Abstract

The solution of partial differential equations has been studied by several scientists since the nineteenth century. Several intuitive attempts have been made using constructive methods such as the variable separation method. A more general study was carried out by George Green in 1828 for the needs of electromagnetism. He was able to develop a general study of elementary solution or fundamental solution; of a linear differential equation with constant coefficients, or of a linear partial differential equation with constant coefficients. This made it possible to show that the solutions have a particular integral writing and form an affine space. Thus the general solution of a PDE is the sum of a particular solution and the solution of the homogeneous equation, i.e. the second member is zero. This article presents demonstrations of such results.