# Solving two-dimensional Volterra-Fredholm integral equations of the second kind by using Bernstein polynomials

• M. Sh. Dahaghin
• Sh. Eskandari
Article

## Abstract

In this paper, we present a numerical method for solving two-dimensional Volterra-Fredholm integral equations of the second kind (2DV-FK2). Our method is based on approximating unknown function with Bernstein polynomials. We obtain an error bound for this method and employ the method on some numerical tests to show the efficiency of the method.

## Keywords

Bernstein polynomial two-dimensional Volterra-Fredholm integral equations error estimation

## Notes

### Acknowledgments

We would like to thank the anonymous reviewers for their valuable suggestions and all those who have been associated with this article.

## References

1. [1]
M A Abdou, A A Badr, M B Soliman. On a method for solving a two-dimensional nonlinear integral equation of the second kind, J Comput Appl Math, 2011, 235: 3589–3598.
2. [2]
H Almasieh, J Nazari Meleh. Numerical solution of a class of mixed two-dimensional nonlinear Volterra-Fredholm integral equations using multiquadric radial basis functions, J Comput Appl Math, 2014, 260: 173–179.
3. [3]
K E Atkinson. The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.
4. [4]
Z Avazzadeh, M Heydari. Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind, Comput Appl Math, 2012, 31: 127–142.
5. [5]
E Babolian, K Maleknejad, M Mordad, B Rahimi. A numerical method for solving Fredholm-Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix, J Comput Appl Math, 2011, 235: 3965–3971.
6. [6]
M Idrees Bhatti, P Bracken. Solutions of differential equations in a Bernstein polynomial basis, J Comput Appl Math, 2007, 205: 272–280.
7. [7]
H Brunner. Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, 2004.
8. [8]
A Cardone, E Messina, E Russo. A fast iterative method for discretized Volterra-Fredholm integral equations, J Comput Appl Math, 2006, 189: 568–579.
9. [9]
A Cardone, E Messina, A Vecchio. An adaptive method for Volterra-Fredholm integral equations on the half line, J Comput Appl Math, 2009, 228: 538–547.
10. [10]
M V K Chari, S J Salon. Numerical Methods in Electromagnetism, Academic Press, 2000.Google Scholar
11. [11]
Z Cheng. Quantum effects of thermal radiation in a Kerr nonlinear blackbody, J Opt Soc Amer B, 2002, 19: 1692–1705.
12. [12]
W C Chew, M S Tong, B Hu. Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan and Claypool, 2009.Google Scholar
13. [13]
G Q Han, J Wang. Extrapolation of Nystrom solution for two dimensional nonlinear Fredholm integral equations, J Comput Appl Math, 2001, 134: 259–268.
14. [14]
A J Jerri. Introduction to Integral Equations with Applications, John Wiley and Sons, 1999.
15. [15]
Y Liu, T Ichiye. Integral equation theories for predicting water structure around molecules, Biophys Chem, 1999, 78: 97–111.
16. [16]
K Maleknejad, N Aghazadeh. Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor series expansion method, Appl Math Comput, 2005, 161: 915–922.
17. [17]
K Maleknejad, E Hashemizadeh, R Ezzati. A new approach to the numerical solution of Volterra integral equations by using Bernstein approximation, Commun Nonlinear Sci Numer Simul, 2011, 16: 647–655.
18. [18]
K Maleknejad, Z JafariBehbahani. Applications of two-dimensional triangular functions for solving nonlinear class of mixed Volterra-Fredholm integral equations, Math Comput Modelling, 2012, 55: 1833–1844.
19. [19]
K Maleknejad, M Mahdiani. Solving nonlinear mixed Volterra-Fredholm integral equations with two dimensional block-pulse functions using direct method, Commun Nonlinear Sci Numer Simul, 2011, 19: 3512–3519.
20. [20]
K Maleknejad, M Tavassoli Kajani, Y Mahmoudi. Numerical solution of linear Fredholm and Volterra integral equations of the second kind by using Legendre wavelets, Kybernetes, 2003, 32: 1530–1539.
21. [21]
F Mirzaee, E Hadadiyan. Approximate solutions for mixed nonlinear Volterra-Fredholm type integral equations via modified block-pulse functions, J Assoc Arab Univ Bas Appl Sci, 2012, 12: 65–73.
22. [22]
M Rabbani, R Jamali. Solving nonlinear system of mixed Volterra-Fredholm integral equations by using variational iteration method, J Math Comput Sci, 2012, 4: 280–287.Google Scholar
23. [23]
A Shirin, M S Islam. Numerical solutions of Fredholm integral equations using Bernstein polynomials, J Sci Res, 2010, 2: 264–272.
24. [24]
Q Tang, D Waxman. An integral equation describing an asexual population in a changing environment, Nonlinear Anal, 2003, 53: 683–699.
25. [25]
A Tari, M Y Rahimi, S Shahmorad, F Talati. Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method, J Comput Appl Math, 2009, 228: 70–76.
26. [26]
K F Warnick. Numerical Analysis for Electromagnetic Integral Equations, Artech House, 2008.
27. [27]
W J Xie, F R Lin. A fast numerical solution method for two dimensional Fredholm integral equations of the second kind, Appl Math Comput, 2009, 59: 1709–1719.
28. [28]
S A Yousefi, A Lotfi, M Dehghan. He’s variational iteration method for solving nonlinear mixed Volterra-Fredholm integral equations, Comput Math Appl, 2009, 58: 2172–2176.