A computational technique based on asymptotic analysis for solving singularly perturbed ODE systems involving a small parameter is considered. The focus is on second-order systems, but the procedure is also applicable for first-order systems. Both initial value and boundary value problems will be solved. The application of the method is considered over the entire time domain for a wide range of...

The singularly perturbed boundary blow-up problem −ε2∆u = u(u− a)(1− u) u > 0 in B, u = ∞ on ∂B is studied in the unit ball B ⊂ R (N ≥ 2), a ∈ (1/2, 1) is a constant. It is shown that there exist exactly three positive solutions for the problem and all of them are radially symmetric solutions.

We revisit a textbook example of singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows wealth phenomena that seem to have been overlooked previously, including pitchfork bifurcation in the number solutions as one varies small parameter, and transcendentally terms initial conditions can be calculated by elementary means. Based on our own classroom experience, we believe t...

The numerical solution of a singularly perturbed semilinear reaction-diffusion two-point boundary value problem is addressed. The method considered is adaptive movement of a fixed number (N +1) of mesh points by equidistribution of a monitor function that uses discrete second-order derivatives. We extend the analysis by Kopteva & Stynes (2001) to a new equation and a more intricate monitor func...

For the first time, some hypersingular nonlinear boundary-value problems with a small parameter ε at the highest derivative are described. These problems essentially (qualitatively and quantitatively) differ from the usual linear and quasilinear singularly perturbed boundary-value problems and have the following unusual properties: (i) in hypersingular boundary-value problems, super thin bounda...

In this paper a viral evolution model with specific immune response is considered. By introducing of dimensionless variables and parameters this model can be modified to the singularly perturbed system of partial integro-differential equations with two small parameters. The transition from the initial-boundary value problem of the initial system to the generating problem makes it possible to re...

We consider a singularly perturbed convection-diffusion boundary value problem whose solution contains exponential and characteristic layers. The is numerically solved by the FEM SDFEM method with bilinear elements on graded mesh. For we prove almost uniform convergence superconvergence. use of mesh allows for to yield estimates in SD norm, which not possible Shishkin type meshes. Numerical res...

It has been demonstrated that the ordinary boundary layer elements play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element ...

A numerical method based on cubic spline with exponential fitting factor is given for the selfadjoint singularly perturbed two-point boundary value problems. The scheme derived in this method is second-order accurate. Numerical examples are given to support the predicted theory.