A new method for solving a class of singular two-point boundary value problems. (English) Zbl 1161.65060

Wang, Wenyan; Cui, Minggen; Han, Bo

Appl. Math. Comput. 206, No. 2, 721-727 (2008).

A solution for the two-point boundary value problem
\[ Lu\equiv u^{\prime\prime}(x)+(k/x)u^{\prime}(x)+b(x)u(x) =f(x) \]
with conditions \( u\prime(0)=0, u(1)= 1 \) is presented in form of a series in the special space \(W=W_{2}^{3}[0,1]\), named reproducing kernel space. The construction of an orthonormal system in \( W \) uses the adjoint operator \(L^{*}\), that is why not simple. This approximate method has an analytical form , but is not numerical.

Reviewer: Ivan Secrieru (Chişinău)

MSC:

65L10	Numerical solution of boundary value problems involving ordinary differential equations
34B05	Linear boundary value problems for ordinary differential equations
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

analytical approximate method; orthonormal basis; reproducing kernel; two-point boundary value problem

References:

[1]	Kelevedjiev, P., Existence of positive solutions to a singular second order boundary value problem, Nonlinear anal., 50, 1107-1118, (2002) · Zbl 1014.34013
[2]	Wong, F.; Lian, W., Positive solution for singular boundary value problems, Comput. math. appl., 32, 9, 41-49, (1996) · Zbl 0868.34019
[3]	Liu, Y.; Yu, H., Existence and uniqueness of positive solution for singular boundary value problem, Comput. math. appl., 50, 133-143, (2005) · Zbl 1094.34015
[4]	Xu, X.; Ma, J., A note on singular nonlinear boundary value problems, J. math. anal. appl., 293, 108-124, (2004) · Zbl 1057.34007
[5]	Kadalbajoo, M.K.; Aggarwal, V.K., Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline, Appl. math. comput., 160, 851-863, (2005) · Zbl 1062.65077
[6]	Kanth, A.S.V.R.; Reddy, Y.N., Higher order finite difference method for a class of singular boundary value problems, Appl. math. comput., 155, 249-258, (2004) · Zbl 1058.65078
[7]	Kanth, A.S.V.R.; Reddy, Y.N., Cubic spline for a class of singular boundary value problems, Appl. math. comput., 170, 733-740, (2005) · Zbl 1103.65086
[8]	Mohanty, R.K.; Sachder, P.L.; Jha, N., An O(h4) accurate cubic spline TAGE method for nonlinear singular two point boundary value problem, Appl. math. comput., 158, 853-868, (2004) · Zbl 1060.65080
[9]	Cui, Minggen; Geng, Fazhan, Solving singular two-point boundary value problem in reproducing kernel space, J. comput. appl. math., 205, 6-15, (2007) · Zbl 1149.65057