Exponential asymptotics, the Viscid Burgers' equation, and standing wave solutions for a reaction-advection-diffusion model

Studies in Applied Mathematics

Volumn 102, Issue 2, 1999, Pages 137-172

  Laforgue, Jacques G L ^a   O'Malley, Robert E ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

NONLINEAR EQUATIONS; PERTURBATION TECHNIQUES;

ADVECTION-DIFFUSION MODELS; BOUNDARY-VALUE PROBLEM; BURGER'S EQUATIONS; EVOLUTIONARY EQUATION; EXPONENTIAL ASYMPTOTICS; MONOTONICS; SINGULARLY PERTURBED; SLOW SPEED; STANDING WAVE SOLUTIONS; STEADY STATE;

BOUNDARY VALUE PROBLEMS;

EID: 0040117313     PISSN: 00222526     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9590.00107     Document Type: Article

References (51)

1. J. B. KELLER, Combining analytical and numerical methods, Invited presentation, ICIAM 95, Hamburg, 1995.
2. J. B. KELLER and M. J. WARD, Asymptotics beyond all orders for a low Reynolds number flow, J. Eng. Math. 30:253-265 (1996).
3. F. W. J. OLVER, Error bounds for asymptotic expansions, with applications to cylinder functions of large argument, Asymptotic Solutions of Differential Equations and their Applications (C. H. Wilcox, ed.) Wiley, New York, 1964, pp. 163-183.
4. M. D. KRUSKAL and H. SEGUR, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. Math. 85:129-181, (1991).
5. J.-P. RAMIS, Séries Divergentes et Théories Asymptotiques, Societé Mathematique de France, Paris, (1993).
6. M. BERRY, Asymptotics, superasymptotics, hyperasymptotics,..., Asymptotics Beyond All Orders (H. Segur, S. Tanveer, and H. Levine, eds.), Plenum, New York, 1991, pp. 1-14.
7. A. B. OLDE DAALHUIS and F. W. J. OLVER, Exponentially improved asymptotic solutions of ordinary differential equations II: Irregular singularities of rank one, Proc. R. Soc. London A445:39-56 (1994).
8. D. S. JONES, Introduction to Asymptotics, World Scientific, Singapore, (1997).
9. G. B. WHITHAM, Linear and Nonlinear Wares, Wiley, New York, (1974).
10. C. A. J. FLETCHER, Burgers' equation: A model for all reasons, Numerical Solutions of Partial Differential Equations (J. Noye, ed.), North-Holland, Amsterdam, 1982, pp. 139-225.
11. H.-O. KREISS and J. LORENZ, Initial-Boundary Value Problems and the Navier Stokes Equations, Academic Press, San Diego, (1989).
12. J. KEVORKIAN and J. D. COLE, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York, (1996).
13. G. I. TAYLOR, Conditions necessary for discontinuous motion in gases, Proc. R. Soc. London A84:371-377 (1910).
14. H. BATEMAN, Some recent researches on the motion of fluids, Mon. Weather Rev. 43:163-170 (1915).
15. M. J. LIGHTHILL, Viscosity effects in sound waves of finite amplitude, Surveys in Mechanics, G. K. Batchelor and R. M. Davies, eds., Cambridge University Press, Cambridge, 1956, pp. 250-351.
16. P. A. ZEGELING, Moving-Grid Methods for Time-Dependent Partial Differential Equations, CWI, Amsterdam, (1993).
17. M. J. BAINES, Moving Finite Elements, Clarendon Press, Oxford, (1994).
18. K. W. MORTON, Numerical Solution of Convection-Diffusion Problems, Chapman and Hall, London, 1996.
19. H.-G. ROOS, M. STYNES, and L. TOBISKA, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin, 1996.
20. A. M. IL'IN, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, American Mathematical Society, Providence, 1992.
21. A. M. IL'IN and O. A. OLEINIK, Behavior of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of the time, Am. Math. Soc. Transl. 2 (42):19-23 (1964).
22. D. G. CRIGHTON, Nonlinear Acoustics, Modern Methods in Analytical Acoustics (D. G. Crighton, A. P. Dowling, J. E. Ffowes-Williams, M. Heckl, and F. G. Leppington, eds.), Springer-Verlag, Berlin, 1992, pp. 648-670.
23. G. KREISS and H.-O. KREISS, Convergence to steady-state of solutions of Burgers' equation, Appl. Numer. Math. 2:161-179 (1986).
