The Fourier Transform Method

Abstract

The Fourier transform has been known as one of the most powerful and useful methods of finding fundamental solutions for operators with constant coefficients. Sometimes the application of a partial Fourier transform might be more useful than the full Fourier transform. In this chapter, by the application of the partial Fourier transform, we shall reduce the problem of finding the heat kernel of a complicated operator to a simpler problem involving an operator with fewer variables. After solving the problem for this simple operator, the inverse Fourier transform provides the heat kernel for the initial operator represented under an integral form. In general, this integral cannot be computed explicitly, but in certain particular cases it actually can be worked out. We shall also apply this method to some degenerate operators.