The Heat Equation in the Interior of an Equilateral Triangle

We present the solution of the classical problem of the heat equation formulated in the interior of an equilateral triangle with Dirichlet boundary conditions. This solution is expressed as an integral in the complex Fourier space, i.e., the complex k₁ and k₂ planes, involving appropriate integral transforms of the Dirichlet boundary conditions. By choosing Dirichlet data so that their integral transforms can be computed explicitly, we show that the solution is expressed in terms of an integral whose integrand decays exponentially as . Hence, it is possible to evaluate this integral numerically in an efficient and straightforward manner. Other types of boundary value problems, including the Neumman and Robin problems, can be solved similarly.