.
- Google Scholar

[11] E. S. Lee, Quasilinearization and invariant imbedding, First edition, Academic Press, New York (1968)
- Google Scholar

[12] G. G. Lorentz , Bernstein polynomials, Second edition, Chelsea Publishing Co., New York (1986)
- Google Scholar

[13] K. Maleknejad, E. Najafi, Numerical solution of nonlinear Volterra integral equations using the idea of quasilinearization, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 93–100.

[14] V. B. Mandelzweig, F. Tabakin, Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Comput. Phys. Comm., 141 (2001), 268–281.

[15] S. G. Pandit , Quadratically converging iterative schemes for nonlinear Volterra integral equations and an application, J. Appl. Math. Stochastic Anal., 10 (1997), 169–178.

[16] M. G. Phillips, Interpolation and approximation by polynomials, CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, Springer-Verlag, New York (2003)
- Google Scholar

[17] J. I. Ramos , Piecewise quasilinearization techniques for singular boundary-value problems , Comput. Phys. Comm., 158 (2004), 12–25.

[18] P.-G. Wang, Y.-H. Wu, B. Wiwatanapaphee, An extension of method of quasilinearization for integro-differential equations, Int. J. Pure Appl. Math., 54 (2009), 27–37.