Method of matched asymptotic expansions

About: Method of matched asymptotic expansions is a research topic. Over the lifetime, 4233 publications have been published within this topic receiving 73311 citations.

Asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations

Abstract: We study asymptotic behavior of solutions of a class of second order quasilinear ordinary differential equations. All solutions are classified into six types by means of their asymptotic behavior. Necessary and/or sufficient conditions are given for such equations to possess a solution of each of the six types.

Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing

Abstract: The method of averaging and the two-time procedure are outlined for a class of hyperbolic second-order partial differential equations with small nonlinearities. It is shown that they both lead to the same integro-differential equation for the lowest-order approximation to the solution. For the special case of the nonlinear wave equation, this lowest-order solution consists of the superposition of two modulated travelling waves, and separate integro-differential equations are derived for the amplitudes of these two waves. As an example, the wave equation with Van der Pol type of nonlinearity is considered.

Singular Perturbation Techniques: A Comparison of the Method of Matched Asymptotic Expansions with that of Multiple Scales

Abstract: A comparison is made between the uniformly valid asymptotic representations which can be developed for the solution of a singular perturbation boundary value problem involving a linear second order differential equation by using both the technique of matched asymptotic expansions and the method of multiple scales. Next, there is a discussion of some of the subtle features as well as the relative advantages, limitations, logical extensions, and typical applications of these two methods of obtaining uniformly valid asymptotic representations when applied to slightly more general singular perturbation problems which arise from investigations of various phenomena in the natural sciences. Finally, a parameter identification example relevant to biological population dynamics is presented to illustrate the fact that the mere knowledge of these singular perturbation techniques can be a powerful analytical tool.

Sharp interface limits for diffuse interface models

Abstract: In this thesis we rigorously prove that the Cahn-Larche system converges to a modified Hele-Shaw problem. For the proof we construct an approximate solution of the Cahn-Larche system by the method of matched asymptotic expansions. Then we can show that the approximate solutions of Cahn-Larche system converge to the solution of the modified Hele-Shaw problem. For the modified Hele-Shaw problem we prove the existence of a classical solution in a sufficiently small time interval [0;T]. By reducing the system to a single evolution equation for the distance function, we show the assertion. Furthermore, we prove an existence result for classical solution to a linearized Hele-Shaw problem used in the higher order expansions. By the same methods as for the Cahn-Larche system, we show the sharp interface limit of a convective Cahn-Hilliard equation to an evolution equation for the interface. Here and for the Cahn-Larche system the main problem is the construction of the approximate solutions. Finally, we obtain that the surface tension term in the “model H” with mobility constant converging sufficiently fast to 0 does generally not converge to the mean curvature of the interface.