A uniformly convergent finite difference method for a singularly perturbed initial value problem

Abstract

Abstract:Initial value problem for linear second order ordinary differential equation with small parameter by the first and second derivatives is considered. An exponentially fined difference scheme with constant fitting factors is developed in a uniform mesh, which gives first-order uniform convergence in the sense of discrete maximum norm. Numerical results are also presented.

Uniform Difference Scheme on the Singularly Perturbed System

“…Uniform convergence is proved in the discrete maximum norm. The approach to the construction of the discrete problem and the error analysis for the approximate solution are similar to those in [8,9].…”

“…These include fitted finite difference methods, finite element methods using special elements such as exponential elements, and methods which use a priori refined or special non-uniform grids which condense in the boundary layers in a special manner. The various approaches to the design and analysis of appropriate numerical methods for singularly perturbed differential equations can be found in [3][4][5][6][7][8] (see also references cited in them).…”

Finite‐difference method for parameterized singularly perturbed problem

“…In Section 4, we formulate the iterative algorithm for solving the discrete problem and give the illustrative numerical results. The approach to construct discrete problem and error analysis for approximate solution is similar to those ones from [1,2,3].…”

Fitted finite difference method for singularly perturbed delay differential equations

“…The difference scheme is constructed by the method of integral identities with the use of exponentially basis functions and interpolating quadrature rules with weight and remainder terms integral form [11,12]. This method of approximation has the advantage that the schemes can also be effective in the case when the continuous problem is considered under certain restrictions.…”

“…methods that are uniformly convergent with respect to the perturbation parameter [8][9][10]. One of the simplest ways to derive such methods consists of using exponentially fitted difference schemes (see, e.g., [8][9][10][11][12] for motivation for this type of mesh). In the direction of numerical treatment for first order singularly perturbed delay differential equations, several can be seen in [13][14][15][16].…”