## Parameter-uniform fitted operator B-spline collocation method for self-adjoint singularly perturbed two-point boundary value problems

Mohan K. Kadalbajoo and Devendra Kumar

### Abstract

In this paper, we develop a B-spline collocation method for the numerical solution of a self-adjoint singularly perturbed boundary value problem of the form $-\epsilon(a(x)y')'+b(x)y(x)=f(x),\quad a(x)\geq a^*>0,\;b(x)\geq b^*>0,\;a'(x)\geq 0,\quad y(0)=\alpha,\quad y(1)=\beta.$ We construct a fitting factor and use the B-spline collocation method, which leads to a tridiagonal linear system. The method is analyzed for parameter-uniform convergence. Several numerical examples are reported which demonstrate the efficiency of the proposed method.

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### Key words

B-spline collocation method, self-adjoint singularly perturbed boundary value problem, parameter-uniform convergence, boundary layer, fitted operator method

### AMS subject classifications

34D15, 30E25, 20B40

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