Uniform difference method for parameterized singularly perturbed delay differential equations

Abstract

This paper deals with the singularly perturbed initial value problem for quasilinear first-order delay differential equation depending on a parameter. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform meshes on each time subinterval. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Some numerical experiments illustrate in practice the result of convergence proved theoretically.

A parameter uniform difference scheme for the parameterized singularly perturbed problem with integral boundary condition

“…The values of ε and N , for which we solve the test problem, are ε = 2 -i , i = 0, 4,8,12,16; N = 64, 128, 256, 512, 1024. From Tables 1 and 2 we observe that ε-uniform experimental rates of convergence monotonically increase towards one, which is in agreement with the theoretical rate given by Theorem 3.1.…”

“…where the truncation errors R i and r are given by (8) and (9), respectively. Before estimating errors of the approximate solution, we need the known equalities for the first order difference equation, namely, the solution of…”

“…Under these assumptions, problem (1)-(3) has a unique solution u(t). For ε 1, the function u(t) has in general a boundary layer of width O(ε) near t = 0 (see [8][9][10]). …”

“…Uniform convergent finite-difference schemes for solving parameterized singularly perturbed two-point boundary value problems have been considered in [8,10,[15][16][17][18][19][20][21][22] (see also references therein). In [8,10,16,17,19,20] authors used the boundary layer technique for solving an analogous problem. A methodology based on the homotopy analysis technique to approximate the analytic solution was investigated in [15,21,22].…”

The numerical solution of the singularly perturbed differential-difference equations based on the Meshless method

“…There exist several numerical studies for approximating the solution of singularly perturbed differential-difference equations. For example Kadalbajoo and Sharma [5][6][7][8], Kadalbajoo and Ramesh [9], Amiraliyeva and Erdogan [10], Amiraliyeva and Amiraliyev [11], Rao and Chakravarthy [12] developed robust numerical schemes for dealing with singularly perturbed differential-difference equations. In recent years, much interest of scientists and engineeres has been paid on meshless based methods, particularly moving least squares (MLS) method [13][14][15][16][17].…”