A brief survey on numerical methods for solving singularly perturbed problems

Preface 1. Asymptotics of Laplace-type integrals 2. Hadamard expansion of Laplace integrals 3. Hadamard expansion of Laplace-type integrals 4. Applications Appendix A Appendix B Appendix C References Index.

Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents

Laplace's method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion, arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron's formula gives them in terms of derivatives of an explicit function; Campbell, Fr\"oman and Walles simplified Perron's method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fr\"oman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients, which contains ordinary potential polynomials. The proof is based on Perron's formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.

An explicit formula for the coefficients in Laplace's method

Homoclinic connections of unstable planewaves are constructed for a novel integrable generalized nonlinear Schrodinger equation studied by Lenells and Fokas (2009 Nonlinearity 22 11–27). Three instability regions are identified and compared using explicit examples in the case of a single unstable wavenumber.

http://iopscience.iop.org/article/10.1088/0951-7715/22/11/003/pdf

Some homoclinic connections of a novel integrable generalized nonlinear Schrödinger equation

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Froman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Froman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.

An Explicit Formula for the Coefficients in Laplace’s Method

A simplification of Laplace's method: Applications to the Gamma function and Gauss hypergeometric function

Asymptotic behaviour of three-dimensional singularly perturbed convection-diffusion problems with discontinuous data

A three-dimensional elliptic singular perturbation problem with discontinuous boundary values is considered. The solution of the problem is written in terms of a double integral. A saddle point analysis is used to obtain a first approximation, which is expressed in terms of a function that can be viewed as a generalization of the complementary error function

First order approximation of an elliptic 3D singular perturbation problem

We show the importance of the error function in the approximation of the solution of singularly perturbed convection-diffusion problems with discontinuous boundary conditions. It is observed that the error function (or a combination of them) provides an excellent approximation and reproduces accurately the effect of the discontinuities on the behaviour of the solution at the boundary and interior layers.

/pdf/the-error-function-in-the-study-of-singularly-perturbed-3by7423vdm.pdf

E. Pérez Sinusía

Papers

A simplification of Laplace's method: Applications to the Gamma function and Gauss hypergeometric function

Asymptotic behaviour of three-dimensional singularly perturbed convection-diffusion problems with discontinuous data

First order approximation of an elliptic 3D singular perturbation problem

The Error Function in the Study of Singularly Perturbed Convection-Diffusion Problems with Discontinuous Boundary Data