# Applying Haar-Sinc Spectral Method for Solving time-fractional Burger Equation

Document Type : Original Article

Author

Department of Computer Engineering , Islamic Azad University, Marivan Branch, Marivan, Iran

Abstract

Haar-Sinc spectral method is used for the numerical approximation of time fractional Burgers’
equations with variable and constant coefficients. The main idea in this method is using a linear discretization of time and space by combination of Haar and Sinc functions, respectively. While implementing the method, the operational matrices of the fractional integral of the fractional Haar functions are made, and by using them, an algebraic equation is obtained. Then, using the collocation method, the algebraic equation is converted into a system of equations, and after solving the system with Maple software, the numerical results of the problem is obtained.
The accuracy and speed of the proposed algorithm are tested by obtaining L, L2 error and the convergence rate.

Keywords

Main Subjects

#### References

[1] I. Ali, S. Haq, S.F. Aldosary, K.S. Nisar, F. Ahmad, Numerical solution of one- and twodimensional time-fractional burgers equation via lucas polynomials coupled with finite difference method, Alexandria Engineering Journal, 61, 2022, 6077-6087.
[2] J.M. Burgers, A Mathematical Model Illustrating the Theory of Turbulence, Adv. Appl. Mech. Academic Press, New York, 1948.
[3] C.F. Chen, C. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEEE. Proc. Cont.Theo. Appl., 144 1997, 87-94.
[4] L. Chen, S. Lu, T. Xu, Fourier spectral approximation for time fractional burgers equation with non smooth solutions, Applied Numerical Mathematics, 169 2021, 164-178.
[5] Y. Chen, M. Yi, C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, J. Comput. Sci., 3 2012, 367-373.
[6] A. Duangpan, R. Boonklurb, and T. Treeyaprasert, Finite integration method with shifted chebyshev polynomials for solving time-fractional burgers’ equations, Mathematics, 7(12) 2019, 1201.
[7] M.S. Hashmi, M. Wajiha, S.W. Yao, A. Ghaffar, M. Inc, Cubic spline based differential quadrature method: A numerical approach for fractional burger equation, Results in Physics, 26(3) 2021, 104415.
[8] M.H. Heydari, Z. Avazzadeh, A. Atangana, Orthonormal shifted discrete legendre polynomials for solving a coupled system of nonlinear variable-order time fractional reaction-advection-diffusion equations, Applied Numerical Mathematics, 161 2021, 425-436.
[9] M.H. Heydari, Z. Avazzadeh, N. Hosseinzadeh, Haar wavelet method for solving high-order differential equations with multi-point boundary conditions, Journal of Applied and Computational Mechanics, 8(2) 2022, 528-544.
[10] M. Hussain, S. Haq, Weighted meshless spectral method for the solutions of multi-term time fractional advection-diffusion problems arising in heat and mass transfer, Int. J. Heat Mass Transf., 129, 2019, 1305-1316.
[11] M. Hussain, S. Haq, A. Ghafoor, I. Ali, Numerical solutions of time-fractional coupled viscous burgers’ equations using meshfree spectral method, Comput. Appl. Math., 39(1) 2020, 6.
[12] M. Kashif, K. D. Dwivedi, T. Som, Numerical solution of coupled type fractional order burgers equation using finite difference and fibonacci collocation method, Chinese Journal of Physics, 77 2022, 2314-2323.
[13] A.A. Kilbas, H. Srivastava, J. Trujillo, A Mathematical Model Illustrating the Theory of Turbulence, San Diego: Elsevier, 2006.
[14] C. Li, D. Li, Z. Wang, L1/ldg method for the generalized time-fractional burgers equation, Mathematics and Computers in Simulation, 187 2021, 357-378.
[15] Y. Li, W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput., 216, 2010, 2276-2285.
[16] J. Lund, K. Bowers, Sinc methods for quadrature and differential equations. SIAM, Philadelphia,1992.
[17] M. Onal, A. Esen, A crank-nicolson approximation for the time fractional burgers equation, Appl. Math. Nonlin. Sci., 5(2) 2020, 177-184.
[18] O. Oruc, A. Esen, F. Bulut, A unified finite difference chebyshev wavelet method for numerically solving time fractional burgers’ equation, Disc. Cont. Dyn. Syst., 12(3) 2019, 533-542.
[19] K. Parand, M. Dehghan, A. Pirkhedri, Sinc-collocation method for solving the blasius equation, Phy. Let. A., 373(4) 2009, 4060-4065.
[20] K. Parand, M. Dehghan, A. Pirkhedri, The Sinc-collocation method for solving the thomas-fermi equation, Appl. Math. Comput., 237(1) 2013, 244-252 .
[21] A. Pirkhedri ,H.H. S. Javadi, Solving the time-fractional diffusion equation via Sinc–Haar collocation method, Applied Mathematics and Computation, 257 2015, 317-325.
[22] M. Razzaghi, Y. Ordokhani, Solution of differential equations via rationalized Haar functions, Math. Comput. Simul., 56 2001, 235-246.
[23] B. K. Singh, M. Gupta, Trigonometric tension b-spline collocation approximations for time fractional burgers’ equation, Journal of Ocean Engineering and Science, In Press, 2022.
[24] F. Stenger, Integration formulas via the trapezoidal formula, J. Inst. Math. App., 12 1973, 103-114.
[25] V.K. Tamboli, P.V. Tandel, Solution of the time-fractional generalized burger–fisher equation using the fractional reduced differential transform method, Journal of Ocean Engineering and Science, 7(4) 2022, 399-407.
[26] D. Tavares, R. Almeida, D.F. Torres, Caputo derivatives of fractional variable order: Numerical approximations, communications in nonlinear science and numerical simulation, 35 2016, 69-87.
[27] A. Yokus, D. Kaya, Numerical and exact solutions for time fractional burgers equation, J. Nonlinear Sci. Appl, 10 2017, 3419-3428.

### History

• Receive Date: 14 October 2023
• Revise Date: 24 March 2024
• Accept Date: 24 March 2024