Abstract
We consider a singularly perturbed convection—diffusion equation, −∈Δu+v⋅∇ u=0, defined on a half-infinite strip, (x,y)∈(0,∞)×(0,1) with a discontinuous Dirichlet boundary condition: u(x,0)=1, u(x,1)=u(0,y)=0. Asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) as the singular parameter ∈→0+ (with fixed distance r to the discontinuity point of the boundary condition) and (b) as that distance r→0+ (with fixed ∈). It is shown that the first term of the expansion at ∈=0 contains an error function or a combination of error functions. This term characterizes the effect of discontinuities on the ∈-behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the point of discontinuity of the boundary condition, the solution u(x,y) is approximated by a linear function of the polar angle at the point of discontinuity (0,0).
Similar content being viewed by others
References
Abramowitz, M. and Stegun, I. A. (1970) Handbook of Mathematical Functions, Dover, New York.
Aziz, A. and Na, T. Y. (1984) Perturbation Methods in Heat Transfer, Hemisphere Publishing Corporation, New York.
Bejan, A. (1984) Convection Heat Transfer, Wiley, New York.
Cook, L. P., Ludford, G. S. S. and Walker, J. S. (1972) Corner regions in the asymptotic solution of ∈ ∇2 u = ∂u/∂y with reference to MHD duct flow, Proc. Cambridge Philos. Soc. 72, 117–121.
Cook, L. P. and Ludford, G. S. S. (1973) The behavior as ∈ → 0+ of solutions to ∈ ∇2 w = ∂w/∂y on the rectangle 0 ⩽ x ⩽ l, |y| ⩽ 1, SIAM J. Math. Anal. 4(1), 161–184.
Eckhaus, W. (1973) Matched Asymptotic Expansions and Singular Perturbations, North-Holland, Amsterdam.
Eckhaus, W. and de Jager, E. M. (1966) Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rational Mech. Anal. 23, 26–86.
Gold, R. R. (1962) Magnetohydrodynamic pipe flow. Part I, J. Fluid Mech. 13, 505–512.
Grasman, J. (1974) An elliptic singular perturbation problem with almost characteristic boundaries, J. Math. Anal. Appl. 46, 438–446.
Hedstrom, G. W. and Osterheld, A. (1980) The effect of cell Reynolds number on the computation of a boundary layer, J. Comput. Phys. 37, 399–421.
Hemker, P. W. and Shishkin, G. I. (1993) Approximation of parabolic PDEs with a discontinuous initial condition, East-West J. Numer. Math. 1(4), 287–302.
Il'in, A. M. (1992) Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Amer. Math. Soc., Providence.
Kevorkian, J. (1990) Partial Differential Equations. Analytical Solution Techniques, Springer-Verlag, New York.
Kevorkian, J. and Cole, J. D. (1996) Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York.
Knowles, J. K. and Messic, R. E. (1964) On a class of singular perturbation problems, J. Math. Anal. Appl. 9, 42–58.
López, J. L. (2000) Asymptotic expansions of symmetric standard elliptic integrals, SIAM J. Math. Anal. 31(4), 754–775.
Lu, P.-C. (1973) Introduction to the Mechanics of Viscous Fluids, Holt, Rinehart and Winston, New York.
Morton, K. W. (1996) Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, London.
O'Malley, R. E. (1974) Introduction to Singular Perturbation, Academic Press, New York.
Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1990) Integrals and Series, Vol. 1, Gordon and Breach Science Pub.
Shercliff, J. A. (1953) Steady motion of conducting fluids in pipes under transverse magnetic fields, Proc. Cambridge Philos. Soc. 49, 136–144.
Shih, S.-D. (1996) A Novel uniform expansion for a singularly perturbed parabolic problem with corner singularity, Meth. Appl. Anal. 3(2), 203–227.
Temme, N. M. (1971) Analytical methods for a singular perturbation problem. The quarter plane, CWI Report, 125.
Temme, N. M. (1974) Analytical methods for a singular perturbation problem in a sector, SIAM J. Math. Anal. 5(6), 876–887.
Temme, N. M. (1998) Analytical methods for a selection of elliptic singular perturbation problems, In: Recent Advances in Differential Equations (Kunming, 1997), Pitman Res. Notes Math. Ser. 386, Longman, Harlow, pp. 131–148.
Van Dyke, M. (1964) Perturbation Methods in Fluid Dynamics, Academic Press, New York.
Vasil'eva, A. B., Butuzov, V. F. and Kalachev, L. V. (1995) The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia.
Wong, R. (1989) Asymptotic Approximations of Integrals, Academic Press, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
López, J.L., Pérez Sinusía, E. Analytic Approximations for a Singularly Perturbed Convection—Diffusion Problem with Discontinuous Data in a Half-Infinite Strip. Acta Applicandae Mathematicae 82, 101–117 (2004). https://doi.org/10.1023/B:ACAP.0000026696.46018.d6
Issue Date:
DOI: https://doi.org/10.1023/B:ACAP.0000026696.46018.d6