Fourier-Transform Method for Partial Differential Equations: Formulas for Representing Solutions to the Cauchy Problem

The paper proposes a method for solving the Cauchy problem for linear partial differential equations with variable coefficients of a special form, allowing, after applying the (inverse) Fourier transform, the original problem to be rewritten as a Cauchy problem for first-order partial differential equations. The resulting problem is solved by the method of characteristics and the (direct) Fourier transform is applied to its solution. For this purpose, it is necessary to know the solution of the Cauchy problem for a first-order equation in the entire domain. This leads to the requirement for the support of the (inverse) Fourier transform of the initial function of the original problem to be compact, and, to describe the class of initial functions, it is necessary to use Paley-Wiener-Schwartz-type theorems on Fourier images, including distributions. The presentation of solutions in the form of the Fourier transform of some function (distribution), determined by the initial function, is presented. The general form of the evolutionary equation is written, which, when the described method is applied, leads to the consideration of a homogeneous first-order equation, and a formula for solution of the Cauchy problem in this general case is derived. The general form of the equation is presented, which leads to consideration of a first-order inhomogeneous equation, and a formula for its solutions is derived. Particular cases of these equations are well-known equations that are encountered in the description of various processes in physics, chemistry, and biology.