Shifted Legendre Tau Method for Solving the Fractional Stochastic Integro-Differential Equations

Document Type : Research Paper

Authors

1 Associate Professor, Mathematics and Computer Science Department, Adib mazandaran institute of higher education, Sari, Iran

2 Associate Professor,Mathematica and Computer Science Department, Adib mazandaran institute of higher education, Sari, Iran

3 School of Mathematics and Computer Science, Damghan University, Damghan, Iran

Abstract

‎In this paper‎, ‎the Tau method based on shifted Legendre polynomials is proposed for solving a class of fractional stochastic integro-differential equations‎. ‎For this purpose‎, ‎shifted Legendre polynomials and their properties are introduced‎. ‎By using the operational matrices of integration and stochastic Ito-integration we transform the problem into the corresponding linear system of algebraic equations‎. ‎Finally the efficiency of the proposed method is confirmed by some examples‎. ‎The results show that this method is very accurate and efficient‎.

Keywords

References

  1. M. Asgari, Block pulse approximation of fractional stochastic integro-differential equation, Comm. Num. Anal., 1–7 (2014).
  2. E. Babolian, A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, J. Comput. Appl. Math., 225, 87–95 (2009).
  3. A. H. Bhrawy, A. S. Alofi. The operational matrix of fractional integration for shifted Chebyshev polynomials. Applied Mathematics Letters, 26, 25–31 (2013).
  4. A. Boggess, F. J. Narcowich, A frst course in wavelets with Fourier analysis, John Wiley and Sons, (2001).
  5. M. H. Heydari, M. R.Hooshmandasl, Gh. Barid Loghmani and C.Cattani, Wavelets Galerkin method for solving stochastic heat equation, J. Comput. Math., 1–18 (2015).
  6. D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43, 525–546 (2001).
  7. K. Krishnaveni, K. Kannan and S. Raja Balachandar, A New Polynomial Method for Solving Fredholm–Volterra Integral Equations, (IJET), 1747–1483 (2013).
  8. U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul., 68, 127–143 (2005).
  9. J. J. Levin and J. A. Nohel, J. Math. Mech., 9, 347–368 (1960).
  10. Y. Li, N. Sun. Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Computers and Mathematics with Applications, 62, 1046–1054 (2011).
  11. Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216, 2276–2285 (2010).
  12. K. Maleknejad, M. Khodabin and M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Model., 55, 791–800 (2012).
  13. M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided spacefractional partial differential equations, Appl. Numer. Math., 56, 80–90 (2006).
  14. R. K. Miller, J. SIAM Appl. Math., 14, 446–452 (1966).
  15. F. Mohammadi, A Chebyshev wavelet operational method for solving stochastic Volterra-Fradholm integral equations, Int. J. Appl. Math., 215–227 (2015).
  16. F. Mohammadi, Numerical solution of stochstic Ito-Volterra integral equations by Harr wavelets, Num. Math., 416–431 (2016).
  17. M. N. Oguztoreli, Time Lag Controll Systems ,Academic Press, New York, (1966).
  18. E. L. Ortiz, L. Samara, An opperational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing, 27, 15–25 (1981).
  19. I. Podlubny, Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applica- tions, Academic Press, New York, (1998).
  20. M. Rehman and R.A. Kh, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 16, 4163–4173 (2011).
  21. A. Saadatmandi , M. Dehghan. A new operational matrix for solving fractional-order differen- tial equations. Computers and mathematics with applications, 59, 1326–1336 (2010).
  22. G. Strang, Wavelets and dilation equations, SIAM, 31, 614–627 (1989).
  23. D. W. Stroock, Probability Theory An Analytic View, 2nd Edition, Cambridge University Press, (2011).
  24. M. P. Tripathi, V. K Baranwal, R. K Pandey and O. P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions. Commun. Nonlinear Sci. Numer. Simul., 18, 1327–1340 (2013).
  25. A. Yousefi, T. Mahdavi-Rad and S.G. Shafiei, A quadrature Tau method for solving fractional integro-differential Equations in the Caputo Sense, J. math. comput. Science, 15, 97–107 (2015).
Global Analysis and Discrete Mathematics
Volume 6, Issue 2
November 2021
Pages 221-241

Files

History

  • Receive Date: 13 January 2022
  • Revise Date: 14 February 2022
  • Accept Date: 23 February 2022

Share

How to cite

Statistics