A numerical treatment for singularly perturbed differential equations with integral boundary condition

Accelerated Exponentially Fitted Operator Method for Singularly Perturbed Problems with Integral Boundary Condition

“…As a result, numerical analysis of singular perturbation cases has been far from trivial because of the boundary layer behavior of the solution. e solutions of the problems with boundary layer undergo rapid changes within very thin layers near the boundary or inside the problem domain [2,[8][9][10], and hence classical numerical methods for solving such problems are unstable and fail to give good results when the perturbation parameter is small (i.e., for h ≥ ε) [10]. erefore, it is important to develop a numerical method that gives good results for small values of the perturbation parameter where others fails to give good result and convergent independent of the values of the perturbation parameter.…”

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

“…Proof One can prove this result following the method given in [9], Lemma 2.1, and in [10], Lemma 2.1.…”

“…Note that, in [10], the first order convergent difference scheme in Bakhvalov type mesh under the first type boundary conditions for equation (1.1) was presented. Also, in the above-mentioned work [9] that includes integral boundary condition, while conditions (2.1) and (4.8) are generally provided for sufficiently small values of ε, as the integral boundary condition of our work is more general, and the convergence is uniform for both small and moderate values of perturbation parameter ε. The paper is organized as follows.…”

“…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]). …”

“…In recent years, many researchers have considered the singularly perturbed cases for these problems. In [9] authors developed a finite difference scheme on Shishkin mesh for a problem with integral boundary conditions and proved that the method is nearly first order convergent except for a logarithmic factor. A hybrid scheme, which is second order convergent on Shishkin mesh, was discussed in [30] (see also [20,33]).…”