A new transform method I: domain-dependent fundamental solutions and integral representations

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In this paper, we present a simplified way of obtaining these representations for elliptic PDEs; namely, we introduce an algorithm for constructing particular, domain-dependent, IRs of the associated fundamental solutions, which are then substituted into Green's IRs. Furthermore, we extend this new method from BVPs in polygons to BVPs in polar coordinates. In the sequel to this paper, these results are used to solve particular BVPs, which elucidate the fact that this method has substantial advantages over the classical transform method.