Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem

Mathematical Modelling and Numerical Analysis

Volumn 33, Issue 5, 1999, Pages 895-922

  Aziz, Kadir A ^a   French, Donald A ^a,b   Jensen, Soren ^a   Kellogg, R Bruce ^c  

^a UNIVERSITY OF MARYLAND BALTIMORE COUNTY   (United States)

^b UNIVERSITY OF CINCINNATI   (United States)

^c University of Maryland   (United States)

Author keywords

Brownian motion; Degenerate parabolic problem; Electron and neutron scattering; Forward backward; Heat equation; Separated flow boundary layers

Indexed keywords

EID: 0041169170     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:1999125     Document Type: Article

References (49)

1. A.K. Aziz and J.-L. Liu, A Galerkin, method for the forward-backward heat equation. Math. Comp. 56 (1991) 35-44.
2. A.K. Aziz and J.-L. Liu, A weighted least squares method for the backward-forward heat equation. SIAM J. Numer. Anal. 28 (1991) 156-167.
3. M.S. Baouendi and P. Grisvard, Sur une équation d'évolution changeant de type. J. Funct. Anal. 2 (1968) 352-367.
4. R. Beals, On an equation of mixed type from electron scattering. J. Math. Anal. Appl. 58 (1977) 32-45.
5. H.A. Bethe, M.E. Rose and L.P. Smith, The multiple scattering of electrons. Proc. Amer. Philos. Soc. 78 (1938) 573-585.
6. W. Bothe, Die Streuabsorption der Elektronenstrahlen (electron scattering). Zeit. Physik 54 (1929) 161-178.
7. T. Cebeci, H.B. Keller and P.G. Williams, Separating boundary-layer flow calculations. J. Comput. Physics (1979) 363-378.
8. T. Cebeci and K. Stewartson, On the calculation of separation of bubbles. J. Fluid Mech. 133 (1983) 287-296.
9. P.G. Ciarlet, The Finite Element Methods for Elliptic Problems. North Holland (1980).
10. C.N. Dawson, C.J. Van Duijn and R.E. Grundy, Large time asymptotics in contaminant transport in porous media. SIAM J. Appl. Math. 56 (1996) 965-993.
11. J.A. Franklin and E.R. Rodemich, Numerical analysis of an elliptic-parabolic partial differential equation. SIAM J. Numer. Anal. 5 (1968) 680-716.
12. M. Freidlin and H. Weinberger, On a backward-forward parabolic equation and its regularization. J. Differential Equation 105 (1993) 264-295.
13. D.A. French, Continuous Galerkin Finite Element Methods for a Forward-Backward Heat Equation. Numer. Methods Partial Differential Equations 15 (1999) 257-265.
14. D.A. French, Discontinuous Galerkin Finite Element Methods for a Forward-Backward Heat Equation. Appl. Numer. Math. 27 (1998) 1-8.
15. M. Gevrey, Sur certaines équations aux dérivées partielles du type parabolique. Comptes Rendues 154 (1912) 1785-1788.
16. M. Gevrey, Sur les équations aux dérivées partielles du type parabolique, IV. J. Math. Pures Appl. 10 (1914) 105-137.
17. Ju.P. Gor'kov, A formula for the solution of a boundary-value problem for the stationary equation of Brownian motion. Dokl. Akad. Nauk SSSR 223 (1975), also Soviet Math. Dokl. 16 (1975) 904-908.
18. Ju.P. Gor'kov, A formula for the solution of a boundary-value problem for the stationary equation of Brownian motion. Dokl. Akad. Nauk SSSR 223 (1975), also Soviet Math. Dokl. 16 (1975) 904-908.
19. P.S. Hagan and J.R. Ockendon, Half-range analysis of a counter-current separator. J. Math. Anal. Appl. 160 (1991) 358-378.
20. E. Hopf. Mathematical problems of radiative equilibrium, Cambridge tracts in mathematics and mathematical physics, No. 31. Cambridge University Press (1934).
21. J.B. Keller and H.F. Weinberger, Boundary and initial boundary-value problems for separable backward-forward parabolic problems. J. Math. Phys. B 38 (1997) 4343-4353.
