Solving shock wave with discontinuity by enhanced differential transform method (EDTM)

ZOU Li; WANG Zhen; ZONG Zhi; ZOU Dong-yang; ZHANG Shuo

doi:10.3879/j.issn.1000-0887.2012.12.008

Volume 33 Issue 12

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Article Navigation > Applied Mathematics and Mechanics > 2012 > 33(12): 1465-1476

ZOU Li, WANG Zhen, ZONG Zhi, ZOU Dong-yang, ZHANG Shuo. Solving shock wave with discontinuity by enhanced differential transform method (EDTM)[J]. Applied Mathematics and Mechanics, 2012, 33(12): 1465-1476. doi: 10.3879/j.issn.1000-0887.2012.12.008

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Solving shock wave with discontinuity by enhanced differential transform method (EDTM)

doi: 10.3879/j.issn.1000-0887.2012.12.008

ZOU Li^1,2,
WANG Zhen⁴,
ZONG Zhi^1,3,
ZOU Dong-yang^1,2,
ZHANG Shuo^1,2

¹ State Key Laboratory of Structure Analysis for Industrial Equipment, Dalian 116085, Liaoning Province, P. R. China;² School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116085, Liaoning Province, Liaoning Province, P. R. China;³ School of Mathematics, Dalian University of Technology, Dalian 116085, Liaoning Province, P. R. China;⁴ School of Naval Architecture, Dalian University of Technology, Dalian 116085, Liaoning Province, P. R. China

Funds: Project supported by the National Natural Science Foundation of China (Nos. 50909017, 51109031, 50921001, 11072053, and 51009022), the Doctoral Foundation of Ministry of Education of China (No. 20100041120037), the Fundamental Research Funds for the Central Universities (Nos. DUT12LK52 and DUT12LK34), and the Major State Basic Research Development Program of China (973 Program) (Nos. 2010CB832704 and 2013CB036101)

Received Date: 2011-10-21
Rev Recd Date: 2012-06-14
Publish Date: 2012-12-15

Abstract

Abstract

An enhanced differential transform method (EDTM), which introduces the Padé technique into the standard differential transform method (DTM), is proposed. The enhanced method is applied to the analytic treatment of the shock wave. It accelerates the convergence of the series solution and provides an exact power series solution.
- enhanced differential transform method (EDTM),
- shock wave,
- Padé tech-nique,
- Burgers equation

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References(30)

