Immersed boundary method: The existence of approximate solution of the two-dimensional heat equation

This paper deals with the heat equation in which the source term involves a Dirac function and describes the elastic reaction of the immersed boundary. We analyze the existence of the approximate solution in two-dimensional case with Dirac function approximated by differentiable function. We obtain the result via finite element method, the Banach Fixed-Point Theorem and a theorem in nonlinear ordinary differential equations in abstract space.