Singularly perturbed problems of convection-diiusion type are dif-cult to solve using standard numerical methods. This shows that it is desirable to construct robust methods that converge uniformly in the singular perturbation parameter. In this paper we derive new conditions that schemes of this type must satisfy, with particular reference to the case of parabolic layers.

The numerical solution of boundary value problems for certain stiff ordinary differential equations is studied. The methods developed use singular perturbation theory to construct approximate numerical solutions which are valid asymptotically; hence, they have the desirable feature of becoming more accurate as the equations become stiffer. Several numerical examples are presented which demonstr...

In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.

We examine the local convergence of the Levenberg–Marquardt method for the solution of nonlinear least squares problems that are rank-deficient and have nonzero residual. We show that replacing the Jacobian by a truncated singular value decomposition can be numerically unstable. We recommend instead the use of subset selection. We corroborate our recommendations by perturbation analyses and num...

In this paper, by using the theories and methods of mathematical analysis and computer algebra, a reliable algorithm of reduction of order approximation method for solving singular perturbation problems was established, a new Maple procedure redordproc was established, too. The procedure redordproc give not only the approximate analytic solutions but also the numerical solutions of the kinetics...

The problems of swinging up an inverted pendulum on a cart and controlling it around the upright position have been traditionally treated as two separate problems. This paper proposes a strategy that is globally asymptotically stabilizing under actuator saturation and, in addition, locally exponentially stabilizing. The proposed methodology performs swing up and control simultaneously and uses ...

We apply the modified variational iteration method MVIM for solving the singular and nonsingular initial and boundary value problems in this paper. The proposed modification is made by introducing Adomian’s polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution wi...

A class of boundary value problems for nonlinear diiusion equations is studied. Using singular perturbation theory and matched asymptotic expansions, the author analyzes the interactions of compact-support solutions of the porous media equation with xed boundaries and with other solutions. The boundary layer analysis yields results on how \stopping" and \merging" disturbances at the interface p...

We consider the approximation of a coupled system of two singularly perturbed reaction-diffusion equations by the finite element method. The solution to such problems contains boundary layers which overlap and interact, and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We pres...

This paper is devoted to the numerical resolution of an anisotropic non-linear diffusion problem involving a small parameter ǫ, defined as the anisotropy strength reciprocal. In this work, the anisotropy is carried by a variable vector function b. The equation being supplemented with Neumann boundary conditions, the limit ǫ→0 is demonstrated to be a singular perturbation of the original diffusi...