Efficient accurate numerical treatment of the modified Burgers’ equation

References

• Cole , JD . 1951 . On a quasi-linear parabolic equation occurring in aerodynamics . Q. Appl. Math. , 9 : 225 – 236 .
   Web of Science ®Google Scholar
• Burgers , JM . 1948 . “ A mathematical model illustrating the theory of turbulence ” . In Adv. in App. Mech. I , 171 – 199 . New York : Academic Press .
   Google Scholar
• El-Danaf , TS . 2002 . Numerical solution of the Korteweg–Vries Burgers’ equation by quintic spline method. Studia Unv. Babes–Bolyai Mathematica , XLVII : 41 – 55 .
   Google Scholar
• Gandarias , ML . 1997 . Nonclassical potential symmetries of the Burgers’ equation . Symmetry in Nonlinear Mathematical Physics , 1 : 130 – 137 .
   Google Scholar
• Harris , SL . 1996 . Sonic shocks governed by the modified Burgers’ equation . EJAM , 6 : 75 – 107 .
   Google Scholar
• Krstic , M . 1999 . On global stabilization of Burgers’ equation by boundary control . Syst. Cont. Lett. , 37 : 123 – 141 .
   Web of Science ®Google Scholar
• Ly , HV , Mease , KD and Tite , ES . 1997 . Distributed and boundary control of the viscous Burgers’ equation . Numer. Funct. Anal. Optim. , 18 : 143 – 188 .
   Web of Science ®Google Scholar
• Ramadan , MA and El-Danaf , TS . 2005 . Numerical treatment for the modified Burgers’ equation . Math. Comp. Simulat. , 70 : 90 – 98 .
   Web of Science ®Google Scholar
• Weijiu , L . 2002 . A asymptotic behavior of solutions of time-delayed Burgers’ equation . Discrete. Cont. dyn.-B , 2 : 47 – 56 .
   Web of Science ®Google Scholar
• Daele , MVG , Berghe , V and Meyer , HD . 1994 . A smooth approximation for the solution of a fourth-order boundary value problem based on non-polynomial splines . J. Comput. Appl. Math. , 51 : 383 – 394 .
   Web of Science ®Google Scholar
• Islam , SU , Khan , MA , Tirmizi , IA and Twizell , EH . 2005 . Non-polynomial spline approach to the solution of a system of third-order boundary-value problems . Appl. Math. , 168 : 152 – 163 .
   Google Scholar
• Ramadan , MA , Lashien , IF and Zahra , WK . 2007 . Polynomial and non polynomial spline approaches to the numerical solution of second order boundary value problems . Appl. Math. Comput. , 184 : 476 – 484 .
   Web of Science ®Google Scholar
• Ramadan , MA , Lashien , IF and Zahra , WK . 2008 . A class of methods based on septic non-polynomial spline function for the solution of sixth order two-point boundary value problems . Inter. J. Comput. Math , 855 5 May : 759 – 770 .
   Web of Science ®Google Scholar
• El-Danaf , TS and Abd Alaal , FEL . The use of non-polynomial splines for solving a fourth-order parabolic partial differential equation . Proceeding of the Mathematical and Physical Society of Egypt . (in press)
   Google Scholar
• Rahidinia , J , Jalilian , R and Kazemi , V . Spline method for the solution of hyperbolic equations . Appl. Math. Comput. , doi:10.1016lj. amc.2007.082
   Web of Science ®Google Scholar
• Dag , I , Irk , D and Saka , B . 2005 . A numerical solution of the Burgers’ equation using cubic B-splines . Appl. Math. Comput. , 163 : 199 – 211 .
   Web of Science ®Google Scholar
• Nguyen , H and Reynen , J . 1982 . “ A space-time finite element approach to Burgers’ equation ” . In Numerical Methods for Non-Linear Problems , Edited by: Taylor , C , Hinton , E , Owen , DRJ and Onate , E . Vol. 2 , 718 – 28 . Swansea : Pineridge Publisher .
   Google Scholar
• Ramadan , MA , El-Danaf , TS and Alaal , Faisal EI . 2005 . A numerical solution of the Burgers’ equation using septic B-splines , 26 : 795 – 804 . Chaos, Solitons & Fractals
   Google Scholar