Abstract
Boundary value problems are considered on a simplex <em>F</em> in the real Euclidean space R<sup>2</sup>. The recent discovery of new families of special functions, orthogonal on <em>F</em>, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on <em>F</em>, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of <em>F</em> a Dirichlet condition is fulfilled and on the other Neumann’s works.
Highlights
The boundary value problems, considered in this paper, occurring in a real Euclidean space R2 on finite region F ⊂ R2 that is half of a square or half of an equilateral triangle.The main idea of this paper is to study the solutions of Helmholtz equation with the mixed boundary value problems
A surprising variety of recently emerged suitable new families of special functions makes that the realization of this idea is relatively simple and straightforward in any dimension
The general formula for special functions corresponding to the Weyl group [5] is given by σ(w)e2πi wλ|x, w∈G
Summary
The boundary value problems, considered in this paper, occurring in a real Euclidean space R2 on finite region F ⊂ R2 that is half of a square or half of an equilateral triangle. The main idea of this paper is to study the solutions of Helmholtz equation with the mixed boundary value problems. The boundary value conditions play an important role in describing the physical phenomena. They are used, inter alia, in the theory of elasticity, electrostatics and fluid mechanics [2, 4, 16]. For the case A1 × A1, there is no hybrid functions, the mixed boundary value problem occurs. We present this case in details in Appendix
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