# Numerical Solution of Nonlinear Stochastic Integral Equation of the Third Kind by Stochastic Operational Matrix Based on Bernstein Polynomials

Document Type : Original Paper

Authors

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Abstract

In the present research, we were numerically solved nonlinear stochastic integral equation of the third kind by stochastic operational matrix based on Bernstein polynomials. For this aim, we were obtaining the Bernstein polynomials operation matrix and the stochastic operation matrix. Also we approximated all the functions in the Volterra integral equation of the third kind using the Bernstein polynomials series and then use the Bernstein polynomials operation matrix. By doing this, solving the third kind of stochastic Volterra integral equation turns into solving a system of algebraic equations, which could be a more suitable solution. Then we were analysed the convergence of the proposed method and provide several numerical examples to evaluate the accuracy and efficiency of this method. The current results were obtained by running a program written in Mathematica software.

Keywords

#### References

 Allaei, S.S., Z.-w. Yang, and H. Brunner, Existence, uniqueness and regularity of solutions to a class
of third-kind Volterra integral equations. Journal of Integral Equations and Applications, 2015. 27(3):
p. 325-342.
 Allaei, S.S., Z.-W. Yang, and H. Brunner, Collocation methods for third-kind VIEs. IMA Journal of
Numerical Analysis, 2017. 37(3): p. 1104-1124.
 Asanov, A., K. Matanova, and R. Asanov, A class of linear and nonlinear Fredholm integral equations
of the third kind. Kuwait Journal of Science, 2017. 44(1).
 Cao, Y. and Y. Xu, Singularity preserving Galerkin methods for weakly singular Fredholm integral
equations. The Journal of Integral Equations and Applications, 1994: p. 303-334.
 Chen, X., Y. Qi, and C. Yang, New existence theorems about the solutions of some stochastic integral
equations. arXiv preprint arXiv:1211.1249, 2012.
 Hashemi, B.H., M. Khodabin, and K. Maleknejad, Numerical method for solving linear stochastic Itô-
Volterra integral equations driven by fractional Brownian motion using hat functions. Turkish Journal
of Mathematics, 2017. 41(3): p. 611-624.
 Hu, Y. and B. Øksendal, Linear Volterra backward stochastic integral equations. Stochastic Processes
and their Applications, 2019. 129(2): p. 626-633.
 Klebaner, F.C., Introduction to stochastic calculus with applications. 2012: World Scientific Publishing
Company.
 Kloeden, P.E. and E. Platen, Time Discrete Approximation of Deterministic Differential Equations, in
Numerical Solution of Stochastic Differential Equations. 1992, Springer. p. 277-303.
 Maleknejad, K., J. Rashidinia, and T. Eftekhari, A new and efficient numerical method based on
shifted fractional￿order Jacobi operational matrices for solving some classes of two￿dimensional
nonlinear fractional integral equations. Numerical Methods for Partial Differential Equations, 2021.
37(3): p. 2687-2713.
 Mandal, B.N. and S. Bhattacharya, Numerical solution of some classes of integral equations using
Bernstein polynomials. Applied Mathematics and computation, 2007. 190(2): p. 1707-1716.
 Mirzaee, F. and E. Hadadiyan, A new numerical method for solving two-dimensional Volterra–
Fredholm integral equations. Journal of Applied Mathematics and Computing, 2016. 52(1): p. 489-
513.
 Mirzaee, F. and S.F. Hoseini, Numerical approach for solving nonlinear stochastic Itô-Volterra integral
equations using Fibonacci operational matrices. Scientia Iranica, 2015. 22(6): p. 2472-2481.
 Nemati, S. and P.M. Lima. Numerical solution of a third-kind Volterra integral equation using an
operational matrix technique. in 2018 European Control Conference (ECC). 2018. IEEE.
 Okayama, T., T. Matsuo, and M. Sugihara, Sinc-collocation methods for weakly singular Fredholm
integral equations of the second kind. Journal of Computational and Applied Mathematics, 2010.
234(4): p. 1211-1227.
 Oksendal, B., Stochastic differential equations: an introduction with applications. 2013: Springer
 Pandey, R.K. and B. Mandal, Numerical solution of a system of generalized Abel integral equations
using Bernstein polynomials. J. Adv. Res. Sci. Comput, 2010. 2(2): p. 44-53.
 Pedas, A. and G. Vainikko, Smoothing transformation and piecewise polynomial projection methods
for weakly singular Fredholm integral equations. Communications on Pure & Applied Analysis, 2006.
5(2): p. 395.
 Powell, M.J.D., Approximation theory and methods. 1981: Cambridge university press.
 Saito, Y. and T. Mitsui, Simulation of stochastic differential equations. Annals of the Institute of
Statistical Mathematics, 1993. 45(3): p. 419-432.
 Singh, S. and S. Saha Ray, Stochastic operational matrix of Chebyshev wavelets for solving multidimensional
stochastic Itô–Volterra integral equations. International Journal of Wavelets, Multiresolution
and Information Processing, 2019. 17 (03): p. 1950007.
 Song, H., Z. Yang, and H. Brunner, Analysis of collocation methods for nonlinear Volterra integral
equations of the third kind. Calcolo, 2019. 56(1): p. 7.
 SUSAN MILTON, J. and C.P. TSOKOS, A stochastic system for communicable diseases. International
Journal of Systems Science, 1974. 5(6): p. 503-509.
 Tripathi, M.P., et al., A new numerical algorithm to solve fractional differential equations based on
operational matrix of generalized hat functions. Communications in Nonlinear Science and Numerical
Simulation, 2013. 18(6): p. 1327-1340.
 Yaghoobnia, A., M. Khodabin, and R. Ezzati, Numerical solution of stochastic Itˆo-Volterra integral
equations based on Bernstein multi-scaling polynomials. Applied Mathematics-A Journal of Chinese
Universities, 2021. 36(3): p. 317-329.

### History

• Receive Date: 15 April 1400
• Revise Date: 20 September 1400
• Accept Date: 13 October 1400
• First Publish Date: 13 October 1400