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Zbl 1157.65047
Rao, S.Chandra Sekhara; Kumar, Mukesh
Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems.
(English)
[J] Appl. Numer. Math. 58, No. 10, 1572-1581 (2008). ISSN 0168-9274

The authors study numerical method for solving the self-adjoint boundary value problem of a singularly perturbed second order linear differential equation $$Lu \equiv \varepsilon u''(x)+q(x)u(x)=f(x),\quad 0\leq x\leq 1,$$ with homogeneous boundary conditions $u(0)=u(1)=0$, where $\varepsilon$ is a small parameter and $f(x),\ q(x)$ are smooth functions and satisfy $q(x)\geq q_*,\, \forall x\in [0,1]$ for some positive constant $q_*$. For small $\varepsilon$, the solution $u(x)$ may exhibit exponential boundary layers at both ends of the interval $[0,1]$, which gives rise to difficulties in numerical solutions. Using a B-splines basis for the space of exponential spline, the authors propose a spline collocation method which leads to an easily solvable tridiagonal algebraic system. It is shown that the method can be implemented on uniform meshes, i.e. there is no need of introducing more nodal points in the boundary layers. Further, the second order uniform convergence of the numerical solution is proven. The efficiency of the method is demonstrated by several numerical experiments where the present method is shown to be superior to the scheme using cubic B-splines on fitted meshes that was suggested previously by {\it M. K. Kadalbajoo} and {\it V. K. Aggarwal} [Appl. Math. Comput. 161, No.~3, 973--987 (2005; Zbl 1073.65062)].
[Vu Hoang Linh (Hanoi)]
MSC 2000:
*65L10 Boundary value problems for ODE (numerical methods)
65L20 Stability of numerical methods for ODE
65L60 Finite numerical methods for ODE
34B05 Linear boundary value problems of ODE
34E15 Asymptotic singular perturbations, general theory (ODE)

Keywords: singular perturbation; exponential B-spline; collocation method; uniform mesh; self-adjoint boundary value problem; second order linear differential equation; exponential boundary layers; convergence; numerical experiments

Citations: Zbl 1073.65062

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