In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction–diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions...

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.

This paper deals with uniform convergent methods for solving singularly perturbed two-point boundary value problems with piecewise smooth coefficients. Construction of the numerical methods is based on locally exact schemes or on local Green’s functions. Uniform convergent properties of the proposed methods on arbitrary meshes are proven. Numerical experiments are presented.

Recommended by Donal O'Regan This paper is concerned with solving nonlinear singularly perturbed boundary value problems. Robust monotone iterates for solving nonlinear difference scheme are constructed. Uniform convergence of the monotone methods is investigated, and convergence rates are estimated. Numerical experiments complement the theoretical results.

This paper is concerned with solving nonlinear monotone difference schemes of the parabolic type. The monotone Jacobi and monotone Gauss– Seidel methods are constructed. Convergence rates of the methods are compared and estimated. The proposed methods are applied to solving nonlinear singularly perturbed parabolic problems. Uniform convergence of the monotone methods is proved. Numerical experi...

Second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered. The behavior of the ordinary Galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed.

A continuous-time anti-integrable limit is considered in the context of time-dependent Lagrangian systems on the torus. The anti-integrable limit is the singular (or adiabatic) limit in the singularly (adiabatically, resp.) perturbed problems. This paper presents an implicit function theorem version of the results of Bolotin and MacKay (1997 Nonlinearity 1

A singularly perturbed time-dependent convection-diffusion problem is examined on non-rectangular domains. The nature of the boundary and interior layers that arise depends on the geometry of the domains. For problems with different types of layers, various numerical methods are constructed to resolve the layers in the solutions and the numerical solutions are shown to converge independently of...

In this article, we consider singularly perturbed reaction-diffusion Robin boundary-value problems. To solve these problems we construct a numerical method which involves both the cubic spline and classical finite difference schemes. The proposed scheme is applied on a piece-wise uniform Shishkin mesh. Truncation error is obtained, and the stability of the method is analyzed. Also, parameter-un...

A wavelet optimized finite difference (WOFD) method is presented for adaptively solving a class of singularly perturbed elliptic and parabolic problems. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. Adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples for elliptic and parabolic proble...