Mathematical Physics Vol 1

7.6 The variable separation method

365

that is

X ′′ X

X ′ X

Y ′′

Y ′ Y −

a 11 (7.168) As the left hand side of the equation (7.168) is a function only of x , and the right hand side a function only of y , it means that these expressions are constants, namely + b 1 + F ( x )= − a 22 Y − b 2 G ( y ) .

X ′′ X

X ′ X

Y ′′

Y ′ Y −

G ( y )= − λ = const .

a 11

+ b 1

+ F ( x )= − a 22

b 2

(7.169)

Y −

In addition to this, the initial conditions must also be satisfied

u ( x , 0 )= X ( x ) · Y ( 0 )= f ( x ) , u y ( x , 0 )= X ( x ) · Y ′ ( 0 )= g ( x )

(7.170)

as well as boundary conditions

aX ( 0 ) Y ( y )+ bX ′ ( 0 ) Y ( y )= 0 ⇒ aX ( 0 )+ bX ′ ( 0 )= 0 , cX ( l ) Y ( y )+ dX ′ ( l ) Y ( y )= 0 ⇒ cX ( l )+ dX ′ ( l )= 0 . (7.172) The given equation is now decomposed into two ordinary second order differential equations a 11 X ′′ + b 1 X ′ + F ( x ) X = − λ X ⇒ a 11 X ′′ + b 1 X ′ +( F + λ ) X = 0 , (7.173) a 22 Y ′′ + b 2 Y ′ + G ( y ) Y =+ λ Y ⇒ a 22 Y ′′ + b 2 Y ′ +( G − λ ) Y = 0 . (7.174) It can be shown, by applying the Sturm – Liouville theory, that there is an infinite number of the so-called eigenvalues λ 1 , λ 2 ,... , for which there exist nontrivial solutions (trivial solutions would be X ≡ 0 , Y ≡ 0) of the equations (7.173) and (7.174). Let X n ( n = 1 , 2 ,... ) be the solutions of equation (7.173), for λ = λ n . The solution of equation (7.174) can be represented in the form Y n ( y )= A n Y n ( y )+ B n Y n ( y ) , (7.175) where A n and B n are arbitrary constants, and Y n and Y n linearly independent particular solutions of equation (7.174), for λ = λ n . These function are determined from the condition Y n ( 0 )= 1; Y ′ n ( 0 )= 0; Y n ( 0 )= 0; Y ′ n ( 0 )= 1 . (7.176) Applying the superposition principle , we obtain (7.171)

X n ( x ) h A n Y n ( y )+ B n Y n ( y ) i .

∞ ∑ n = 1

u ( x , y )=

(7.177)

This solution must additionally satisfy the conditions ∞ ∑ n = 1 A n X n ( x )= f ( x ) ∞ ∑ n = 1

B n X n ( x )= g ( x ) ,

(7.178)

by which the problem is reduced to expanding functions f and g into series of eigenfunctions X n . Solving equations of hyperbolic and parabolic type by Fourier method

Solving equations using the Fourier method will be demonstrated on some examples.