Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 22

Throughout the book, we focus on the study of some of the basic PDEs of mathematical physics. These equations involve differential operators as the gradient and Laplacian, acting on :

(1.5.1)

These equations describe fundamental physical phenomena as follows:

(1.5.2)

Here is a given function. In the special case, when , i.e. when Eq. (1.5.2) are homogeneous, the first equation is called the Laplace equation and, being time‐independent, has some special features: besides the fact that it describes the steady‐state heat transfer and the standing wave equations (loosely speaking, the time‐independent versions of the other two equations), the Laplace equation arises in describing several physical phenomena such as electrostatic potential in regions with no electric charge, gravitational potential in the regions with no mass distribution, in elasticity, etc.