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The Conformal Mapping Method for the Helmholtz Equation

In: Integral Methods in Science and Engineering, Volume 1

Author

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  • N. Khatiashvili

    (Iv. Javakhishvili Tbilisi State University)

Abstract

The Helmholtz equation describes a lot of physical processes. For example, in quantum chaos some model systems are described by the Helmholtz equation with appropriate boundary conditions. One of them is the quantum billiard problem (see [Bu01], [Gr01], [Gu90], [KoSc97], [Si00], and [Si70]). Generic billiards are one of the simplest examples of conservative dynamical systems with chaotic classical trajectories. According to this model, the particle is trapped inside the simply corrected region D with the boundary S, in which it can move freely and this movement is ballistic. In this case, the Schrödinger equation for a free particle assumes the form of the Helmholtz equation (see [Gr01], [Gu90], [Si00], and [Si70]). This chapter deals with the two-dimensional homogeneous problem for the Helmholtz equation in the finite domain D with the boundary S. The following problem is considered.

Suggested Citation

  • N. Khatiashvili, 2010. "The Conformal Mapping Method for the Helmholtz Equation," Springer Books, in: Christian Constanda & M.E. Pérez (ed.), Integral Methods in Science and Engineering, Volume 1, chapter 17, pages 173-177, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4899-2_17
    DOI: 10.1007/978-0-8176-4899-2_17
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