Mathematical Physics Vol 1

Chapter 7. Partial differential equations

362

Classification of second order LPE with n variables

It has been previously shown how the classification of second order linear partial equations with two variables is done. In this section, we will generalize this procedure to n variables. Observe the quadratic form Φ ≡ n ∑ i , j = 1 a ◦ i j y i y j , (7.142) where a ◦ i j are constant coefficients, which correspond to coefficients a i j from differential equation (7.81), at point M 0 ( x ◦ 1 ,..., x ◦ n ) . It is shown in linear algebra that, for the quadratic form (7.142), a linear transformation can always be chosen y i = n ∑ k = 1 α ik η k , (7.143) where α ik are real numbers, so that the quadratic form is reduced to a canonical form 10 where A i are real numbers. The differential equation (7.81) at point M 0 is called: 1 ◦ an equation of elliptic type , if all coefficients A i are of the same sign; 2 ◦ an equation of hyperbolic type or normal-hyperbolic type , if n − 1 coefficients A i are of the same sign, while one is of the opposite sign; 3 ◦ an equation of ultra hyperbolic type , if m coefficients A i are of the same sign, while n − m are of the opposite sign, for m > 1 and n − m > 1; 4 ◦ an equation of parabolic type , if at least on of he coefficients A i is equal to zero. 7.4.4 Examples of classification of some equations of mathematical physics Let us now present examples of the most commonly used partial differential equations. Φ = n ∑ i = 1 A i η 2 i , (7.144)

Example 238 Equation of oscillation in the plane (wave equation)

∂ 2 u ∂ x 2

∂ 2 u ∂ x 2 2

+ λ 2 u = 0 ,

(7.145)

1 −

that is, in the space

∂ 2 u ∂ x 2

∂ 2 u ∂ x 2 2

∂ 2 u ∂ x 2 3

+ λ 2 u = 0

(7.146)

1 −

+

is, according to 2 ◦ and u = u ( x 1 , x 2 , x 3 ) , of hyperbolic type.

10 Canonical or diagonal form. The latter term is more common in linear algebra.