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Numerical Solution of Stochastic Ito-Volterra Integral Equations using Haar Wavelets

  • Fakhrodin Mohammadi (a1)

Abstract

This paper presents a computational method for solving stochastic Ito-Volterra integral equations. First, Haar wavelets and their properties are employed to derive a general procedure for forming the stochastic operational matrix of Haar wavelets. Then, application of this stochastic operational matrix for solving stochastic Ito-Volterra integral equations is explained. The convergence and error analysis of the proposed method are investigated. Finally, the efficiency of the presented method is confirmed by some examples.

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*Corresponding author. Email address:f.mohammadi62@hotmail.com (F. Mohammadi)

References

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[1]Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, New York, 1999.
[2]Oksendal, B., Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer-Verlag, New York, 1998.
[3]Maleknejad, K., Khodabin, M., and Rostami, M., Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Model., 55 (2012), pp. 791800.
[4]Maleknejad, K., Khodabin, M., and Rostami, M., A numerical method for solving m-dimensional stochastic ItoVolterra integral equations by stochastic operational matrix, Comput. Math. Appl., 63 (2012), pp. 133143.
[5]Cortes, J. C., Jodar, L., and Villafuerte, L., Numerical solution of random differential equations: a mean square approach, Math. Comput. Model., 45 (2007), pp. 757765.
[6]Cortes, J. C., Jodar, L., and Villafuerte, L., Mean square numerical solution of random differential equations: facts and possibilities, Comput. Math. Appl., 53 (2007), pp. 10981106.
[7]Murge, M. G. and Pachpatte, B. G., Succesive approximations for solutions of second order stochastic integrodifferential equations of Ito type, Indian J. Pure Appl. Math., 21, (1990) pp. 260274.
[8]Khodabin, M., Maleknejad, K., Rostami, M., and Nouri, M., Numerical solution of stochastic differential equations by second order Runge-Kutta methods, Math. Comput. Model., 53 (2011), pp. 19101920.
[9]Khodabin, M., Maleknejad, K., Rostami, M., and Nouri, M., Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput. Math. Appl., 64 (2012), pp. 19031913.
[10]Zhang, X., Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, J. Funct. Anal., 258 (2010), pp. 13611425.
[11]Zhang, X., Euler schemes and large deviations for stochastic Volterra equations with singular kernels, J. Differ. Equ., 244 (2008), pp. 22262250.
[12]Jankovic, S. and Ilic, D., One linear analytic approximation for stochastic integro-differential equations, Acta Math. Sci., 30 (2010), pp. 10731085.
[13]Heydari, M. H., Hooshmandasl, M. R., Maalek, F. M. and Cattani, C., A computational method for solving stochastic Ito-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys., 270 (2014), pp. 402415.
[14]Strang, G., Wavelets and dilation equations: A brief introduction, SIAM, 31 (1989), pp. 614627.
[15]Mallat, S., A Wavelet Tour of Signal Processing, 2nd ed., Academic Press, 1999.
[16]Boggess, A. and Narcowich, F. J., A First Course in Wavelets with Fourier Analysis, Wiley, 2001.
[17]Mohammadi, F. and Hosseini, M. M., A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst., 348, 8 (2011), pp. 17871796.
[18]Lepik, U., Numerical solution of differential equations using Haar wavelets, Math. Comput. Simulat., 68 (2005), pp. 127143.
[19]Li, Y. and Zhao, W., Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216, no. 8 (2010), pp. 22762285.
[20]Arnold, L., Stochastic Differential Equations: Theory and Applications, Wiley, 1974.
[21]Jiang, Z. H. and Schaufelberger, W., Block Pulse Functions and Their Applications in Control Systems, Springer-Verlag, New York, 1992.
[22]Rao, G. P., Piecewise Constant Orthogonal Functions and Their Application to Systems and Control, Springer-Verlag, Heidelberg, 1983.

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Numerical Solution of Stochastic Ito-Volterra Integral Equations using Haar Wavelets

  • Fakhrodin Mohammadi (a1)

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