Mathematical Physics Vol 1

Chapter 7. Partial differential equations

356

7.4.1 Some properties of homogeneous second order partial LDE

Property 1. If u i ( x 1 , x 2 ) ( i = 1 , 2 ,..., n ) are solutions of homogeneous equation (7.83), then their arbitrary linear combination u = m ∑ i = 1 C i u i ( x 1 , x 2 ) , (7.85) where C i are arbitrary constants, is also a solution of equation (7.83). Proof:

C i u i ! =

n ∑ i = 1

n ∑ i = 1

n ∑ i = 1

L ( u )= L

L ( C i u i )=

C i L ( u i )= 0 .

(7.86)

If u 1 , u 2 ,..., u n ,... are solutions of the initial equation (7.83), and C 1 , C 2 ,..., C n , ··· are constants, then the function u given by the infinite series

∞ ∑ i = 1

C i u i , in Ω

u =

(7.87)

is also a solution of equation (7.83). Understandably, under the condition that the series on the right had side of the equation (7.87), as well as the series obtained by differentiating it formally member by member, including all possible second order derivatives, are convergent in Ω . Such a solution u is said to be obtained by superposition from solutions u i . Property 2. If u o ( x 1 , x 2 , α 1 , α 2 ) is a solution of equation (7.83), where α i are parameters independent of x i , then the following functions are also solutions u = Z C ( α 1 ) · u o ( x 1 , x 2 , α 1 ) d α 1 , (7.88) u = x C ( α 1 , α 2 ) · u o ( x 1 , x 2 , α 1 , α 2 ) d α 1 d α 2 , (7.89) where C ( α 1 ) and C ( α 1 , α 2 ) are arbitrary functions, and is is assumed that the integrals above can be differentiated. This solution, obtained from solution u o , is said to be obtained by integration , by parameter α 1 , that is, by parameters α 1 , α 2 . If we assume that a i j , a i and b are constants, then the following properties are obtained Property 3. If u 1 ( x 1 , x 2 ) is a solution of the equation (7.83), then the function u = u 1 ( x 1 − α 1 , x 2 − α 2 ) (7.90) is also a solution of the equation. This property can be proved by introducing the substitution x i = ξ i + α i ,

in the differential equation (7.83).