Legendre cardinal functions and their applications in solving nonlinear stochastic differential equations

[1] A. Alipanah, Spectral Methods using cardinal functions, PHD Thesis in applied Mathematic, AmirKabir University of
Technology, 2006.
[2] M. Asgari, E. Hashemizadeh, M. Khodabin and K. Maleknedjad, Numerical solution of nonlinear stochastic integral equation
by stochastic operational matrix based on Bernstein polynomials, Bull. Math. Soc. Sci. Math. Rouman. Tome. 57 (2014),
3–12.
[3] W.F Blyth, R.L May and P. Widyaningsih, Volterra integral equations Solved in Fredholm form using Walsh functions,
ANZIAM. J. 45 (2004), 269–282.
[4] J.P. Boyd, Chebychev and Fourier spectral methods, Dover Publications, Inc., 2000.
[5] C. Canuto, M. Hussaini, A. Quarteroni and T. Zang, Spectral methods in fluid dynamics, Springer, Berlin, 1988.
[6] F. Mohamadi, A wavelet-based computational method for solving stochastic Itˆo-Volterra integral equations, J. Comput.
Phys. 298 (2015), 254–265.[7] M.E.A. El-Mikkawy and G.S. Cheon, Combinatorial and hypergeometric identities via the Legendre polynomials- A computational approach, App. Math. Comput. 166 (2005), 181–195.
[8] D. Funaro, Polynomial approximation of differential equations, Springer Verlag, New York, 1992.
[9] M.H. Heydari, M.R. Mahmoudi, A. Shakiba and Z. Avazzadeh, Chebyshev cardinal wavelets and their application in solving
nonlinear stochastic differential equations with fractional Brownian motion, Commun. Nonlinear Sci. Numer. Simul. 64
(2018), 98—121.
[10] M. Heydari, Z. Avazzadeh and G.B. Loghmani, Chebychev cardinal functions for solving Volterra-Fredholm integro differential equations using operational matrices, Iran. J. Sci. Technol. 36 (2012), no.1, 13–24.
[11] M. Ghasemi and C.M.T. Kajani, Application of Hes homotopy perturbation method to nonlinear integro-differential equations :Wavelet- Galerkin method and homotopy perturbation method, Appl. Math. Comput. 18 (2007), no. 1, 450–455.
[12] A. Gil, J. Segura and N.M. Temme, Numerical Methods for Special Functions, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.
[13] Gulsu, M. and Sezer, M. The approximate solution of high-order linear difference equation with variable coefficients in
terms of Taylor polynomials , App. Math. Comput. 168(2005) 76-88.
[14] M.M. Kader, N.H. Sweilam and W.Y. Kota, Cardinal functions for Legendre pseudo-spectral method for solving the integrodifferential equations, J. Egypt. Math. Soc. 22 (2014), 511–516.
[15] P.E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Springer, Berlin, 1992.
[16] K. Parand and M. Delkhosh, Operational matrix to solve nonlinear Riccati differential equations of arbitrary order, St.
Pertersburg Polytech. Univ. J. Phys. Math. 3 (2017), 242–254.
[17] M. Lakestani and M. Deghan, The use of Chebychev cardinal functions for the solution of of a partial differential equation
with an unknown time-independent coefficient subjectto an extra measurement, J. Comput. Appl. Math. 235 (2010), no. 3,
669–678.
[18] M. Lakestani and M. Deghan, Numerical solution of fourth-order integro-differential equations using Chebychev cardinal
functions, Int. J. Comput. Math. 87 (2010), no. 6, 1389–1394.
[19] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of nonlinear integral equation, App. Math. Comput. 167
(2005), no. 2, 1119–1129.
[20] K. Maleknejed, R. Mollapourasl and M. Alizadeh, Numerical solution of Volterra type integral equation of the first kind
with wavelet basis, Appl. Math. Comput. 194 (2007), no. 2, 400–405.
[21] K. Maleknejad, M. Khodabin and Rostami, Numerical method for solving m− dimensional stochastic Itˆo Volterra integral
equations by stochastic operational matrix based on block-pulse functions, Comput. Math. Appl. 63 (2012), 133–143.
[22] X. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays-part II, Numer. Funct.
Anal. Optim. 15 (1994), no. 1-2, 65–76.
[23] G.N. Milstein, Numerical integration of stochastic differential equations, Math. Appl. Kluwer, Dordrecht, 1995.
[24] F. Mirzaee and E. Hadadiyan, A collocation technique for solving nonlinear stochastic Itˆo-Volterra integral equation, Appl.
Math. Comput. 247 (2014), 1011–1020.
[25] F. Mirazee, S. Alipour and N. Samadyar, Numerical solution based on hybrid of block-pulse and parabolic function for
solving a system of nonlinear stochastic Itˆo-Volterra integral equations of fractional order, J. Comput. Appl. Math. 349
(2019), 157–171.
[26] F. Mirazee and N. Samadyar, On the numerical solution of stochastic quadratic integral equations via operational matrix
method, Math. Meth. Appl. Sci. 41 (2018), no. 12, 4465–4479.
[27] F. Mirazee and N. Samadyar, Application of hat basis functions for solving two-dimensional stochastic fractional integral
equations, Comput. Appl. Math. 37 (2018), no. 4, 4899–4916.
[28] F. Mirazee and S. Alipour, Approximation solution of nonlinear quadratic integral equations of fractional order via piecewise
linear functions, J. Comput. Appl. Math. 331 (2018), 217–227.
[29] F. Mirazee and A. Hamzeh, A computational method for solving nonlinear stochastic Volterra integral equations, J. Comput.
Appl. Math. 306 (2016), 166–178.
[30] F. Mirazee and N. Samadyar, Numerical solutions based on two- dimensional orthonormal Bernstein polynomials for solving
some classes of two-dimensional nonlinear integral equations of fractional order, Appl. Math. Comput. 344 (2019), 191–203.[31] F. Mirazee and N. Samadyar, Using radial basis functions to solve two dimansional linear stochastic integral equations on
non-rectangular domains, Engin. Anal. Bound. Elem. 92 (2018), 180–195.
[32] F. Mirazee and N. Samadyar, Numerical solution of nonlinear stochastic Itˆo-Volterra integral equations driven by fractional
Brownian motion, Math. Meth. Appl. Sci. 41 (2018), no. 4, 1410–1423.
[33] F. Mirazee, N. Samadyar and S.F. Hoseini, Euler polynomial solutions of nonlinear stochastic Itˆo-Volterra integral equations,
J. Comput. Appl. Math. 330 (2018), 574–585.
[34] M.H. Heydari, A new direct method based on the Chebyshev cardinal functions for variable-order fractional optimal control
problems, J. Franklin Inst. 355 (2018), 4970—4995.
[35] M.H. Reihani and Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J.
Comput. Appl. Math. 200 (2007), 12–20.
[36] X. Shang and D. Ha, Numerical solution of Fredholm integral equations of the first kind by using linear Legendre multiwavelets, Appl. Math. Comput. 191 (2007), no. 2, 440–444.
[37] A.M. Wazwaz, A first in integral equations, New Jersey, World Scientific, 1997.
[38] S. Yousefi and M. Razzaghi, Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simul. 70 (2005), no. 1, 1–8.
[39] S.A. Yousefi, A. Lotfi and M. Dehghan, He’s variational iteration method for solving nonlinear mixed Volterra-Fredholm
integral equations, Comput. Math. Appl. 58 (2009), no. 11-12, 2172–2176.