Mousavi, B., Askari Hemmat, A., Heydari, M. (2017). Wilson wavelets for solving nonlinear stochastic integral equations. Wavelet and Linear Algebra, 4(2), 33-48. doi: 10.22072/wala.2017.59458.1106

Bibi Khadijeh Mousavi; Ataollah Askari Hemmat; Mohammad Hossien Heydari. "Wilson wavelets for solving nonlinear stochastic integral equations". Wavelet and Linear Algebra, 4, 2, 2017, 33-48. doi: 10.22072/wala.2017.59458.1106

Mousavi, B., Askari Hemmat, A., Heydari, M. (2017). 'Wilson wavelets for solving nonlinear stochastic integral equations', Wavelet and Linear Algebra, 4(2), pp. 33-48. doi: 10.22072/wala.2017.59458.1106

Mousavi, B., Askari Hemmat, A., Heydari, M. Wilson wavelets for solving nonlinear stochastic integral equations. Wavelet and Linear Algebra, 2017; 4(2): 33-48. doi: 10.22072/wala.2017.59458.1106

Wilson wavelets for solving nonlinear stochastic integral equations

^{1}Department of Pure Mathematica, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman

^{2}Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman

^{3}Shiraz University of Technology, Shiraz,

Abstract

A new computational method based on Wilson wavelets is proposed for solving a class of nonlinear stochastic It\^{o}-Volterra integral equations. To do this a new stochastic operational matrix of It\^{o} integration for Wilson wavelets is obtained. Block pulse functions (BPFs) and collocation method are used to generate a process to forming this matrix. Using these basis functions and their operational matrices of integration and stochastic integration, the problem under study is transformed to a system of nonlinear algebraic equations which can be simply solved to obtain an approximate solution for the main problem. Moreover, a new technique for computing nonlinear terms in such problems is presented. Furthermore, convergence of Wilson wavelets expansion is investigated. Several examples are presented to show the efficiency and accuracy of the proposed method.