22. 2 + m+1/cursive Greek chi∂z/∂cursive Greek chi - n/cursive Greek chi∂z/∂t = 0. Math. Annal. 61 (1905) 397-405.
23. J.J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order. Comm. Pure Appl. Math. 20 (1967) 797-872.
24. T. LaRosa, The Propagation of an Electron Beam Through the Solar Corona. Ph.D. thesis, Physics and Astronomy, University of Maryland, College Park (1986).
25. T. LaRosa, The spatial structure of a nonthermal electron beam: conditions for stabilization via strong turbulence. Astrophys. J. 335 (1988) 425-440.
26. J.-L. Lions, Équations différentielles opérationelles et problèmes aux limites. Springer, Berlin (1961).
27. J.-L. Liu, Weak residual error estimates for symmetric positive systems. Numer. Funct. Anal. Optim. 14 (1993) 607-619.
28. J.-L. Liu, A finite difference method for symmetric positive differential equations. Math. Comp. 62 (1994) 105-118.
29. H. Lu, Galerkin and weighted Galerkin methods for the forward-backward heat equation. Numer. Math. 75 (1997) 339-56.
30. H. Lu and Z.-H. Wen, Solution of a forward-backward heat equation, preprint, Department of Mathematics, University of Nijmegen, NL (1994).
31. D.B. Melrose, Plasma astrophysics: non-thermal processes in diffuse magnetized plasmas. Vol. 1: The emission, absorption, and transfer of waves in plasmas. Vol. 2: Astrophysical application. Gordon and Breach, New York (1978).
32. A.F. Messiter and R.L. Enlow, A model for laminar boundary-layer flow near a separation point. SIAM J. Appl. Math. 25 (1973) 655-670.
33. W. Myller-Lebedeff, Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen. Math. Annal. 66 (1910) 388-416.
34. K. Nickel, Die Prandtlschen Grenzschichtdifferentialgleichungen als asymptotischer Grenzfall der Navier-Stokesschen und der Eulerschen differentialgleichungen. Arch. Rational Mech. Anal. 13 (1963) 1-14.
35. O.A. Oleǐnik, Mathematical problems of boundary layer theory. Uspekhi Mat. Nauk 23 (1968) 3-65.
36. O.A. Oleǐnik and E.V. Radkevič, Second order equations with nonnegative characteristic form. Plenum Press, New York-London (1973).
37. y. Ann. Scuola Norm. Sup. Pisa, Ser. IV 2 (1975) 219-263.
38. C.M. Pagani, On forward-backward parabolic equations in bounded domains. Boll. Un. Mat. Ita. B (5) 13 (1976) 336-354.
39. W.R.C. Phillips and J.T. Ratnanather, The outer region of a turbulent boundary layer. Phys. Fluids A, Fluid dynamics 2 (1990) 427.
40. J.T. Ratnanather and P.G Daniels, Solution of the thermal boundary layer equations in regions of flow reversal. SIAM J. Appl. Math. 55 (1995) 192-204.
41. L.B. Schiff and J.L. Steger, Numerical simulation of steady supersonic viscous flow. AIAA J. 18 (1980) 1421-1430.
42. F.T. Smith, Steady and unsteady boundary-layer separation. Annu. Rev. Fluid Mech 18 (1986) 197-220.
43. F.T. Smith and P. W. Duck, Separation of jets or thermal boundary layers from a wall. Quart. J. Mech. Appl. Math. 30 (1977) 143-156.
44. K. Stewartson, Multistructural boundary layers on flat plates and related bodies. Adv. in Appl. Mech. 14 (1974) 145-239.
45. K. Stewartson, D'Alembert's paradox. SIAM Rev. 23 (1981) 308-343.
46. K. Stewartson and P.G. Williams, Self-induced separation. Proc. Royal Soc. Ser. A 312 (1969) 181-206.
47. V. Vanaja, Numerical solution of a simple Fokker-Planck equation. Appl. Numer. Math. 9 (1992) 533-540.
48. V. Vanaja and R.B. Kellogg, Iterative methods for a forward-backward heat equation. SIAM J. Numer. Anal. 27 (1990) 622-635.
49. G. Milton Wing, An introduction to transport theory. John Wiley and Sons Inc., New York-London (1962).