Operational matrix and their applications for solving time-varying delay systems

[1] N. Aghazadeh and A.A. Khajehnasiri, Solving nonlinear two-dimensional Volterra integro-differential equations by block-pulse functions, Math. Sci. 7 (2013), 1–6.

[2] Q.H. Ansari, I.V. Konnov and J.C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl. 110 (2001), 481–492.

[3] Q.H. Ansari, S. Schaible and J.C. Yao System of vector equilibrium problems and its applications, J. Optim. Theory Appl. 107 (2000), no. 3, 547–557

[4] A.H. Bhrawy and S.S. Ezz-Eldien, A new Legendre operational technique for delay fractional optimal control problems, Calcolo 53 (2016), 521–543.

[5] K.B. Datta and B. M. Mohan, Orthogonal Functions in system and control, World Scientific Publishing Co. Ltd, 1995.

[6] A. Ebadian and A.A. Khajehnasiri. Block-pulse functions and their applications to solving systems of higher-order nonlinear Volterra integro-differential equations, Electron. J. Diff. Equ. 54 (2014), 1–9.

[7] A. Ebadian, H. Rahmani Fazli and A.A. Khajehnasiri, Solution of nonlinear fractional diffusion-wave equation by triangular functions, SeMA J. 72 (2015), 37–46.

[8] W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995), 46–53.

[9] M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini and C. Cattani, A computational method for solving stochastic It Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys. 270 (2014), 402–415.

[10] M.H. Heydari, M.R. Hooshmandasl, F.M. Maalek Ghaini and C. Cattani, An efficient computational method for solving nonlinear stochastic Ito integral equations: Application for stochastic problems in physics, J. Comput. Phys. 283 (2015), 148–168.

[11] S.M. Hoseini and H.R. Marzban, Analysis of time-varying delay systems via triangular functions, Appl. Math. Comput. 217 (2011), 7432–7441.

[12] C. Hwang and M.Y. Chen, Analysis of time-delay systems using the Galerkin method, Int. J. Control 44 (1986), 847–866.

[13] A.A Khajehnasiri and R. Ezzati, Boubaker polynomials and their applications for solving fractional two-dimensional nonlinear partial integro-differential Volterra integral equations, Comput. Appl. Math. 41 (2022),

1–18

[14] A.A. Khajehnasiri, R. Ezzati and M. Afshar Kermani, Solving systems of fractional two-dimensional nonlinear partial Volterra integral equations by using Haar wavelets, J. Appl. Anal. 21 (2021), 1–22.

[15] A.A. Khajehnasiri and M. Safavi, Solving fractional Black-Scholes equation by using Boubaker functions, Math. Meth. Appl. Sci. 44 (2021), 1–12.

[16] K. Kothari, U. Mehta and J. Vanualailai, A novel approach of fractional-order time delay system modeling based on Haar wavelet, ISA Trans. 80 (2018), 371–380.

[17] H.R. Marzban and M. Razzaghi, Solution of piecwise constant delay systems using hybrid of block-pulse and Chebyshev polynomials, Optim. Control Appl. Meth. 32 (2011), 647–659.

[18] H.R. Marzban and M. Razzaghi, Solution of time-varying delay systems by hybrid functions, Math. Comput. Simul. 64 (2004), 597–607.

[19] F. Mirzaee and E. Hadadiyan, Application of two-dimensional hat functions for solving space-time integral equations, J. Appl. Math. Comput. 4 (2015), 1–34.

[20] S. Najafalizadeh and R. Ezzati, A block pulse operational matrix method for solving two-dimensional nonlinear integro-differential equations of fractional order, J. Comput. Appl. Math. 326 (2017), 159–170.

[21] P.N. Paraskevopoulos, Legendre series approach to identification and analysis of linear systems, IEEE Trans. Automat. Control 30 (1985), 585–589.

[22] R.K. Pandey, N. Kumar and R.N. Mohaptra, An Approximate method for solving fractional delay differential equations, Int. J. Appl. Comput. Math. 3 (2017), 1395–1405.

[23] P. Rahimkhani, Y. Ordokhani and E. Babolian, An efficient approximate method for solving delay fractional optimal control problems, Nonlinear Dyn. 82 (2016), 1649—1661.

[24] H. Rahmani Fazli, F. Hassani, A. Ebadian and A. . Khajehnasiri, National economies in state-space of fractional-order financial system, Afr. Mat. 10 (2015), 1–12.

[25] M. Saeedi and M.M. Moghadam, Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS Wavelets, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1216–1226.

[26] H. Saeedi, N. Mollahasani, M.M. Moghadam and G.N. Chuev, An operational haar wavelet method for solving fractional Volterra integral equations, Int. J. Appl. Math. Comput. Sci. 21 (2011), 535–547.

[27] E. Safaie, M.H. Farahi and M. Farmani Ardehaie, A new Legendre operational technique for delay fractional optimal control problems, Comp. Appl. Math. 34 (2015), 831–846.

[28] M. Safavi, A.A. Khajehnasiri, A. Jafari and J. Banar, A new approach to numerical solution of nonlinear partial mixed Volterra-Fredholm integral equations via two-dimensional triangular functions, Malays. J. Math. Sci. 15 (2021), 489—507.

[29] M.P. Tripathi, V.K. Baranwal, R.K. Pandey and O. P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1327–1340.

[30] X.T.Wang, Numerical solutions of optimal control for linear time-varying systems with delays via hybrid functions, J. Franklin Inst. 344 (2007), 941–953.