Orthogonality Of Eigen Functions
Orthogonality Of Eigen Functions:-
Definition :
Two distinct continuous functions f and φ on [a,b] are said to be orthogonal with respect to a continuous weight function γ if
b
∫ f(x) φ(x) γ(x) dx = 0 ...(14)
a
An infinite set of functions defined on [a,b] is to be an orthogonal system with respect to the weight function γ on [a,b] if every pair of distinct functions of the set are orthogonal with respect to γ .
Example :
The set of functions {φₙ} , where φₙ(x) = sin nx , n=1,2,.... on [0,π] is an orthogonal system with respect to the weight function having the constant value 1 on [0,π] , for
π
∫ (sin mx)(sin nx)(1) dx
0
π
=[sin(m-n)x / 2(m-n) - sin(m+n)x / 2(m+n)]
0
= 0 for m≠n
The infinite set of eigen functions of the Sturm - Liouville Problem (1) and (2) form an orthogonal system with respect to weight function γ , as is evident from the following theorem .
Theorem :-
If φₘ and φₙ are eigen functions of the Sturm - Liouville Problem(1) and (2) corresponding to the distinct eigen values λₘ and λₙ respectively , then they are orthogonal with respect to the weight function γ .
Proof :
Since φₘ is an eigen function corresponding to λₘ . It satisfies equation (1) with λ = λₘ . Hence we have,
d/dx [p(x)φ'ₘ(x)] + [q(x)+λₘ γ(x)] φₘ(x) = 0
Similarly for φₙ we get
d/dx [p(x)φₙ'(x)] + [q(x) + λₙ γ(x)]φₙ(x) =0
Multiply equation(15) by φₙ(x) and equation(16) by φₘ(x) and subtracting we obtain
(λₘ-λₙ) γ(x) φₘ(x)φₙ(x) =
d/dx [p(x)φₙ'(x)]φₘ(x)
- d/dx [p(x)φₘ'(x)]φₙ(x),
that is
(λₘ- λₙ) γ(x) φₘ(x) φₙ(x) =
d/dx [p(x){φ'ₙ(x) φₘ(x) - φₘ'(x) φₙ(x)}];
which on integration yields
b
( λₘ- λₙ) ∫ γ(x) φₘ(x) φₙ(x) dx
a
= p(b){φ'ₙ(b) φₘ(b) - φ'ₘ(b) φₙ(b)}
- p(a){φₙ'(a) φₘ(a) - φ'ₘ(a) φₙ(a)}
The orthogonality of φₙ and φₘ is established if we show that the right side of equation(17 ) vanishes .
From the first equation of (2) we get
a₁ φₘ(a) + a₂ φₘ'(a) =0 and a₁φₙ(a) + a₂φₙ'(a)
= 0
Elimination of a₂ from these two equations leads to
a₁[φₙ(a) φₘ'(a) - φₘ(a) φₙ'(a)] = 0
and if a₁ ≠ 0 then
φₙ(a) φ'ₘ(a) - φₘ(a) φ'ₙ(a)=0 ..........(18)
Similarly if b₁≠0 then from 2nd equation of (2) we get
φₙ(b) φ'ₘ(b) - φₘ(b) φₙ'(b) = 0 ...............(19)
By (18) and (19) the right side of (17) vanishes, establishing that φₘ and φₙ are orthogonal .
If either a₁= 0 or b₁= 0 or a₁= b₁= 0 it can be checked that the right side of(17) vanishes .
Corollary (Eigen function Expansion) :-
Let g(x) be a piece wise continuous function defined on [a,b] satisfying the boundary conditions (2) . Let φ₁,φ₂,......φₙ be the set of eigen functions of the Sturm - Liouville Problems (1) and (2) . Then
g(x) = c₁φ₁(x) + c₂φ₂(x) + ....+cₙφₙ(x)+... ...(20)
Where cₙ's are given by
b b
cₙ ∫ γ(x) φₙ²(x) dx = ∫γ(x) g(x) φₙ(x) dx
a a
n= 1,2,3.... ........(21)
Proof :
By the above theorem φ₁,φ₂,...φₙ form an orthogonal system with respect to γ on [a,b] . Hence
b
∫ γ(x) φₙ(x) φₘ(x) dx = 0 for n≠m .....(22)
a
From (20) we have
b
∫ g(x) γ(x) φₙ(x) dx =
a
b ∞
∫( Σ cₖ φₖ(x) ) γ(x) φₙ(x) dx
a k=1
∞ b
= Σ cₖ (∫ γ(x) φₖ(x) φₙ(x) dx )
k=1 a
b
= cₙ ∫ γ(x) φₙ²(x) dx , by (22)
a
Here we have assumed the uniform convergence of the series in the right side of (20) .
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