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**

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Volterra integral equations driven by fractional Brownian motion using hat functions. Turkish Journal

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[10] Maleknejad, K., J. Rashidinia, and T. Eftekhari, A new and efficient numerical method based on

shifted fractionalorder Jacobi operational matrices for solving some classes of twodimensional

nonlinear fractional integral equations. Numerical Methods for Partial Differential Equations, 2021.

37(3): p. 2687-2713.

[11] 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.

of third-kind Volterra integral equations. Journal of Integral Equations and Applications, 2015. 27(3):

p. 325-342.

[2] 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.

[3] 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).

[4] 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.

[5] Chen, X., Y. Qi, and C. Yang, New existence theorems about the solutions of some stochastic integral

equations. arXiv preprint arXiv:1211.1249, 2012.

[6] 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.

[7] Hu, Y. and B. Øksendal, Linear Volterra backward stochastic integral equations. Stochastic Processes

and their Applications, 2019. 129(2): p. 626-633.

[8] Klebaner, F.C., Introduction to stochastic calculus with applications. 2012: World Scientific Publishing

Company.

[9] 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.

[10] Maleknejad, K., J. Rashidinia, and T. Eftekhari, A new and efficient numerical method based on

shifted fractionalorder Jacobi operational matrices for solving some classes of twodimensional

nonlinear fractional integral equations. Numerical Methods for Partial Differential Equations, 2021.

37(3): p. 2687-2713.

[11] 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.

[12] 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.

[13] 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.

[14] 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.

[15] 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.

[16] Oksendal, B., Stochastic differential equations: an introduction with applications. 2013: Springer

Science & Business Media.

[17] 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.

[18] 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.

[19] Powell, M.J.D., Approximation theory and methods. 1981: Cambridge university press.

[20] Saito, Y. and T. Mitsui, Simulation of stochastic differential equations. Annals of the Institute of

Statistical Mathematics, 1993. 45(3): p. 419-432.

[21] 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.

[22] 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.

[23] SUSAN MILTON, J. and C.P. TSOKOS, A stochastic system for communicable diseases. International

Journal of Systems Science, 1974. 5(6): p. 503-509.

[24] 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.

[25] 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.

Fredholm integral equations. Journal of Applied Mathematics and Computing, 2016. 52(1): p. 489-

513.

[13] 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.

[14] 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.

[15] 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.

[16] Oksendal, B., Stochastic differential equations: an introduction with applications. 2013: Springer

Science & Business Media.

[17] 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.

[18] 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.

[19] Powell, M.J.D., Approximation theory and methods. 1981: Cambridge university press.

[20] Saito, Y. and T. Mitsui, Simulation of stochastic differential equations. Annals of the Institute of

Statistical Mathematics, 1993. 45(3): p. 419-432.

[21] 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.

[22] 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.

[23] SUSAN MILTON, J. and C.P. TSOKOS, A stochastic system for communicable diseases. International

Journal of Systems Science, 1974. 5(6): p. 503-509.

[24] 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.

[25] 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.

December 2021

Pages 739-749

**Receive Date:**15 April 1400**Revise Date:**20 September 1400**Accept Date:**13 October 1400