Application of DJ method to Ito stochastic differential equations
Subject Areas : Difference and functional equations
1 - Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
Keywords: Stochastic differential equations, iterative methods, Ito calculus,
Abstract :
This paper develops iterative method described by [V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] to solve Ito stochastic differential equations. The convergence of the method for Ito stochastic differential equations is assessed. To verify efficiency of method, some examples are expressed.
[1] M. Abundo, A stochastic model for predator-pray systems: basic properties, stability and computer simulation, J. Math. Biol. 29 (1991), 495-511.
[2] M. Asgari, E. Hasheminejad, M. Khodabin, K. Maleknejad, Numerical solution of nonlinear stochastic integral equation by stochastic operational matrix based on Bernstein polynomials, Bull. Math. Soc. Sci. Math. Roumanie. 57 (2014), 3-12.
[3] S. Bhalekar, V. Daftardar-Gejji, New iterative method: application to partial differntial equations, Appl. Math. Comput. 203 (2008), 778-783.
[4] S. Bhalekar, V. Daftardar-Gejji, Solving evolution equations using a new iterative method, Numer. Methods Partial Differ. Equat. 26 (2010), 906-916.
[5] J. C. Cortes, L. Jodar, L. Villafuerte, Mean square numerical solution of random differential equations: facts and possibilities, Comput. Math. Appl. 53 (2007), 1098-1106.
[6] J. C. Cortes, L. Jodar, L. Villafuerte, Numerical solution of random differential equations: a mean square approach, Math. Comput. Model. 45 (2007), 757-765.
[7] V. Daftardar-Gejji, S. Bhalekar, An iterative method for solving fractional differential equations, Proc. Appl. Math. Mech. 7 (2008), 2050017-2050018.
[8] V. Daftardar-Gejji, S. Bhalekar, Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method, Compt. Appl. Math. 59 (2010), 1801-1809.
[9] V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006), 753-763.
[10] M. R. Yaghouti, H. Deilami, Numerical solution of singular differential-difference equations, World Appl. Program. 3 (2013), 182-189.
[11] H. Jafari, S. J. Johnston, S. M. Sani, D. Baleanu, A decomposition method for solving q-difference equations, Appl. Math. Inf. Sci. 9 (2015), 2917-2920.
[12] H. Jafari, S. Seifi, A. Alipoor, M. Zabihi, An iterative method for solving linear and nonlinear fractional diffusion-wave equation, Int. J. Numer. Anal. Relat. Topics. 3 (2009), 20-32.
[13] M. Khodabin, K. Maleknejad, F. Hosseini Shekarabi, Application of triangular functions to numerical solution of stochastic Volterra integral equations, Int. J. Appl. Math. 43 (2011), 1-9.
[14] 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.
[15] Y. Liang, D. Greenhalgh, X. Mao, A stochastic differential equation model for the spread of HIV amongst people who inject drugs, Comput. Math. Methods. Med. (2016), 14 pages.
[16] 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. Comput. Phys. 270 (2014) 402-415.
[17] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, Siam. Rev. 43 (2001), 525-546.
[18] Y. Hu, D. Nualart, J. Song, A nonlinear stochastic heat equation: Holder continuity and smoothness of the density of the solution, Stoch. Proc. Appl. 123 (2013), 1083-1103.
[19] S. Jankovic, D. Ilic, One linear analytic approximation for stochastic integro-differential equations, Acta Math. Sci. 30 (2010), 1073-1085.
[20] 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.
[21] F. Mohammadi, A Chebyshev wavelet operational method for solving stochastic Volterra-Fredholm integral equations, Int. J. Appl. Math. Res. 4 (2015), 217-227.
[22] F. Mohammadi, A computational wavelet method for numerical solution of stochastic Volterra-Fredholm integral equations, Wavelets. Linear. Algebr. 3 (2016), 13-25.
[23] F. Mohammadi, A wavelet-based computational method for solving stochastic Ito-Volterra equations, J. Comput. Math. 298 (2015), 254-265.
[24] F. Mohammadi, Numerical solution of stochastic Volterra-Fredholm integral equations using Haar wavelets, U.P.B. Sci. Bull. (Series A). 78 (2016), 111-126.
[25] F. Mohammadi, Second kind Chebyshev wavelet Galerkin method for stochastic Ito-Volterra integral equations, Mediterr. J. Math. 13 (2016), 2613-2631.
[26] M. G. Murge, B. G. Pachpatte, Successive approximations for solutions of second order stochastic integro-differential equations of Ito type, Indian J. Pure Appl. Math. 21 (1990), 260-274.
[27] K. Nouri, Study on efficiency of Adomian decomposition method for stochastic differential equations, Int. J. Nonlinear Anal. Appl. 8 (2017), 61-68.
[28] B. Oksendal, Stochastic differential equations: An introduction with applications, Springer, 1998.
[29] K. P. Sharp, Stochastic differential equations in finance, Appl. Math. Comput. 39 (1990), 207-224.
[30] M. Yaseen, M. Samraiz, A modified new iterative method for solving linear and nonlinear Klein-Gordon Equations, Appl. Math. Sci. 6 (2012), 2979-2987.
[31] M. Yaseen, M. Samraiz, S. Naheed, Exact solutions of Laplace equation by DJ method, Results Phys. 3 (2013), 38-40.