A Efficient Computational Method for Solving Stochastic Itô-Volterra Integral Equations

Year 2015, Volume: 5 Issue: 2, 286 - 297, 01.12.2015

F. Mohammadi

Abstract

In this paper, a new stochastic operational matrix for the Legendre wavelets is presented and a general procedure for forming this matrix is given. A computational method based on this stochastic operational matrix is proposed for solving stochastic It^o-Voltera integral equations. Convergence and error analysis of the Legendre wavelets basis are investigated. To reveal the accuracy and eciency of the proposed method some numerical examples are included.

Keywords

Legendre wavelets, It^o integral, Stochastic operational matrix, Stochastic It^o-Volterra integral equations.

References

• Kloeden, P. E. and Platen, E., (1992), Numerical Solution of Stochastic Differential Equations, Springer-Verlag. New York.
• Oksendal, B., (2013), Stochastic differential equations: An introduction with applications, Springer Science and Business Media.
• Maleknejad, K., Khodabin, M. and Rostami, M., (2012), Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions. Mathematical and Computer Modelling, 55(3), pp. 791-800.
• Maleknejad, K., Khodabin, M. and Rostami, M., (2012), A numerical method for solving m- dimensional stochastic Itˆo-Volterra integral equations by stochastic operational matrix. Computers and Mathematics with Applications, 63(1), pp. 133-143.
• Khodabin, M., Maleknojad, K. and Hossoini Shckarabi, F., (2013), Application of triangular func- tions to numerical solution of stochastic Volterra integral equations. IAENG International Journal of Applied Mathematics, 43(1), pp. 1-9.
• Khodabin, M., Maleknejad, K., Rostami, M. and Nouri, M., (2012), Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix. Computers and Mathematics with Applications, 64(6), pp. 1903-1913.
• Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M. and Cattani, C., (2014), A computational method for solving stochastic Itˆo-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions. Journal of Computational Physics, 270, pp. 402-415.
• Cortes, J. C., Jodar, L. and Villafuerte, L., (2007), Numerical solution of random differential equations: a mean square approach. Mathematical and Computer Modelling, 45(7), pp. 757-765.
• Cortes, J. C., Jodar, L. and Villafuerte, L., (2007), Mean square numerical solution of random differ- ential equations: Facts and possibilities. Computers and Mathematics with Applications, 53(7), pp. 1098-1106.
• Jankovic, S. and Ilic, D., (2010), One linear analytic approximation for stochastic integrodifferential equations. Acta Mathematica Scientia, 30(4), pp. 1073-1085.
• Strang, G., (1989), Wavelets and dilation equations: A brief introduction. SIAM review, 31(4), pp. 614-627.
• Boggess, A. and Narcowich, F. J., (2009), A first course in wavelets with Fourier analysis. John Wiley and Sons.
• Razzaghi, M. and Yousefi, S., (2001), The Legendre wavelets operational matrix of integration. Inter- national Journal of Systems Science, 32(4), pp. 495-502.
• Razzaghi, M. and Yousefi, S., (2000), Legendre wavelets direct method for variational problems. Mathematics and Computers in Simulation, 53(3), pp. 185-192.
• Mohammadi, F., Hosseini, M. M. and Mohyud-Din, S. T., (2011), Legendre wavelet galerkin method for solving ordinary differential equations with non-analytic solution. International Journal of Systems Science, 42(4), pp. 579-585.
• Mohammadi, F. and Hosseini, M. M., (2011), A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations. Journal of the Franklin Institute, 348(8), pp. 1787-1796.
• Jiang, Z., Schoufelberger, W., Thoma, M. and Wyner, A., (1992), Block pulse functions and their applications in control systems. Springer-Verlag New York, Inc.
• Liu, N. and Lin, E. B., (2010), Legendre wavelet method for numerical solutions of partial differential equations. Numerical Methods for Partial Differential Equations, 26(1), pp. 81-94.