References

[1]	Natfeg A H. Introduction to Perturbation Techniques[M]. New York: John Wiley & Sons, 1981.
[2]	Natfeg A H. Problems in Perturbation[M]. New York: John Wiley & Sons, 1985.
[3]	Lyapunov A M. General Problem on Stability of Motion[M]. London: Taylor & Francis, 1992.
[4]	Adomain G. Nonlinear stochastic differential equations[J]. Journal Mathematical Analysis and Application, 1976, 55(2): 441-452.
[5]	Adomain G. A global method for solution of complex systems[J]. Mathematical Model, 1984, 5(4): 521-568.
[6]	Adomain G. Solving Frontier Problems of Physics: the Decomposition Method[M]. Boston and London: Kluwer Academic Publishers, 1994.
[7]	廖世俊.同伦分析方法:一种新的求解非线性问题的近似解析方法[J]. 应用数学和力学, 1998, 19(10): 885-890.（LIAO Shi-jun. Homotopy analysis method: a new analytic method for nonlinear problems[J].Applied Mathematics and Mechanics(English Edition), 1998, 19(10): 957-962.）
[8]	廖世俊.超越摄动:同伦分析方法导论[M]. 科学出版社, 2007.（LIAO Shi-jun. Beyond Perturbation: Introduction to the Homotopy Analysis Method[M]. Science Press, 2007.(in Chinese)）
[9]	卢东强. 自由表面与粘性尾迹的相互作用[J]. 应用数学和力学, 2004, 25(6): 591598.（LU Dong-qiang. Interaction of viscous waves with a free surface[J]. Applied Mathematics and Mechanics(English Edition), 2004, 25(6): 647-655.）
[10]	卢东强, 戴世强, 张宝善. 一个二流体系统中非线性水波的Hamilton描述[J]. 应用数学和力学, 1999, 20(4): 331-336.(LU Dong-qiang, DAI Shi-qiang, ZHANG Bao-shan. Hamiltonian formulation of nonlinear water waves in a twofluid system[J]. Applied Mathematics and Mechanics(English Edition), 1999, 20(4): 343-349.）
[11]	Zhou J K. Differential Transform and Its Applications for Electrical Circuits[M]. Wuhan, China: Huazhong University Press, 1986.
[12]	Ravi Kanth A S V, Aruna K. Differential transform method for solving the linear and nonlinear KleinGordon equation[J]. Computer Physics Communication, 2009, 180(5): 708-711.
[13]	Chen C K, Ho S H. Solving partial differential equations by twodimensional differential transform method[J]. Applied Mathematics Computation, 1999, 106(2/3): 171-179.
[14]	Jang M J, Chen C L, Liu Y C. Two-dimensional differential transformation method for partial differential equation[J]. Applied Mathematics Computation, 2001, 121(2/3): 261-270.
[15]	Adbel-Halim Hassan I H. Different applications for the differential transformation in the differential equations[J]. Applied Mathematics Computation, 2002, 129(2/3): 183-201.
[16]	Ayaz F. On the two-dimensional differential transform method[J]. Applied Mathematics Computation, 2003, 143(2/3): 361-374.
[17]	Ayaz F. Solutions of the system of differential equations by differential transform method[J]. Applied Mathematics Computation, 2004, 147(2): 547-567.
[18]	Wang Z, Zou L, Zhang H Q. Applying homotopy analysis method for solving differentialdifference equation [J]. Physics Letters A, 2007, 369(1): 77-84.
[19]	Zou L, Zong Z, Wang Z, He L. Solving the discrete KdV equation with homotopy analysis method[J]. Physics Letters A, 2007, 370(3/4): 287-294.
[20]	Adbel-Halim Hassan I H. Comparison differential transformation technique with adomian decomposition method for linear and nonlinear initial value problems[J]. Chaos, Solutions & Fractals, 2008, 36(1): 53-65.
[21]	Figen K O, Ayaz F. Solitary wave solutions for the KdV and mKdV equations by differential transform method[J]. Chaos, Solution & Fractals, 2009, 41(1): 464-472.
[22]	Cole J D.On a quasi-linear parabolic equation occurring in aerodynamics[J]. Quart Applied Mathematics, 1951, 9: 225-236.
[23]	Bateman H. Some recent researches on the motion of fluids[J]. Monthly Weather Review, 1915, 43(4): 163-170.
[24]	Burgers J M. A mathematical model illustrating the theory of turbulence[J]. Advance Applied Mechanics, 1948, 1: 171-199.
[25]	Zhang X H, Ouyang J, Zhang L. Elementfree characteristic Galerkin method for Burgers equation[J]. Engineering Analysis with Boundary Elements, 2009, 33(3): 356-362.
[26]	Kutluay S, Eeen A, Dag I. Numerical solutions of the Burgers equation by the leastsquares quadratic B-spline finite element method[J]. Journal of Computational and Applied Mathematics, 2004, 167(1): 21-33.
[27]	Whitham G B. Linear and Nonlinear Waves[M]. New York: John Wiley & Sons, 1974.
[28]	Rosenau P, Hyman J M. Compactons: solitons with finite wavelength[J]. Physical Review Letters, 1993, 70(5): 564567.
[29]	Tian L X, Yin J L. Shockpeakon and shockcompacton solutions for K(p,q)-equation by variational iteration method[J]. Journal of Mathematical Analysis and Applications, 2007, 207(1): 46-52.
[30]	Camassa R, Holm D D. An integrable shallow water equation with peaked solitons[J]. Physical Review Letters, 1993, 71(11): 1661-1664.