1.Ali, M.R. and Hadhoud, A.R. Hybrid orthonormal Bernstein and Block-pulse functions wavelet scheme for solving the 2D Bratu problem, Results in Physics, 12 (2019) 525–530.
2. Ali, M.R., Hadhoud, A.R. and Ma, W.X. Evolutionary numerical approach for solving nonlinear singular periodic boundary value problems, J. Intell. Fuzzy Syst. 39(5) (2020) 7723–7731.
3. Ali, M.R., Hadhoud, A.R. and Srivastava, H.M. Solution of fractional Volterra–Fredholm integro–differential equations under mixed boundary conditions by using the HOBW method, Adv. Differ. Equ. 2019, 115 (2019).
4. Ali, M.R. and Ma, W.X. Detection of new multi-wave solutions in an unbounded domain, Modern Physics Letters B, 33(34) (2019) 1950425.5. Ali, M.R., Ma, W.X. and Sadat, R. Lie symmetry analysis and invariant solutions for (2+1) dimensional Bogoyavlensky–Konopelchenko equation with variable–coeﬀicient in wave propagation, Journal of Ocean Engineer-ing and Science (JOES), (2021).
6. Ali, M.R. and Sadat, R. Construction of Lump and optical solitons solutions for (3+1) model for the propagation of nonlinear dispersive waves in inhomogeneous media, Opt. Quantum Electron. 53 (5) (2021) 1–13.
7. Ali, M.R. and Sadat, R. Lie symmetry analysis, new group invariant for the (3+1)-dimensional and variable coeﬀicients for liquids with gas bubbles models, Chin. J. Phys. 71 (2021) 539–547.
8. Alipour, M., Baleanu, D. and Babaei, F. Hybrid Bernstein Block-pulse functions method for second kind integral equations with convergence analysis, Abstract and Applied Analysis. Vol. 2014. Hindawi, 2014.9. Ayub, A., Sabir, Z., Altamirano, G.C., Sadat, R. and Ali, M.R. Characteristics of melting heat transport of blood with time-dependent cross–
nanofluid model using Keller-Box and BVP4C method, Eng. Comput. (2021) 1–15.
10. Baleanu, D., Sadat, R. and Ali, M.R. The method of lines for solution of the carbon nanotubes engine oil nanofluid over an unsteady rotating disk, Eur. Phys. J. Plus. 135(10) (2020) 1–13.
11. Bass, R.F. Real analysis for graduate students, Createspace Ind Pub, 2013.
12. Bayram, M. Automatic analysis of the control of metabolicnetworks, Comput. Biol. Med. 26 (1996) 401–408.
13. Bellucci, M.A. On the explicit representation of orthonormal Bernstein Polynomials, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A, 2014.
14. Chen, Y., Yi, M. and Yu, C. Error analysis for numerical solution of fractional differential equation by Haar wavelets method, J. Comput. Sci3 (2012) 367–373.
15. Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl. 62 (2011) 2364–2373.
16. Feng, Y., Lin, X., Zhou, S. and Li, H. Chaos in a fractional-order neutral differential system, Appl. Math. Inf. Sci. (AMIS), 7 (2013) 233–238.
17. Gasca, M. and Sauer, T. On the history of multivariate polynomial interpolation, J. Comput. Appl. Math. 122 (2000) 23–35.
18. Gülsu, M., Öztürk, Y. and Anapalı, A. Numerical approach for solving fractional relaxation-oscillation equation, Appl. Math. Model. 37 (2013) 5927–5937.
19. Guzel, N., Emiroglu, I., Guler, C., Tasci F. and Sivri, M. A solution proposal to the interval fractional transportation problem, Appl. Math. Inf. Sci. (AMIS), 6 (2012) 567–571.
20. Hilfer, R. Applications of fractional calculus in physics, Academic Press, Orlando, 1999.
21. Javadi, Sh. and Taheri, Z. Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials, Journal of Computational and Applied Mathematics, 2016.
22. Li, X. Numerical solution of fractional differential equations using cubicspline wavelet collocation method, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 3934–3946.
23. Li, Y. and Sun, N. Numerical solution of fractional differential equations using the generalized Block-pulse operational matrix, Comput. Math. Appl. 62 (2011) 1046–1054.
24. Ma, W.X., Ali, M.R. and Sadat, R. Analytical solutions for nonlinear dispersive physical model, Complexity 2020 (2020).
25. Maleknejad, K., Hashemizadeh, E. and Basirat, B. Computational method based on Bernstein operational matrices for nonlinear VolterraFredholm–Hammerstein integral equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 52–61.
26. Marzban, H.R. and Malakoutikhah, F. Solution of delay fractional optimal control problems using a hybrid of Block-pulse functions and orthonormal Taylor polynomials, J. Franklin Inst. 356(15) (2019) 8182–8215.
27. Mashayekhi, S. and Razzaghi, M. Numerical solution of nonlinear fractional integro–differential equations by hybrid functions, Eng. Anal. Bound Elem. 56 (2015) 81–89.
28. Mason, J.C. and Handscomb, D.C. Chebyshev polynomials, CRC Press LLC, 2003.
29. Moghaddam, B.P. and Aghili, A. A numerical method for solving linear non-homogenous fractional ordinary differential equation by using the operational matrices of the Bernstein polynomials, Appl. Math. Inf. Sci. 6 (2012) 441–445.
30. Mollahasani, N., Moghadam, M.M. and Afrooz, K. A new treatment based on hybrid functions to the solution of telegraph equations of fractional order, Appl. Math. Model. 40 (2016), no. 4, 2804–2814.
31. Mousa, M.M., Ali, M.R. and Ma, W.X. A combined method for simulating MHD convection in square cavities through localized heating by method of line and penalty–artificial compressibility, J. Taibah Univ. Sci. 15(1) (2021) 208–217.
32. Podlubny, I. Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, some methods of their solution and some of their applications. San Diego: Academic Press, 1999.
33. Ramadan, M. and Osheba, H.S. A new hybrid orthonormal Bernstein and improved Block-pulse functions method for solving mathematical physics and engineering problems, Alex. Eng. J. 59(5) (2020) 3643–3652.
34. Sabir, Z., Ali, M.R., Zahoor Raja, M.A., Shoaib, M., Sandoval Nunez, R.A. and Sadat, R. Computational intelligence approach using Levenberg–Marquardt backpropagation neural networks to solve the fourth-order nonlinear system of Emden–Fowler model, Eng. Comput. (2021) 1–17.
35. Salimi Shamloo, A. and Babolian, E. Numerical solution of fractional
differential, integral and integro–differential equations by using piecewise constant orthogonal functions, PAMM: Proceedings in Applied Mathematics and Mechanics. Vol. 7. No. 1. Berlin: WILEY‐VCH Verlag, 2007.
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