Numerical solution of the time fractional nonlinear burgers equation using the quintic B-Spline method

Document Type : Research Article

Authors

1 Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq.

2 Department of Mathematics, College of Science, University of Basrah, Basrah, Iraq

Abstract

This paper introduced a novel approach for resolving fractional partial differential equations. The time fractional nonlinear Burgers equation of order $\kappa$ was solved to illustrate the efficacy of the technique, where $\kappa\in(0,1]$. The quintic B-spline method facilitated spatial partitioning, while the finite difference method addressed the fractional Caputo derivative, which simulates anomalous diffusion processes influenced by memory effects. The proposed methods stability is demonstrated utilizing the von Neumann technique; it has been shown to be unconditionally stable. Additionally, a convergence study is shown, demonstrating that the approach exhibits uniform convergence of $(\gamma h^{4}+\sigma(\Delta \eta^{2}))$. We validated the methods correctness through numerical tests by comparing it with the exact solution and alternative numerical methods. Based on $L^{2}$ and $L^{\infty}$  error norms, the quintic B-spline approach exhibits improved convergence rates and reduced computing costs.

Keywords

Main Subjects

References

[1] S.U. Arifeen, I. Ali, I. Ahmad, S. Shaheen, Computational study of time-fractional non-linear kawahara equations using quintic b-spline and galerkin’s method, Partial Differ. Equ. Appl. Math. 12 (2024) 100779.
[2] N. Biranvand, A.R. Vahidi, E. Babolian, An improved model along with a spectral numerical simulation for fractional predator-prey interactions, Eng. Comput. 38 (2021) 2467–2480.
[3] D.T. Blackstock, Generalized burgers equation for plane waves, J. Acoust. Soc. Am. 77 (1985) 2050–2053.
[4] A. Boggess, A First Course in Wavelets with Fourier Analysis, Wiley, Hoboken, New Jersey, Second edition, 2009.
[5] X. Chi, H. Zhang, Numerical study for the unsteady space fractional magnetohydrodynamic free convective flow and heat transfer with hall effects, Appl. Math. Lett. 120 (2021) 107312.
[6] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhauser Boston, 1997.
[7] A. Esen, O. Tasbozan, Numerical solution of time fractional burgers equation, Acta Univ. Sapientiae Math. 7 (2015) 167–185.
[8] A. Esen, O. Tasbozan, Numerical solution of time fractional burgers equation by cubic b-spline finite elements, Mediterr. J. Math. 13 (2015) 1325–1337.
[9] W. Ford, Numerical Linear Algebra with Applications, Academic Press, London, 2015.
[10] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng. 167 (1998) 57–68.
[11] J.H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng. 178 (1999) 257–262.
[12] M. Irodotou-Ellina, E.N. Houstis, Ano(h 6) quintic spline collocation method for fourth order two-point boundary value problems, BIT 28 (1988) 288–301.
[13] L.W. Johnson, R.D. Riess, Numerical Analysis, Addison-Wesley, Reading, Mass, 1982.
[14] S.L. Khalaf, K.K. Kassid, A.R. Khudair, A numerical method for solving quadratic fractional optimal control problems, Results Control Optim. 13 (2023) 100330.
[15] A.A. Kilbas Theory and Applications of Fractional Differential Equations, North-Holland mathematics studies, Elsevier, Boston, 2006.
[16] D. Li, C. Zhang, M. Ran, A linear finite difference scheme for generalized time fractional burgers equation, Appl. Math. Model. 40 (2016) 6069–6081.
[17] R.K. Lodhi, H.K. Mishra, Quintic b-spline method for solving second order linear and nonlinear singularly perturbed two-point boundary value problems, J. Comput. Appl. Math. 319 (2017) 170–187.
[18] N.K. Mahdi, A.R. Khudair, Linear fractional dynamic equations hyers-ulam stability analysis on time scale, Results Control Optim. 14 (2024) 100347.
[19] N.K. Mahdi, A.R. Khudair, Some delta q-fractional linear dynamic equations and a generalized delta q-mittag-leffler function, Comput. Methods Differ. Equ. 12 (2024) 502–510.
[20] H.K. Mishra, R.K. Lodhi, Two-parameter singular perturbation boundary value problems via quintic b-spline method, Proceedings of the National Academy of Sciences, India Section A: Physical Sciences 92 (2022) 541–553.
[21] J. Mohapatra, S. Mohapatra, A. Nath, An approximation technique for a system of time-fractional differential equations arising in population dynamics, J. Math. Model. 13(3) (2025) 519–531.
[22] A. Mohebbi, Analysis of a numerical method for the solution of time fractional burgers equation, Bull. Iran. Math. Soc. 44 (2018) 457–480.
[23] F.K. Nashmi, B.A. Taha, Numerical study of the time-fractional partial differential equations by using quartic b-spline method, Partial Differ. Equ. Appl. Math. 12 (2024) 101008.
[24] H. Or-Roshid, M. Rashidi, Multi-soliton fusion phenomenon of burgers equation and fission, fusion phenomenon of sharma-tasso-olver equation, J. Ocean Eng. Sci. 2 (2017) 120–126.
[25] J.L.D. Palencia , On the existence of non-radial normalized solutions for coupled fractional nonlinear schr¨odinger systems with potential, J. Math. Model. 13(2) (2025) 357–373.
[26] R.G. Rice, Applied Mathematics and Modeling for Chemical Engineers, Wiley, Hoboken, Third edition, 2023.
[27] P. Roul, A high accuracy numerical method and its convergence for time-fractional black-scholes equation governing european options, Appl. Numer. Math. 151 (2020) 472–493.
[28] A.H. Siddiqi, Functional Analysis and Applications, Springer Singapore, 2018.
[29] B.K. Singh, Homotopy perturbation new integral transform method for numeric study of space-and time-fractional (n + 1)-dimensional heat- and wave-like equations, Waves, Wavelets Fractals 4 (2018) 19–36.
[30] B.K. Singh, M. Gupta, Trigonometric tension b-spline collocation approximations for time fractional burgers’ equation, J. Ocean Eng. Sci. 9 (2024) 508–516.
[31] B.K. Singh, P. Kumar, Frdtm for numerical simulation of multi-dimensional, time-fractional model of navier-stokes equation, Ain Shams Eng. J. 9 (2018) 827–834.
[32] J. Singh, A. Babarit, A fast approach coupling boundary element method and plane wave approximation for wave interaction analysis in sparse arrays of wave energy converters, Ocean Eng. 85 (2014) 12–20.
[33] G.D. Smith, Numerical Solution of Partial Differential Equations, Clarendon Press, 1992.
[34] L. Song, S. Xu, J.Yang, Dynamical models of happiness with fractional order, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 616–628.
[35] J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, 2004.
[36] H. Yang, A. Przekwas, A comparative study of advanced shock-capturing shcemes applied to burgers’ equation, J. Comput. Phys. 102 (1992) 139–159.
[37] A. Yokus, D. Kaya , Numerical and exact solutions for time fractional burgers’ equation, J. Nonlin. Sci. Appl. 10 (2017) 3419–3428.
[38] F. Zakipour, A. Saadatmandi, A novel fractional bernoulli-picard iteration method to solve fractional differential equations, J. Math Model. 13(1) (2025) 139–152.
[39] S. Zerbib, K. Hilal, A. Kajouni , Nonlocal caputo generalized proportional fractional integro-differential systems: an existence study, J. Math. Model. 13(2) (2025) 375–391.
Journal of Mathematical Modeling
Volume 14, Issue 1
March 2026
Pages 253-271

Files

Share

How to cite

Statistics