Mathematical Physics Vol 1

Chapter 7. Partial differential equations

334

be emphasized that approximate (numerical) methods for solving PDE have advanced a lot in recent decades. In this chapter, some of the more important partial differential equations that appear in technology will be discussed. We will develop some of these equations and show one of the methods for solving them (Fourier method of separation of variables). As solving a partial equation can become more complicated, when the initial or boundary conditions are changed, it is often resorted to solving these equations by numerical methods. Some of the numerical methods will be presented in the next chapter.

7.1 Definitions and notation

Definition Let u be a function of variables x 1 , x 2 ,..., x n , which has continuous partial derivatives up to order m , in the observed region Ω ⊂ R n . Any relation between the variables x i ( i = 1 ,..., n ), the function u and its partial derivatives F x 1 , x 2 ,..., x n , u , ∂ u ∂ x 1 ,..., ∂ u ∂ x n ,..., ∂ m u ∂ x m 1 ,..., ∂ m u ∂ x m n = 0 , (7.1)

is called a partial differential equation .

if the highest derivative in (7.1) is of order m ( m ≤ n ), then the equation is called partial differential equation of order m .

Definition Each function u , of variables x 1 , x 2 ,..., x k , of which partial derivatives of the necessary order exist, and which identically satisfies, together with its partial derivatives, the equation (7.1), is called a solution of the partial differential equation (7.1).

Definition The general solution of the equation (7.1) is the solution that contains the number of arbitrary independent functions equal to the order of the equation.

Definition A particular solution is obtained from the general solution by assigning a specific form (expression) to the arbitrary functions.

Example 232 Observe the differential equation ∂ 2 u ∂ x ∂ y

= 2 x − y .