High-frequency asymptotics for the modified Helmholtz equation in a quarter-plane

Abstract

We study the high-frequency asymptotics for the solution of the modified Helmholtz equation, in a quarter plane and subject to a specific Neumann condition, as an illustration of the procedure to extract the asymptotics via Fokas' transform method. The approach is based on Fokas' integral representation of the solution. Full asymptotic expansions of the solution are obtained by using Watson's lemma and the method of steepest descents for definite integrals.

Keywords:

• high-frequency asymptotics
• Fokas' integral representation
• method of steepest descents
• modified Helmholtz equation
• Neumann condition

AMS Subject Classifications::

• 35B40
• 35C15
• 35J05

Acknowledgements

This research was supported in part by the National Natural Science Foundation of China under grant numbers 10471154 and 10871212.