Spline approach to the solution of a singularly-perturbed boundary-value problems

Numerical Solution of Singularly Perturbed Boundary Value Problems Based on Optimal Control Strategy

“…It was proved that for every δ > 0, there exists an approximate solution v δ of B-spline functions such that the least square error of v δ is less than δ > 0, and also that v δ satisfies the exact boundary conditions. Due to the amount of computations required, only second order differential equations with boundary conditions have been considered in the literature [5,6,9,19,21]. However, our method converges for problems with higher order equations as shown in the examples.…”

“…There are many known methods for solving problems in the form of (1) and (2), such as collocation methods [2][3][4], finite difference methods [11], finite element methods [7,9,18,25,26], boundary value techniques [1,5,22], spline methods [10,15,20,21,24,27], and so on. In recent years, a large number of special purpose methods have been developed to provide accurate numerical solutions.…”

Numerical Solution of Singular Perturbation Problems Via Deviating Argument and Exponential Fitting

“…Rao and Kumar [10] presented an exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problem and the method is shown to have second order uniform convergence. Rashidinia et al [11] used spline in compression to develop the numerical methods for singularly perturbed two-point boundary-value problem. They discussed the convergence analysis of the method and show that proposed methods are second-order and fourth order accurate and applicable to problems both in singular and non-singular cases.…”

High accuracy non-equidistant method for singular perturbation reaction-diffusion problem

“…A streamline diffusion method, which is uniformly convergent with order 2, is presented also by Stynes and Tobiska [11] . Rashidiniab et al [12] constructed a high order spline approach method. However, the stability was not discussed in his paper.…”