A finite difference method for the singularly perturbed problem with nonlocal boundary condition

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

“…For small values of ε, the solution y(x) of problems (1)-(3) has in general boundary layers at x � 0 and x � l (see [2]). e linear ordinary differential equation (1) cannot, in general, be solved analytically because of the dependence of a(x) and b(x) on the spatial coordinate x.…”

“…Boundary value problems involving integral boundary conditions have received considerable attention in recent years [1,2]. For a discussion of existence and uniqueness results and for applications of problems with integral boundary conditions, one can refer [3][4][5] and the references therein.…”

“…For a discussion of existence and uniqueness results and for applications of problems with integral boundary conditions, one can refer [3][4][5] and the references therein. In [2,6,7], some approximating or numerical treatment aspects of this kind of problems have been considered. However, the methods or algorithms developed so far mainly concerned with the regular cases (i.e., when the boundary layers are absent).…”

“…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.…”

Solving the forced Duffing equation with integral boundary conditions in the reproducing kernel space

“…In fact, boundary value problems involving integral boundary conditions have received considerable attention in recent years [7,8,10,11,21,22,28]. It is well known that the forced Duffing equation arises in a variety of different scientific fields such as periodic orbit extraction, nonuniformity caused by an infinite domain, nonlinear mechanical oscillators, prediction of diseases, etc.…”