zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method. (English) Zbl 1193.65178
Summary: The Differential Transformation Method (DTM) is employed to obtain the numerical/analytical solutions of the Burgers and coupled Burgers equations. We begin by showing how the differential transformation method applies to the linear and nonlinear parts of any PDE and give some examples to illustrate the sufficiency of the method for solving such nonlinear partial differential equations. We also compare it against three famous methods, namely the homotopy perturbation method, the homotopy analysis method and the variational iteration method. These results show that the technique introduced here is accurate and easy to apply.
MSC:
65M99Numerical methods for IVP of PDE
76L05Shock waves; blast waves (fluid mechanics)
References:
[1]Burgers, J. M.: Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. R. Nether. acad. Sci. Amsterdam 43, 2-12 (1940) · Zbl 0061.45710
[2]Aksan, E. N.: Quadratic B-spline finite element method for numerical solution of the Burgers equation, Appl. math. Comput. 174, 884-896 (2006) · Zbl 1090.65108 · doi:10.1016/j.amc.2005.05.020
[3]Kutluay, S.; Esen, A.: A lumped Galerkin method for solving the Burgers equation, Int. J. Comput. math. 81, No. 11, 1433-1444 (2004) · Zbl 1063.65105 · doi:10.1080/00207160412331286833
[4]Abbasbandy, S.; Darvishi, M. T.: A numerical solution of Burgers equation by modified Adomian method, Appl. math. Comput. 163, 1265-1272 (2005) · Zbl 1060.65649 · doi:10.1016/j.amc.2004.04.061
[5]Sirendaoreji: Exact solutions of the two-dimensional Burgers equation, J. phys. A: math. Gen. 32, 6897-6900 (1999) · Zbl 0959.35148 · doi:10.1088/0305-4470/32/39/313
[6]Bateman, H.: Some recent researches on the motion of fluids, Mon. weather rev. 43, 163-170 (1915)
[7]Hopf, E.: The partial differential equation ut+uux=uxx, Commun. pure appl. Math. 3, 201-230 (1950) · Zbl 0039.10403 · doi:10.1002/cpa.3160030302
[8]Cole, J. D.: On a quasilinear parabolic equation occurring in aerodynamics, Quart. appl. Math. 9, 225-236 (1951) · Zbl 0043.09902
[9]Benton, E. R.; Platzman, G. W.: A table of solutions of the one-dimensional Burgers equation, Quart. appl. Math. 30, 195-212 (1972) · Zbl 0255.76059
[10]Karpman, V. I.: Nonlinear waves in dispersive media, (1975)
[11]Burgers, J.: Advances in applied mechanics, (1948)
[12]Caldwell, J.; Wanless, P.; Cook, A. E.: A finite element approach to Burgers equation, Appl. math. Modelling 5, 189-193 (1981) · Zbl 0476.76054 · doi:10.1016/0307-904X(81)90043-3
[13]Herbst, B. M.; Schoombie, S. W.; Mitchell, A. R.: Int. J. Numer. methods eng., Int. J. Numer. methods eng. 18, 1321-1336 (1982)
[14]Esipov, S. E.: Coupled Burgers equations: A model of polydispersive. Sedimentation, Phys. rev. E 52, 3711-3718 (1995)
[15]Nee, J.; Duan, J.: Limit set of trajectories of the coupled viscous Burgers equations, Appl. math. Lett. 11, No. 1, 57-61 (1998) · Zbl 1076.35537 · doi:10.1016/S0893-9659(97)00133-X
[16]Hizel, E.; Küçükarslan, S.: Homotopy perturbation method for (2+1)-dimensional coupled Burgers system, Nonlinear anal. 10, No. 3, 1932-1938 (2009) · Zbl 1168.35301 · doi:10.1016/j.nonrwa.2008.02.033
[17]Zhou, J. K.: Differential transformation and its application for electrical circuits, (1986)
[18]Ayaz, Fatma: Solutions of the system of differential equations by differential transform method, Appl. math. Comput. 147, 547-567 (2004) · Zbl 1032.35011 · doi:10.1016/S0096-3003(02)00794-4
[19]Al-Sawalha, M. Mossa; Noorani, M. S. M.: Application of the differential transformation method for the solution of the hyperchaotic Rössler system, Commun. nonlinear sci. Numer. simul. 14, 509-1514 (2009)
