232
Views
10
CrossRef citations to date
0
Altmetric
Review

Robust numerical scheme for nonlinear modified Burgers equation

&
Pages 1910-1926 | Received 16 Oct 2016, Accepted 30 Apr 2017, Published online: 19 Jun 2017

References

  • S. Albeverio, S.A. Molchanov, and D. Surgailis, Stratified structure of the universe and Burgers equation a probabilistic approach, Probab. Theory Related Fields 100(4) (1994), pp. 457–484. doi: 10.1007/BF01268990
  • T. Aziz and A. Khan, Quintic spline approach to the solution of a singularly-perturbed boundary-value problem, J. Optim. Theory Appl. 112(3) (2002), pp. 517–527. doi: 10.1023/A:1017959915002
  • H. Bachau, E. Cormier, P. Decleva, J. Hansen, and F. Martín, Applications of B-splines in atomic and molecular physics, Rep. Progr. Phys. 64(12) (2001), p. 1815. doi: 10.1088/0034-4885/64/12/205
  • R.H. Bartels, J.C. Beatty, and B.A. Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufmann, California, 1995.
  • H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev. 43 (1915), pp. 163–170. doi: 10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2
  • C. de Boor, On the convergence of odd-degree spline interpolation, J. Approx. Theory 1(4) (1968), pp. 452–463. doi: 10.1016/0021-9045(68)90033-6
  • A.G. Bratsos, A fourth-order numerical scheme for solving the modified Burgers equation, Comput. Math. Appl. 60(5) (2010), pp. 1393–1400. doi: 10.1016/j.camwa.2010.06.021
  • J.M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, Trans. Roy. Netherlands Acad. Sci. Amsterdam 17 (1939), pp. 1–53.
  • H.B. Curry and I.J. Schöenberg, On Pólya frequency functions IV: The fundamental spline functions and their limits, J. Anal. Math. 17(1) (1966), pp. 71–107. doi: 10.1007/BF02788653
  • İ. Dağ, B. Saka, and D. Irk, Galerkin method for the numerical solution of the RLW equation using quintic b-splines, J. Comput. Appl. Math. 190(1) (2006), pp. 532–547. doi: 10.1016/j.cam.2005.04.026
  • Y. Duan, R. Liu and Y. Jiang, Lattice Boltzmann model for the modified Burgers equation, Appl. Math. Comput. 202(2) (2008), pp. 489–497.
  • S. Durrleman and R. Simon, Flexible regression models with cubic splines, Stat. Med. 8(5) (1989), pp. 551–561. doi: 10.1002/sim.4780080504
  • K. Gorschkov, L. Ostrovsky, and E. Pelinovsky, Some problems of asymptotic theory of nonlinear waves, Proc. IEEE 62(11) (1974), pp. 1511–1517. doi: 10.1109/PROC.1974.9657
  • B. Greenshields, J.R. Bibbins, W. Channing, and H. Miller. ‘A study of traffic capacity,’ in Highway Research Board Proceedings, vol. 14, National Research Council (USA), Highway Research Board, 1935.
  • S. Gurbatov, A. Saichev, and S. Shandarin, Large-scale structure of the universe within the framework of model equation for nonlinear diffusion, Sov. Phys. Dokl 30 (1984), pp. 321–324.
  • S. Harris, Sonic shocks governed by the modified Burgers equation, European J. Appl. Math. 7(02) (1996), pp. 201–222. doi: 10.1017/S0956792500002291
  • R.S. Hirsh, Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys. 19(1) (1975), pp. 90–109. doi: 10.1016/0021-9991(75)90118-7
  • D. Irk, Sextic b-spline collocation method for the modified Burgers equation, Kybernetes 38(9) (2009), pp. 1599–1620. doi: 10.1108/03684920910991568
  • G. Kreiss and H.-O. Kreiss, Convergence to steady state of solutions of Burgers equation, Appl. Numer. Math. 2(3–5) (1986), pp. 161–179. doi: 10.1016/0168-9274(86)90026-7
