Mathematical Physics Vol 1

Chapter 7. Partial differential equations

366

Example 241 Find the solution of the equation

2 u

u tt = a

xx , 0 ≤ x ≤ l ,

(7.179)

that satisfies the boundary

u ( 0 , t )= 0 , u ( l , t )= 0 ,

(7.180)

and initial conditions

u ( x , 0 )= ϕ ( x ) , u t ( x , 0 )= ψ ( x ) .

(7.181)

Solution Note that, according to classification (under 2 ◦ , p. 362), this equation is of hyperbolic type. According to the Fourier method, let us look for a solution in the form u ( x , t )= X ( x ) · T ( t ) . (7.182) Now we can write the given equation (7.179) in the following form X ′′ · T = 1 a 2 T ′′ · X ⇒ X ′′ X = 1 a 2 T ′′ T = − λ , (7.183) from where it follows X ′′ + λ · X = 0 ∧ T ′′ + a 2 · λ T = 0 , (7.184) ( X ( x )̸ ≡ 0 ∧ T ( t )̸ ≡ 0, as the solutions are not trivial). The boundary conditions (7.180) come down to u ( 0 , t )= X ( 0 ) · T ( t )= 0 ⇒ X ( 0 )= 0 , (7.185) u ( l , t )= X ( l ) · T ( t )= 0 ⇒ X ( l )= 0 . (7.186) In this way, looking for the function X ( x ) is reduced to the problem of eigenvalues : find the values λ for which we obtain nontrivial solutions for the problem X ′′ + λ · X = 0 , X ( 0 )= X ( l )= 0 , (7.187) as well as the corresponding solutions. The values λ obtained in this way are called eigenvalues , and the solutions X ( x ) – eigenfunctions . This is the Sturm – Liouville task.

Discussion . λ can be negative ( λ < 0), zero ( λ = 0) or positive ( λ > 0). Let us observe these three cases: 1 ◦ λ < 0