A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations
- Abstract
- Keywords
- References
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Abstract
In this paper, the stochastic operational matrix of Ito -integration for the Chebyshev wavelets is applied for solving stochastic Volterra-Fredholm integral equations. The main characteristic of the presented method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Chebyshev wavelets basis is considered. The efficiency and accuracy of the proposed method was demonstrated by some non-trivial examples and comparison with the other existing methods.
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Keywords
Chebyshev wavelets; Ito integral; Brownian motion process; Stochastic Volterra-Fredholm integral equations; Stochastic operational matrix.
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References
[1] P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, Springer, 1992.
[2] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, fifth ed., Springer-Verlag, New York, 1998.
[3] Higham D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM review. 43 (3) (2001) 525-546.
[4] A. Abdulle, G.A. Pavliotis, Numerical methods for stochastic partial differential equations with multiple scales, J. Comp. Phys. 231 (2012) 2482-2497.
[5] R. Mazo, Brownian Motion, International Series of Monographs on Physics, vol. 112, Oxford University Press , New York, 2002.
[6] E. Weinan, D. Liu and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math. 58 (11) (2005) 1544-1585.
[7] K. Maleknejad, M. Khodabin, M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Model. 55 (2012) 791-800.
[8] M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri, Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput. Math. Appl. 64 (2012) 1903-1913.
[9] M. Khodabin, K. Maleknejad, and F. Hosseini Shekarabi, Application of Triangular Functions to Numerical Solution of Stochastic Volterra Integral Equations, IAENG Int. J. Appl. Math. 43 (1) (2011) 1-9.
[10] M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek, C. Cattani, A computational method for solving stochastic Ito-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comp. Phys. 270 (2014) 402-415.
[11] J .C. Cortes, L. Jodar, L. Villafuerte, Numerical solution of random differential equations: a mean square approach, Math. Comput. Model. 45 (2007) 757-765.
[12] M. G. Murge, B. G. Pachpatte, Succesive approximations for solutions of second order stochastic integro-differential equations of Ito type, Indian. J. Pure. Ap. Mat. 21 (3) (1990) 260-274.
[13] S. Jankovic, D. Ilic, One linear analytic approximation for stochastic integro-differential equations, Acta Math. Sci. 30 (2010) 1073-1085.
[14] G. Strang, Wavelets and dilation equations: A brief introduction, SIAM review. 31 (4) (1989) 614-627.
[15] A. Boggess, F. J. Narcowich, A first course in wavelets with Fourier analysis, John Wiley and Sons , 2001.
[16] C. Cattani, Harmonic wavelet approximation of random, fractal and high frequency signals, Telecommun. Syst. 43 (2010) 207-217.
[17] F. Mohammadi, M. M. Hosseini, and Syed Tauseef Mohyud-Din, Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution, Int. J. Syst. Sci. 42 (2) (2011) 579-585.
[18] F. Mohammadi , M. M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst. 348 (8) (2011) 1787-1796.
[19] E. Babolian, F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, it Appl. Math. Comput. 188 (1) (2007) 417-426.
[20] Y. Li, Solving a nonliear fractional differential equation using chebyshev wavelets, Commun Nonlinear. Sci. Numer. Simul. , 15 (9) (2010) 2284-2292.
[21] M. H. Heydari, M. R. Hooshmandasl, F. Mohammadi, C. Cattani, Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations, Commun. Nonlinear. Sci. Numer. Simul., 19 (1) (2014) 37-48.
[22] U. Lepik, Numerical solution of differential equations using Haar wavelets. Math. Comput. Simulat., 68 (2005) 127-143.
[23] H. Adibi, P. Assari, Chebyshev wavelet method for numerical solution of Fredholm integral equations of the first kind, Math. Probl. Eng. 2010 (2010).
[24] Z. H. Jiang, W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, Springer-Verlag , 1992.
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