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B-spline collocation method for the singular-perturbation problem using artificial viscosity. (English) Zbl 1165.65371
Summary: We develop a B-spline collocation method using artificial viscosity for solving singularly-perturbed equations given by $ϵ{u}^{\text{'}\text{'}}\left(x\right)+a\left(x\right){u}^{\text{'}}\left(x\right)+b\left(x\right)u\left(x\right)=f\left(x\right)$, $a\left(x\right)\ge {a}^{*}>0$, $b\left(x\right)\ge {b}^{*}>0$, $u\left(0\right)=\alpha$, $u\left(1\right)=\beta$. We use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh and use B-spline collocation method which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. The design of artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behaviour of the method with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.
##### MSC:
 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE