Abstract
An innovative scheme of collocation having quintic Hermite splines as base functions has been followed to solve Burgers’ equation. The scheme relies on approximation of Burgers’ equation directly in non-linear form without using Hopf–Cole transformation (Hopf in Commun Pure Appl Math 3:201–216, 1950; Cole in Q Appl Math 9:225–236, 1951). The significance of the numerical technique is demonstrated by comparing the numerical results to the exact solution and published results (Asaithambi in Appl Math Comput 216:2700–2708, 2010; Mittal and Jain in Appl Math Comput 218:7839–7855, 2012). Five problems with different initial conditions have been examined to validate the efficiency and accuracy of the scheme. Euclidean and supremum norms have been reckoned to scrutinize the stability of the numerical scheme. Results have been demonstrated in plane and surface plots to indicate the effectiveness of the scheme.
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References
Abbasbandy, S.; Darvishi, M.T.: A numerical solution of Burgers’ equation by modified adomain method. Appl. Math. Comput. 163, 1265–1272 (2005)
Aksan, E.N.: A numerical solution of Burgers’ equation by finite element method constructed on the method of discretization in time. Appl. Math. Comput. 170, 895–904 (2005)
Arora, S.; Kaur, I.: Applications of Quintic Hermite collocation with time discretization to singularly perturbed problems. Appl. Math. Comput. 316, 409–421 (2018)
Arora, S.; Kaur, I.: An efficient scheme for numerical solution of Burgers’ equation using quintic hermite interpolating polynomials. Arab. J. Math. 5, 23–34 (2016)
Arora, S.; Kaur, I.; Potůček, F.: Modelling of Displacement washing of Pulp fibers using the Hermite collocation method. Br. J. Chem. Eng. 32, 563–575 (2015)
Arora, G.; Singh, B.K.: Numerical solution of Burgers’ equation with modified cubic B-spline differential quadrature method. Appl. Math. Comput. 224, 166–177 (2013)
Asaithambi, A.: Numerical solution of the Burgers’ equation by automatic differentiation. Appl. Math. Comput. 216, 2700–2708 (2010)
Burger, J.M.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171–199 (1948)
Chen, F.; Wong, P.J.Y.: Error estimates for discrete spline interpolation: quintic and biquintic splines. J. Comput. Appl. Math. 236, 3835–3854 (2012)
Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9, 225–236 (1951)
Dağ, İ.; Irk, D.; Saka, B.: A numerical solution of the Burgers’ equation using cubic B-splines. Appl. Math. Comput. 163, 199–211 (2005)
Dyksen, W.R.; Lynch, R.E.: A new decoupling technique for the Hermite cubic collocation equations arising from boundary value problems. Math. Comput. Simul. 54, 359–372 (2000)
Finden, W.F.: Higher order approximations using interpolation applied to collocation solutions of two-point boundary value problems. J. Comput. Appl. Math. 206, 99–115 (2007)
Gülsu, M.; Yalman, H.; Öztürk, Y.; Sezer, M.: A new Hermite collocation method for solving differential difference equations. Appl. Appl. Math. 6, 1856–1869 (2011)
Hall, C.A.: On error bounds for spline interpolation. J. Approx. Theory 1, 209–218 (1968)
Hopf, E.: The partial differential equation \(u_{t}+uu_{x}=\nu u_{xx}\). Commun. Pure Appl. Math. 3, 201–216 (1950)
Inc, M.: On numerical solution of Burgers’ equation by homotopy analysis method. Phys. Lett. A 372, 356–360 (2008)
Kadalbajoo, M.K.; Awasthi, A.: A numerical method based on Crank–Nicolson scheme for Burgers’ equation. Appl. Math. Comput. 182, 1430–1442 (2006)
Kutluay, S.; Esen, A.; Dag, I.: Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method. J. Comput. Appl. Math. 167, 21–33 (2004)
Kutluay, S.; Bahadir, A.R.; Özdeş, A.: Numerical Solution of one-dimensional Burgers equation: explicit and exact–explicit finite difference methods. J. Comput. Appl. Math. 103, 251–261 (1999)
Lang, A.W.; Sloan, D.M.: Hermite collocation solution of near-singular problems using numerical coordinate transformations based on adaptivity. J. Comput. Appl. Math. 140, 499–520 (2002)
Liao, W.: An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation. Appl. Math. Comput. 206, 755–764 (2008)
Liu, X.; Liu, G.R.; Tai, K.; Lam, K.Y.: Radial point interpolation collocation method (RPICM) for partial differential equations. Comput. Math. Appl. 50, 1425–1442 (2005)
Ma, H.; Sun, W.; Tang, T.: Hermite spectral methods with a time-dependent scaling for parabolic equations in unbounded domains. SIAM J. Numer. Anal. 43, 58–75 (2006)
Mittal, R.C.; Jain, R.K.: Numerical solutions of non-linear Burgers’ equation with modified cubic B-splines collocation method. Appl. Math. Comput. 218, 7839–7855 (2012)
Orsini, P.; Power, H.; Lees, M.: The Hermite radial basis function control volume method for multi-zones problems; A non-overlapping domain decomposition algorithm. Comput. Methods Appl. Mech. Eng. 200, 477–493 (2011)
Özis, T.; Esen, A.; Kutluay, S.: Numerical solution of Burgers’ equation by quadratic B-spline finite elements. Appl. Math. Comput. 165, 237–249 (2005)
Pandey, K.; Verma, L.; Verma, A.K.: Du Fort-Frankel finite difference scheme for Burgers equation. Arab. J. Math. 2, 91–101 (2013)
Ricciardi, K.L.; Brill, S.H.: Optimal Hermite collocation applied to a one-dimensional convection–diffusion equation using an adaptive hybrid optimization algorithm. Int. J. Numer. Methods Heat Fluid Flow 19, 874–893 (2009)
Wood, W.L.: An exact solution for Burger’ s equation. Commun. Numer. Methods Eng. 22, 797–798 (2006)
Xu, M.; Wang, R.; Zhang, J.; Fang, Q.: A novel numerical scheme for solving Burgers’ equation. Appl. Math. Comput. 217, 4473–4482 (2011)
Yalçinbaş, S.; Aynigül, M.; Sezer, M.: A collocation method using Hermite polynomials for approximate solution of pantograph equations. J. Franklin Inst. 348, 1128–1139 (2011)
Acknowledgements
Mrs. Inderpreet Kaur is thankful to DST for providing INSPIRE fellowship (IF140424).
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Arora, S., Kaur, I. & Tilahun, W. An exploration of quintic Hermite splines to solve Burgers’ equation. Arab. J. Math. 9, 19–36 (2020). https://doi.org/10.1007/s40065-019-0237-9
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DOI: https://doi.org/10.1007/s40065-019-0237-9