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An approximation of the analytic solution of the shock wave equation. (English) Zbl 1091.65104

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

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L67 Shocks and singularities for hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
35Q51 Soliton equations
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References:

[1] Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053
[2] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[3] K. Al-Khaled, Theory and Computations in Hyperbolic Model Problems, Ph.D. Thesis, University of Nebraska-Lincoln, USA, 1996.; K. Al-Khaled, Theory and Computations in Hyperbolic Model Problems, Ph.D. Thesis, University of Nebraska-Lincoln, USA, 1996.
[4] Al-khaled, K.; Allan, F., Construction of solutions of shallow water equation by the decomposition method, J. Math. Comp. Sim., 66, 6, 479-486 (2004) · Zbl 1113.65098
[5] Banta, E. D., Lossless propagation of one-dimensional, finite amplitude sound waves, J. Math. Anal. Appl., 10, 166-173 (1965) · Zbl 0144.13006
[6] Behrens, J., Atmospheric and ocean modeling with an adaptive finite element solver for the shallow-water equations, Appl. Numer. Math., 26, 1-2, 217-226 (1998) · Zbl 0897.76046
[7] Bermudez, A.; Elena Vazquez, M., Upwind methods for hyperbolic conservation laws with course terms, Comput. Fluids, 23, 1049-1071 (1994) · Zbl 0816.76052
[8] Bruno, R., Convergence of approximate solutions of the Cauchy problem for a \(2 \times 2\) non-strictly hyperbolic system of conservation laws, (Shearer, M., Nonlinear Hyperbolic Problems (1993), Vieweg Braunschweig: Vieweg Braunschweig taormina), 487-494 · Zbl 0921.35101
[9] Chow, C. Y., An Introduction to Computational Fluid Mechanics (1979), Wiley: Wiley N.Y.
[10] Cherrualt, Y., Convergence of Adomian’s method, Kybernetes, 18, 31-38 (1989)
[11] Cherrualt, Y.; Adomian, G., Decomposition methods: a new proof of convergence, Math. Comput. Model., 18, 103-106 (1993) · Zbl 0805.65057
[12] Kevorkian, J., Partial Differential Equations, Analytical Solution Techniques (1990), Wadsworth and Brooks: Wadsworth and Brooks N.Y. · Zbl 0697.35001
[13] Smaller, J., Shock Waves and Reaction-Diffusion Equations (1983), Springer: Springer New York
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