# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems. (English) Zbl 1157.65047

The authors study numerical method for solving the self-adjoint boundary value problem of a singularly perturbed second order linear differential equation

$Lu\equiv \epsilon {u}^{\text{'}\text{'}}\left(x\right)+q\left(x\right)u\left(x\right)=f\left(x\right),\phantom{\rule{1.em}{0ex}}0\le x\le 1,$

with homogeneous boundary conditions $u\left(0\right)=u\left(1\right)=0$, where $\epsilon$ is a small parameter and $f\left(x\right),\phantom{\rule{4pt}{0ex}}q\left(x\right)$ are smooth functions and satisfy $q\left(x\right)\ge {q}_{*},\phantom{\rule{0.166667em}{0ex}}\forall x\in \left[0,1\right]$ for some positive constant ${q}_{*}$. For small $\epsilon$, the solution $u\left(x\right)$ may exhibit exponential boundary layers at both ends of the interval $\left[0,1\right]$, 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 M. K. Kadalbajoo and V. K. Aggarwal [Appl. Math. Comput. 161, No. 3, 973–987 (2005; Zbl 1073.65062)].

##### MSC:
 65L10 Boundary value problems for ODE (numerical methods) 65L20 Stability and convergence of numerical methods for ODE 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 34B05 Linear boundary value problems for ODE 34E15 Asymptotic singular perturbations, general theory (ODE)