Skip to main content
SpringerLink
Log in

A New Computational Method Based on Bernstein Operational Matrices for Solving Two-Dimensional Linear Stochastic Volterra Integral Equations

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper is concerned with obtaining the approximate numerical solution of two-dimensional linear stochastic Volterra integral equation by using two-dimensional Bernstein polynomials as basis. Properties of these polynomials and operational matrix of integration together with the product operational matrix are utilized to transform the integral equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Bernstein coefficients. Some theorems are included to show the convergence and advantage of the proposed method. The numerical example illustrates the efficiency and accuracy of the method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Fallahpour, M., Khodabin, M., Maleknejad, K.: Approximation solution of two-dimensional linear stochastic fredholm integral equation by applying the Haar wavelet. Int. J. Math. Model. Comput. 05(04), 361–372 (2015)

    Google Scholar 

  2. Fallahpour, M., Khodabin, M., Maleknejad, K.: Approximation solution of two-dimensional linear stochastic Volterra-Fredholm integral equation via two-dimensional Block-pulse functions. Int. J. Ind. Math. 8(4), IJIM-00774 (2016)

  3. Fallahpour, M., Khodabin, M., Maleknejad, K.: Theoretical error analysis and validation in numerical solution of two-dimensional linear stochastic Volterra-Fredholm integral equation by applying the block-pulse functions. Cogent Math. 4, 1296750 (2017)

    Article  MathSciNet  Google Scholar 

  4. Rivlin, T.J.: An Introduction to the Approximation of Functions. Dover Publications, New York (1969)

    MATH  Google Scholar 

  5. Maleknejad, K., Hashemizadeh, E., Ezzati, R.: A new approach to the numerical solution of Volterra integral equations by using Bernsteins approximation. Commun. Nonlinear Sci. Numer. Simul. 16, 647–655 (2011)

    Article  MathSciNet  Google Scholar 

  6. Doha, E.H., Bhrawy, A.H., Saker, M.A.: Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations. Appl. Math. Lett. 24, 559–565 (2011)

    Article  MathSciNet  Google Scholar 

  7. Hosseini Shekarabi, F., Maleknejad, K., Ezzati, R.: Application of two-dimensional Bernstein polynomials for solving mixed VolterraFredholm integral equations. Springer, Berlin (2014). https://doi.org/10.1007/s13370-014-0283-6

    Book  MATH  Google Scholar 

  8. Lorentz, G.G.: Bernstein Polynomials. Chelsea Publishing Company, New York (1986)

    MATH  Google Scholar 

  9. Asgari, M., Hashemizadeh, E., Khodabin, M., Maleknejad, K.: Numerical solution of nonlinear stochastic integral equation by stochastic operational matrix based on Bernstein polynomials. ull. Math. Soc. Sci. Math. Roumanie Tome 57(05), 3–12 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Heitzinger, C.: Simulation and inverse modeling of semiconductor manufacturing processes. In: Multivariate Bernstein Polynomials. Eingereicht An Der Technischen Universität Wien, Fakultät Für Elektrotechnik Und Informationstechnik, Von (2003)

  11. Popoviciu, T.: Sur iapproximation des fonctions convexes dordre superieur. Mutlzemntica (Cluj) 10, 49–54 (1935)

    MATH  Google Scholar 

  12. Pallini, A.: Bernstein-type approximation of smooth functions. Dipartimento di Statistica e Matematica Applicata All’Economia Università di Pisa, Anno LXV, n. 2 (2005)

  13. Klebaner, F.C.: Introduction to stochastic calculus with applications. Monash University, Imperial College Press, Australia (2005)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Mohsen Fallahpour, Morteza Khodabin & Reza Ezzati

Authors
  1. Mohsen Fallahpour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Morteza Khodabin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Reza Ezzati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Morteza Khodabin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fallahpour, M., Khodabin, M. & Ezzati, R. A New Computational Method Based on Bernstein Operational Matrices for Solving Two-Dimensional Linear Stochastic Volterra Integral Equations. Differ Equ Dyn Syst 30, 873–884 (2022). https://doi.org/10.1007/s12591-019-00474-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-019-00474-y

Keywords

Mathematics Subject Classification

Search

Navigation