Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 33 (1999) no. 5, p. 895-922
@article{M2AN_1999__33_5_895_0,
     author = {Aziz, A. Kadir and French, Donald A. and Jensen, S\o ren and Kellogg, R. Bruce},
     title = {Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {33},
     number = {5},
     year = {1999},
     pages = {895-922},
     zbl = {0962.65082},
     mrnumber = {1726715},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1999__33_5_895_0}
}
Aziz, A. Kadir; French, Donald A.; Jensen, Soren; Kellogg, R. Bruce. Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 33 (1999) no. 5, pp. 895-922. http://www.numdam.org/item/M2AN_1999__33_5_895_0/

[1] A.K. Aziz and J.-L. Liu, A Galerkin method for the forward-backward heat equation. Math. Comp. 56 (1991) 35-44. | MR 1052085 | Zbl 0718.65068

[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. | MR 1083329 | Zbl 0727.65080

[3] M.S. Baouendi and P. Grisvard, Sur une équation d'évolution changeant de type. J. Funct. Anal 2 (1968) 352-367. | MR 252817 | Zbl 0164.12701

[4] R. Beals, On an equation of mixed type from electron scattering. J. Math. Anal. Appl. 58 (1977) 32-45. | MR 492921 | Zbl 0353.35072

[5] H.A. Bethe, M.E. Rose and L.P. Smith, The multiple scattering of electrons. Proc. Amer. Philos. Soc. 78 (1938) 573-585. | JFM 64.0898.03

[6] W. Bothe, Die Streuabsorption der Elektronenstrahlen (electron scattering). Zeit. Physik 54 (1929) 161-178. | JFM 55.0531.03

[7] T. Cebeci, H.B. Keller and P.G. Williams, Separating boundary-layer flow calculations. J. Comput. Physics (1979) 363-378. | MR 534787 | Zbl 0411.76026

[8] T. Cebeci and K. Stewartson, On the calculation of separation of bubbles. J. Fluid Mech. 133 (1983) 287-296. | Zbl 0524.76092

[9] P.G. Ciarlet, The Finite Element Methods for Elliptic Problems. North Holland (1980). | MR 1115235 | Zbl 0999.65129

[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. | MR 1398404 | Zbl 0862.35012

[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. | MR 245210 | Zbl 0261.65069

[12] M. Freidlin and H. Weinberger, On a backward-forward parabolic equation and its regularization. J. Differential Equation 105 (1993) 264-295. | MR 1240397 | Zbl 0780.35065

[13] D.A. French, Continuons Galerkin Finite Element Methods for a Forward-Backward Heat Equation. Numer. Methods Partial Differential Equations 15 (1999) 257-265. | MR 1674286 | Zbl 0948.65095

[14] D.A. French, Discontinuous Galerkin Finite Element Methods for a Forward-Backward Heat Equation. Appl. Numer. Math. 27 (1998) 1-8. | MR 1639826 | Zbl 0931.65101

[15] M. Gevrey, Sur certaines équations aux dérivées partielles du type parabolique. Comptes Rendues 154 (1912) 1785-1788. | JFM 43.0446.03

[16] M. Gevrey, Sur les équations aux dérivées partielles du type parabolique, IV. J. Math. Pures Appl 10 (1914) 105-137. | JFM 45.0571.03

[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. | MR 386033 | Zbl 0324.35013

[18] P.S. Hagan and J.R. Ockendon, Half-range analysis of a counter-current separator. J. Math. Anal. Appl. 160 (1991) 358-378. | MR 1126123 | Zbl 0753.76190

[19] E. Hopf, Mathematical problems of radiative equilibrium, Cambridge tracts in mathematics and mathematical physics, No. 31. Cambridge University Press (1934). | JFM 60.0809.01 | MR 183517

[20] 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. | MR 1459662 | Zbl 0881.35051

[21] S. Kepinski, Über die Differentialgleichung . Math. Annal 61 (1905) 397-405. | JFM 36.0416.01 | MR 1511354

[22] J.J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of second order. Comm. Pure Appl. Math. 20 (1967) 797-872. | MR 234118 | Zbl 0153.14503

[23] T. Larosa, The Propagation of an Electron Beam Through the Solar Corona. Ph.D. thesis, Physics and Astronomy, University of Maryland, College Park (1986).

