Asymptotic and Numerical Approximation of a Nonlinear Singular Boundary Value Problem



In this work, we consider a singular boundary value problem for a nonlinear second-order differential equation of the form g00(u) = ug(u)q=q; (0.1) where 0 < u < 1 and q is a known parameter, q < 0. We search for a positive solution of (0.1) which satisfies the boundary conditions g0(0) = 0; (0.2) lim u!1¡ g(u) = lim u!1¡ (1 ¡ u)g0(u) = 0: (0.3) We analyse the asymptotic properties of the solution of (0.1)-(0.3) near the singularity, depending on the value of q. We show the existence of a one-parameter family of solutions of equation (0.1) which satisfy the boundary condition (0.3) and obtain convergent or asymptotic expansions of these solutions.


[1] C.M. Bender and S. Orszag, “Advanced Methods for Scientists and Engineers”, New York, Mc Graw-Hill, 1978.

E.S. Birger and N.B. Lyalikova (Konyukhova), On finding the solutions for a given condition at infinity of certain systems of ordinary differential equations, II, U.S.S.R. Comput. Maths. Math. Phys., 6 (1966), N.3, 47–57.

E.A. Coddington and N.Levinson, “Theory of Ordinary Differential Equations”, Mc Graw-Hill, New York, 1955.

A.L. Dyshko, M.P. Carpentier, N.B. Konyukhova and P.M. Lima, Singular problems for Emden-Fowler-type second-order nonlinear ordinary differential equations, Comp. Maths. Math. Phys., 41 (2001), N.4 ,557-580.

N.B.Konyukhova, Singular Cauchy problems for systems of ordinary differential equations, U.S.S.R. Comput. Maths. Math. Phys., 23 (1983), N.3, 72-82.

N.B. Konyukhova, On numerical isolation of the solutions tending do zero at infinity of certain two-dimensional non-linear sets of ordinary differential equations, U.S.S.R. Comput. Maths. Math. Phys., 10 (1970), N.1, 95-111.

N.B. Konyukhova, Stationary Lyapunov problem for a system of first-order quasilinear partial differential equations, Diff. Eq., 30 (1994), N.8, 1284-1294.

N.B. Konyukhova, Stable Lyapunov manifolds for autonomous systems of nonlinear ordinary differential equations, Comput. Maths Math. Phys., 34 (1994), N.10, 1179–1195.

P.M. Lima and M.P. Carpentier, Iterative methods for a singular boundaryvalue problem, J. Comp. Appl. Math.,111 (1999) 173-186.

P.M. Lima and M.P. Carpentier, Numerical solution of a singular boundaryvalue problem in non-Newtonian fluid mechanics, Comp. Phys. Commun., 126(2000), N.1/2, 114-120.

C.D. Luning and W.L. Perry, An iterative method for solution of a boundary value problem in non-newtonian fluid flow J. Non-Newtonian Fluid Mechanics, 15 (1984), 145-154.

A.M. Lyapunov, “Probl`eme Général de la Stabilité du Mouvement”, Kharkov, 1892; reprint, GITTL, Moscow, 1950; French. transl. Ann.Fac.Sci.Univ.Toulouse (2) 9 (1907), 203-474; reprint Ann.of Math. Studies, vol.17, Princeton Univ.Press, Princeton, N.J.,1947.

A. Nachman and A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38 (1980), 275-281.

W.Wasov, “Asymptotic Expansions for Ordinary Differential Equations”, New York, Wiley, 1965.

S. Wolfram, “The Mathematica Book”, Cambridge Univ. Press, 1996.


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