[20]Al-Sawalha, M. Mossa; Noorani, M. S. M.: A numeric–analytic method for approximating the chaotic Chen system, Chaos solitons fractals 42, 1784-1791 (2009) · Zbl 1198.65002 · doi:10.1016/j.chaos.2009.03.096
[21]Ayaz, F.: Application of differential transform method to differential–algebraic equations, Appl. math. Comput. 152, 649-657 (2004) · Zbl 1077.65088 · doi:10.1016/S0096-3003(03)00581-2
[22]Arikoglu, A.; Ozkol, I.: Solution of difference equations by using differential transform method, Appl. math. Comput. 174, 1216-1228 (2006) · Zbl 1138.65309 · doi:10.1016/j.amc.2005.06.013
[23]Arikoglu, A.; Ozkol, I.: Solution of differential–difference equations by using differential transform method, Appl. math. Comput. 181, 153-162 (2006) · Zbl 1148.65310 · doi:10.1016/j.amc.2006.01.022
[24]Kangalgil, Figen; Ayaz, Fatma: Solitary wave solutions for the KdV and mkdv equations by differential transform method, Chaos solitons fractals 41, 464-472 (2009) · Zbl 1198.35222 · doi:10.1016/j.chaos.2008.02.009
[25]Chen, C. K.: Solving partial differential equations by two dimensional differential transform, Appl. math. Comput. 106, 171-179 (1999) · Zbl 1028.35008 · doi:10.1016/S0096-3003(98)10115-7
[26]Jang, M. J.; Chen, C. L.; Liy, Y. C.: Two-dimensional differential transform for partial differential equations, Appl. math. Comput. 121, 261-270 (2001) · Zbl 1024.65093 · doi:10.1016/S0096-3003(99)00293-3
[27]A. Borhanifar, Reza Abazari, Exact solutions for non-linear Schrödinger equations by differential transformation method, J. Appl. Math. Comput., doi:10.1007/s12190-009-0338-2.
[28]Momani, S.; Odibat, Z.; Hashim, I.: Algorithms for nonlinear fractional partial differential equations: A selection of numerical methods, Topol. method nonlinear anal. 31, 211-226 (2008) · Zbl 1172.26302
[29]Arikoglu, A.; Ozkol, I.: Solution of fractional differential equations by using differential transform method, Chaos solitons fractals 34, 1473-1481 (2007) · Zbl 1152.34306 · doi:10.1016/j.chaos.2006.09.004
[30]Keskin, Y.; Kurnaz, A.; Kiris, M. E.; Oturanc, G.: Approximate solutions of generalized pantograph equations by the differential transform method, Int. J. Nonlinear. sci. 8, 159-164 (2007)
[31]Arikoglu, A.; Ozkol, I.: Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. math. Comput. 168, 1145-1158 (2005) · Zbl 1090.65145 · doi:10.1016/j.amc.2004.10.009
[32]Odibat, Z. M.: Differential transform method for solving Volterra integral equations with separable kernels, Math. comput. Modelling 48, 1144-1149 (2008) · Zbl 1187.45003 · doi:10.1016/j.mcm.2007.12.022
[33]Abazari, Reza: Solution of Riccati types matrix differential equations using matrix differential transform method, J. appl. Math. inform. 27, 1133-1143 (2009)
[34]Rashidi, M. M.; Domairry, G.; Dinarvand, S.: Approximate solutions for the Burgers and regularized long wave equations by means of the homotopy analysis method, Commun. nonlinear sci. Numer. simul. 14, No. 3, 708-717 (2009)
[35]Ebaid, A.: Exact solitary wave solutions for some nonlinear evolution equations via exp-function method, Phys. lett. A 365, 213-219 (2007) · Zbl 1203.35213 · doi:10.1016/j.physleta.2007.01.009
[36]Biazar, J.; Aminikhah, H.: Exact and numerical solutions for non-linear Burgers equation by VIM, Math. comput. Modelling 49, 1394-1400 (2009) · Zbl 1165.65395 · doi:10.1016/j.mcm.2008.12.006
[37]Bahadir, A. Refik: A fully implicit finite-difference scheme for two-dimensional Burgers equations, Appl. math. Comput. 137, 131-137 (2003) · Zbl 1027.65111 · doi:10.1016/S0096-3003(02)00091-7
[38]Abdou, M. A.; Soliman, A. A.: Variational iteration method for solving Burgers and coupled Burgers equations, J. comput. Appl. math. 181, 245-251 (2005) · Zbl 1072.65127 · doi:10.1016/j.cam.2004.11.032