Solution of the modified Helmholtz equation using mixed boundary conditions in an equilateral triangle

Abstract

The modified Helmholtz equation $q_{xx}+q_{yy}-4{\beta }^{2}q=0$, is one of the basic equations of classical mathematical physics. In this paper we have obtained the solution of the boundary-value problems for the modified Helmholtz equation in an equilateral triangle. An additional mixed boundary condition related to the symmetry of the solution is taken into consideration. We have analysed the Global relation and only used the algebraic techniques to obtain the explicit solution of modified Helmholtz equation bypassing the Riemann Hilbert approach. This solution is applied to the problem of diffusion-limited coalescence, $A+A\rightleftharpoons A$.