24. G. KREISS, Convergence to steady-state of solutions of viscous conservation laws, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations (H. G. Kaper and M. Garbey, eds.), Dekker, New York, 1991.
25. G. KREISS, H.-O. KREISS, and N. A. PETERSSON, On the convergence to steady state of solutions of nonlinear hyperbolic-parabolic systems, SIAM J. Numerical Anal. 31:1577-1604 (1994).
26. xx - f(u), Comm. Pure Appl. Math. 42:523-576 (1989).
27. J. CARR, D. B. DUNCAN, and C. H. WALSHAW, Numerical approximation of a metastable system, IMA J. Numer. Anal. 15:505-521 (1995).
28. J. G. LAFORGUE and R. E. O'MALLEY, JR., Supersensitive Boundary Value Problems, Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters (H. G. Kaper and M. Garbey, eds.), Kluwer, Dordrecht, 1993, pp. 215-223.
29. A. BOHÉ, The existence of supersensitive boundary value problems, Methods Appl. Anal. 3:318-334 (1996).
30. F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, London, (1974).
31. J. M. BURGERS, The Nonlinear Diffusion Equation, Reidel, Dordrecht, (1974).
32. L. G. REYNA and M. J. WARD, On the exponentially slow motion of a viscous shock, Comm. Pure Appl. Math. 48:79-120 (1995).
33. J. G. LAFORGUE and R. E. O'MALLEY, JR., Shock layer movement for Burgers' equation, SIAM J. Appl. Math. 55:332-347 (1995).
34. Z. RAKIB and G. I. SIVASHINSKY, Instabilities in upward propagating flames, Combust. Sci. Technol. 54:69-84 (1987).
35. H. BERESTYCKI, S. KAMIN, and G. SIVASHINSKY, Nonlinear dynamics and metastability in a Burgers type equation, Comptes Rendus Acad. Sci. Paris, 321:185-190 (1995).
36. X. SUN and M. J. WARD, Metastability for a generalized Burgers equation with applications to propagating flame fronts, Eur. J. Appl. Math. 9: (1997).
37. S. E. HARRIS, Sonic shocks governed by the modified Burgers' equation, Eur. J. Appl. Math. 7:201-222 (1996).
38. G. I. BARENBLAIT, Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge, (1996).
39. J. G. LAFORGUE and R. E. O'MALLEY, JR., Viscous shock motion for advection-diffusion equations, Stud. Appl. Math. 95:147-170 (1995).
40. P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, Philadelphia, 1973.
41. J. B. KELLER, Accuracy and validity of the Born and Rytov approximations, J. Opt. Soc. Am. 59:1003-1004 (1969).
42. P. CONSTANTIN, C. FOIAS, B. NICOLAENKO, and R. TEMAM, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New York, 1989.
43. J. G. LAFORGUE and R. E. O'MALLEY, JR., On the metastable and supersensitive motion of shocks for viscous conservation laws in one space dimension, Nonlinear Problems in Applied Mathematics (T. S. Angell, L. P. Cook. R. E. Kleinman, and W. E. Olmstead, eds.), SIAM, Philadelphia, 1996, pp. 169-176.
44. J. G. LAFORGUE and R. E. O'MALLEY, JR., On the motion of viscous shocks and the supersensitivity of their steady state limits, Methods Appl. Anal. 1:465-487 (1994).
45. M. J. WARD, Metastable patterns, layer collapses, and coarsening for a one-dimensional Ginzburg-Landau equation, Stud. Appl. Math. 91:51-93 (1994).
46. N. J. BALMFORTH, Solitary waves and homoclinic orbits, Ann. Rev. Fluid Mech. 27:335-374 (1995).
47. M. SCHATZMAN, Interaction of standing waves, Z. Angew. Math. Mech. 76(suppl. 2):97-100 (1996).
48. M. J. WARD, Dynamic metastability and singular perturbations, in Boundaries, Interfaces, and Transitions, (M. C. Delfour, Ed.), pp. 237-263, American Math Society, Providence, 1998.
49. M. J. WARD and L. G. REYNA, Internal layers, small eigenvalues, and the sensitivity of metastable motion, SIAM J. Appl. Math. 55:425-445 (1995).
50. R. E. O'MALLEY, JR. and M. J. WARD, Exponential asymptotics, boundary layer resonance, and dynamic metastability, Mathematics Is for Solving Problems (L. P. Cook, V. Roytburd, and M. Tulin, eds.), SIAM, Philadelphia, 1996, pp. 189-203.
51. J. LAFORGUE, Using Exponential Asymptotics to Study the Metastability and Supersensitivity of Shock and Transition Layers, Ph.D. Thesis, University of Washington, 1994.