[1] A. Abdulle and A. Blumenthal, Stabilized multilevel Monte Carlo method for stiff stochastic differential equations, J. Comput. Phys., 251 (2013), 445-460. [2] E. Babolian and F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput., 188(1) (2007), 417-426. [3] E. Babolian and A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, J. Comput. Appl. Math., 225(1) (2009), 87-95. [4] M.A. Berger and V.J. Mizel, Volterra equations with Ito integrals I, J. Integral Equations, 2(3) (1980), 187-245. [5] K. Bittner, Wilson bases on the interval, Advances in Gabor Analysis, Birkhäuser Boston, (2003) 197-221. [6] K. Bittner, Linear approximation and reproduction of polynomials by wilson bases, J. Fourier Anal. Appl., 8(1) (2002), 85-108. [7] K. Bittner, Biorthogonal wilson Bases, Proc. SPIE Wavelet Applications in Signal and Image Processing VII, 3813 (1999), 410-421. [8] Y. Cao, D. Gillespie and L. Petzod, Adaptive explicit-implicit tau-leaping method with automatic tau selection, J. Chem. Phys., 126(22) (2007), 1-9. [9] C. Cattani and A. Kudreyko, Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Appl. Math. Comput., 215(12) (2010), 4164-4171. [10] J.C. Cortes, L. Jodar and L. Villafuerte, Numerical solution of random differential equations: a mean square approach, Math. Comput. Modelling, 45(7-8) (2007), 757-765. [11] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [12] I. Daubechies, S. Jaffard and J.L. Journe, A simple wilson orthonormal basis with exponential decay, SIAM J. Math. Anal., 22(2) (1991), 554--573. [13] H.G. Feichtinger and T. Strohmer (eds.), Advances in Gabor analysis, Springer Science and Business Media, Davis, U.S.A, 2012. [14] M.H. Heydari, M.R. Hooshandasl, F.M. Maalek Ghaini and C. Cattani, A computational method for solving stochastic It^{o} Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys., 270(1) (2014), 402-415. [15] M.H. Heydari, M.R. Hooshmandasl, A. Shakiba and C. Cattani, Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations, Nonlinear Dyn., 85(2) (2016), 1185-1202. [16] M.H. Heydari, C. Cattani, M.R. Hooshandasl, F.M. Maalek Ghaini, An efficient computational method for solving nonlinear stochastic It^{o} integral equations: Application for stochastic problems in physics, J. Comput. Phys., 283 (2015), 148-168. [17] M.H Heydari, M.R. Hooshmandasl and F. Mohammadi, Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Appl. Math. Comput., 234 (2014), 267-276. [18] M.H. Heydari, M.R. Hooshmandasl, F.M.M. Ghaini and F. Fereidouni, Two-dimensional Legendre wavelets for solving fractional poisson equation with Dirichlet boundary conditions, Eng. Anal. Bound. Elem., 37(11) (2013), 1331-1338. [19] M.H. Heydari, M.R. Hooshmandasl and F.M. Maleak Ghaini, A good approximate solution for linear equation in a large interval using block pulse functions, J. Math. Ext., 7(1) (2013), 17-32. [20] M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini and M. Li, Chebyshev wavelets method for solution of nonlinear fractional integro-differential equations in a large interval, Adv. Math. Phys., 2013 (2013), DOI. 10.1155/2013/482083. [21] H. Holden, B. Oksendal, J. Uboe and T. Zhang, Stochastic Partial Differential Equations, second ed., Springer, New york, 1998. [22] S.K. Kaushik and S. Panwar, An interplay between gabor and wilson frames, J. Funct. Spaces Appl., 2013 (2013), DOI. 10.1155/2013/610917. [23] M. Khodabin, K. Malekinejad, M. Rostami and M. Nouri, Interpolation solution in generalized stochastic exponential population growth model, Appl. Math. Modelling, 36(3) (2012), 1023-1033. [24] M. Khodabin, K. Malekinejad, M. Rostami and M. Nouri, Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput. Math. Appl., 64(6) (2012), 1903-1913. [25] M. Khodabin, K. Malekinejad, M. Rostami and M. Nouri, Numerical solution of stochastic differential equations by second order Runge- Kutta methods, Appl. Math. Modelling, 53 (2011), 1910-1920. [26] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1999. [27] J.J. Levin and J.A. Nohel, On a system of integro-differential equations occurring in reactor dynamics, J. Math. Mech., 9 (1960), 347-368. [28] K. Maleknejad, M. Khodabin and M. Rostami, Numerical solutions of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Modelling, 55(3-4) (2012), 791-800. [29] K. Maleknejad, M. Khodabin and M. Rostami, A numerical method for solving m-dimensional stochastic Ito-Volterra integral equations by stochastic operational matrix, Comput. Math. Appl., 63(1) (2012), 133-143. [30] J.J. Levin and J.A. Nohel, On a system of integro differential equations occurring in reactor dynamics, J. Math. Mech., 9(3) (1960), 347-36. [31] F. Mohammadi, A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations, Int. J. Appl. Math. Res., 4(2) (2015), 217-227. [32] F. Mohammadi, A wavelet-based computational method for solving stochastic It^{o}-Volterra integral equations, J. Comput. Phys., 298(1) (2015), 254-265. [33] B.KH. Mousavi, A. Askari hemmat and M. H. Heydari, An application of Wilson system in numerical solution of Fredholm integral equations, PJAA, 2 (2017), 61-72. [34] B. Oksendal, Stochastic Differential Equations, fifth ed. in: An introduction with Applications, Springer, New York, 1998. [35] E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, 2010. [36] S. Yousefi and A. Banifatemi, Numerical solution of Fredholm integral equations by using CAS wavelets, Appl. Math. Comput., 183(1) (2006), 458-463.