Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems
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 \textit{M. K. Kadalbajoo} and \textit{V. K. Aggarwal} [Appl. Math. Comput. 161, No.~3, 973--987 (2005; Zbl 1073.65062)].
second order linear differential equation
scientific article; zbMATH DE number 5349253
scientific article