[24] T. Larosa, The spatial structure of a nonthermal electron beam: conditions for stabilization via strong turbulence. Astrophys J. 335 (1988) 425-440.

[25] J.-L. Lions, Équations différentielles opérationelles et problèmes aux limites. Springer, Berlin (1961). | MR 153974 | Zbl 0098.31101

[26] J.-L. Liu, Weak residual error estimates for symmetric positive Systems. Numer. Funct. Anal. Optim. 14 (1993) 607-519. | MR 1248131 | Zbl 0795.65067

[27] J.-L. Liu, A finite difference method for symmetric positive differential equations. Math. Comp, 62 (1994) 105-118. | MR 1208839 | Zbl 0797.65059

[28] H. Lu, Galerkin and weighted Galerkin methods for the forward-backward heat equation. Numer. Math, 75 (1997) 339-56. | MR 1427712 | Zbl 0876.65071

[29] H. Lu and Z.-H. Wen, Solution of a forward-baekward heat equation, preprint, Department of Mathematics, University of Nijmegen, NL (1994). | Zbl 0900.65289

[30] D.B. Melrose, Plasma astrophysics: non-thermal processes in diffuse magnetized plasmas. Vol. 1: The émission, absorption, and transfer of waves in plasmas. Vol. 2: Astrophysical application. Gordon and Breach, New York (1978).

[31] 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. | Zbl 0246.76031

[32] W. Myller-Lebedeff, Die Theorie der Integralgleichungen in Anwendung auf einige Reihenentwicklungen. Math. Annal. 66 (1910) 388-416. | JFM 38.0385.02

[33] K. Nickel, Die Prandtlschen Grenzschichtdifferentialgleichungen als asymptotischer Grenzfall der Navier-Stokesschen und der Eulerschen differentialgleichungen. Arch. Rational Mech. Anal 13 (1963) 1-14. | MR 148315 | Zbl 0115.21004

[34] O.A. Oleǐnik, Mathematical problems of boundary layer theory. Uspekhi Mat. Nauk 23 (1968) 3-65. | MR 228849 | Zbl 0188.41004

[35] O.A. Oleǐnik and E.V. Radkevič, Second order equations with nonnegative characteristic form. Plenum Press, New York-London (1973). | MR 457908

[36] C.M. Pagani, On an initial-boundary value problem for the equation wt = wxx - xwy. Ann. Scuola Norm. Sup. Pisa, Ser. IV 2 (1975) 219-263. | Numdam | MR 380115 | Zbl 0306.35053

[37] C.M. Pagani, On forward-backward parabolic equations in bounded domains. Boll. Un. Mat Ita. B (5) 13 (1976) 336-354. | MR 499718 | Zbl 0345.35053

[38] W.R.C. Phillips and J.T. Ratnanather, The outer region of a turbulent boundary layer, Phys. Fluids A, Fluid dynamics 2 (1990) 427. | Zbl 0708.76078

[39] 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. | MR 1313013 | Zbl 0818.76016

[40] L.B. Schiff and J.L. Steger, Numerical simulation of steady supersonic viscous flow. AIAA J. 18 (1980) 1421-1430. | MR 595898 | Zbl 0462.76058

[41] F.T. Smith, Steady and unsteady boundary-layer separation. Annu. Rev. Fluid Mech 18 (1986) 197-220. | MR 828790 | Zbl 0615.76048

[42] 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. | MR 452113 | Zbl 0381.76031

[43] K. Stewartson, Multistructural boundary layers on flat plates and related bodies. Adv. in Appl. Mech. 14 (1974) 145-239.

[44] K. Stewartson, D'Alembert's paradox. SIAM Rev. 23 (1981) 308-343. | MR 631832 | Zbl 0484.76002

[45] K. Stewartson and P.G. Williams, Self-induced separation. Proc. Royal Soc. Ser. A 312 (1969) 181-206. | Zbl 0184.52903

[46] V. Vanaja, Numerical solution of a simple Fokker-Planck equation. Appl. Numer. Math. 9 (1992) 533-540. | MR 1160631 | Zbl 0756.65145

[47] V. Vanaja and R.B. Kellogg, Iterative methods for a forward-baekward heat equation, SIAM J. Numer. Anal. 27 (1990) 622-635. | MR 1041255 | Zbl 0708.65082

[48] G. Milton Wing, An introduction to transport theory. John Wiley and Sons Inc., New York-London (1962). | MR 155646