  • I. Lee-Bapty and D. Crighton, Nonlinear wave motion governed by the modified Burgers equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 323(1570) (1987), pp. 173–209. doi: 10.1098/rsta.1987.0081
  • R. Mohammadi, B-spline collocation algorithm for numerical solution of the generalized Burger's-Huxley equation, Numer. Methods. Partial Differential Equations 29(4) (2013), pp. 1173–1191. doi: 10.1002/num.21750
  • R. Mohammadi, Quintic b-spline collocation approach for solving generalized Black–Scholes equation governing option pricing, Comput. Math. Appl. 69(8) (2015), pp. 777–797. doi: 10.1016/j.camwa.2015.02.018
  • G. Nariboli and W. Lin, A new type of Burgers equation, ZAMM Z. Angew. Math. Mech. 53(8) (1973), pp. 505–510. doi: 10.1002/zamm.19730530802
  • J. Nimmo and D. Crighton, Backlund transformations for nonlinear parabolic equations: The general results, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 384 (1982), pp. 381–401. doi: 10.1098/rspa.1982.0164
  • D. Panayotounakos and D. Drikakis, On the closed-form solutions of the wave, diffusion and Burgers equations in fluid mechanics, ZAMM Z. Angew. Math. Mech. 75(6) (1995), pp. 437–447. doi: 10.1002/zamm.19950750604
  • M.A. Ramadan and T.S. El-Danaf, Numerical treatment for the modified Burgers equation, Math. Comput. Simulation 70(2) (2005), pp. 90–98. doi: 10.1016/j.matcom.2005.04.002
  • M.A. Ramadan, T.S. El-Danaf, and F.E.A. Alaal, A numerical solution of the Burgers equation using septic B-splines, Chaos Solitons Fractals 26(4) (2005), pp. 1249–1258. doi: 10.1016/j.chaos.2005.02.019
  • K. Raslan and S.M. Hassan, Solitary waves for the MRLW equation, Appl. Math. Lett. 22(7) (2009), pp. 984–989. doi: 10.1016/j.aml.2009.01.020
  • L.G. Reyna and M.J. Ward, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math. 48(2) (1995), pp. 79–120. doi: 10.1002/cpa.3160480202
  • Z. Rong-Pei, Y. Xi-Jun, and Z. Guo-Zhong, Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations, Chin. Phys. B 20(11) (2011), pp. 110205-1.
  • W. Rudin. Principles of Mathematical Analysis. vol. 3, McGraw-Hill, New York, 1964.
  • S.G. Rubin and R.A. Graves, Cubic Splines Approximation for Problems in Fluid Mechanics, Nasa TR R-436, Washington, DC, 1975.
  • P. Sachdev, C. Rao, and B.O. Enflo, Large-time asymptotics for periodic solutions of the modified Burgers equation, Stud. Appl. Math. 114(3) (2005), pp. 307–323. doi: 10.1111/j.0022-2526.2005.01551.x
  • B. Saka and İ Dağ, A numerical study of the Burgers equation, J. Franklin Inst. 345(4) (2008), pp. 328–348. doi: 10.1016/j.jfranklin.2007.10.004
  • L. Schueremans and D. Gemert, Splines and Other Metamodels in Reliability Analysis, Wiley Online Library, New Jersey, 2007.
  • I.J. Schöenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math 4(2) (1946), pp. 45–99. doi: 10.1090/qam/15914
  • N. Sugimoto, Y. Yamane, and T. Kakutani, Torsional shock waves in a viscoelastic rod, J. Appl. Mech. 51(3) (1984), pp. 595–601. doi: 10.1115/1.3167679
  • M. Unser, Splines: A perfect fit for signal and image processing, IEEE. Signal Process. Mag. 16(6) (1999), pp. 22–38. doi: 10.1109/79.799930
  • H. Yang and A. Przekwas, A comparative study of advanced shock-capturing shcemes applied to Burgers equation, J. Comput. Phys. 102(1) (1992), pp. 139–159. doi: 10.1016/S0021-9991(05)80012-9
  • X. Yu, Analysis of the stability and density waves for traffic flow, Chin. Phys. 11(11) (2002), pp. 1128–1134. doi: 10.1088/1009-1963/11/11/307

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.