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He’s homotopy perturbation method for solving the shock wave equation. (English) Zbl 1172.76042

Summary: We discuss the analytic solution of fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for \(0 < t < \infty \). The results presented converge very rapidly, indicating that the method is reliable and accurate.

MSC:

76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
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[1] DOI: 10.1016/S0168-9274(97)00090-1 · Zbl 0897.76046 · doi:10.1016/S0168-9274(97)00090-1
[2] DOI: 10.1016/0045-7930(94)90004-3 · Zbl 0816.76052 · doi:10.1016/0045-7930(94)90004-3
[3] Bruno, R. 1993.Convergence of approximate solutions of the Cauchy problem for a 2 {\(\times\)} 2 non-strictly hyperbolic system of conservation laws,inNonlinear Hyperbolic Problems, Edited by: Shearer, M. 487–494. Taormina: Vieweg Braunschweig. · Zbl 0921.35101
[4] Smaller J, Shockwaves and Reaction–Diffusion Equations (1983)
[5] DOI: 10.1016/j.cam.2005.05.009 · Zbl 1091.65104 · doi:10.1016/j.cam.2005.05.009
[6] DOI: 10.1016/S0096-3003(03)00341-2 · Zbl 1039.65052 · doi:10.1016/S0096-3003(03)00341-2
[7] DOI: 10.1515/IJNSNS.2005.6.2.207 · Zbl 1401.65085 · doi:10.1515/IJNSNS.2005.6.2.207
[8] DOI: 10.1016/j.physleta.2005.10.005 · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005
[9] DOI: 10.1016/S0045-7825(99)00018-3 · Zbl 0956.70017 · doi:10.1016/S0045-7825(99)00018-3
[10] DOI: 10.1016/S0020-7462(98)00085-7 · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[11] DOI: 10.1016/S0096-3003(01)00312-5 · Zbl 1030.34013 · doi:10.1016/S0096-3003(01)00312-5
[12] He JH, Topol. Methods Nonlinear Anal. 31 pp 205– (2008)
[13] DOI: 10.1142/S0217979206033796 · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[14] DOI: 10.1142/S0217979206034819 · doi:10.1142/S0217979206034819
[15] DOI: 10.1016/j.jsv.2006.10.001 · Zbl 1242.70044 · doi:10.1016/j.jsv.2006.10.001
[16] DOI: 10.1016/j.physleta.2007.04.072 · Zbl 1209.65120 · doi:10.1016/j.physleta.2007.04.072
[17] DOI: 10.1088/0031-8949/75/4/031 · Zbl 1110.35354 · doi:10.1088/0031-8949/75/4/031
[18] DOI: 10.1088/0031-8949/75/6/007 · Zbl 1117.35326 · doi:10.1088/0031-8949/75/6/007
[19] DOI: 10.1515/IJNSNS.2007.8.3.353 · Zbl 06942281 · doi:10.1515/IJNSNS.2007.8.3.353
[20] DOI: 10.1016/j.chaos.2006.04.013 · doi:10.1016/j.chaos.2006.04.013
[21] DOI: 10.1515/IJNSNS.2007.8.2.243 · Zbl 06942269 · doi:10.1515/IJNSNS.2007.8.2.243
[22] DOI: 10.1515/IJNSNS.2007.8.2.239 · Zbl 06942268 · doi:10.1515/IJNSNS.2007.8.2.239
[23] DOI: 10.1016/j.camwa.2008.07.020 · Zbl 1165.65377 · doi:10.1016/j.camwa.2008.07.020
[24] Yıldırım A, Int. J. Comput. Math. pp 1– (2008)
[25] Yıldırım A, Zeitschr. Naturforsch. A J. Phys. Sci. 63 pp 621– (2008)
[26] Yıldırım A, Commun. Numer. Methods Eng. (2008)
[27] Al-Khaled K, Ph.D. thesis (1996)
[28] Chow CY, An Introduction to Computational Fluid Mechanics (1979)
[29] Kevorkian J, Partial Differential Equations, Analytical Solution Techniques (1990)
[30] DOI: 10.1016/0022-247X(65)90153-8 · Zbl 0144.13006 · doi:10.1016/0022-247X(65)90153-8
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