Sturm - Liouville Problem

Sturm - Liouville Problem:

  The Sturm - Liouville Problem is a special class of linear homogeneous boundary value problem given in the following definition. 

Definition :

     Consider a second order homogeneous linear differential equation of the form 

 d/dx [p(x) dy/dx ] + [q(x) + λ γ(x)]y=0, a≤x≤b

                                                         ...............(1)

  where p, q and γ are real - valued continuous functions on [a,b] and λ  is real parameter . Further consider the following two sets of prescribed conditions 

      a1 y(a) + a2 y'(a) = 0}

                                           }...................(2)

      b1 y(b) + b2 y'(b) = 0}

   y(a) = y(b) , y'(a) = y'(b) , p(a) = p(b) .........(3)

 where the real constants a1 and a2 are not both zero and b1 and b2  are not both zero .

    A boundary value problem consisting of equation(1) with either the boundary condition (2) or the boundary condition (3) is called Sturm - Liouville Problem (or Sturm - Liouville System ).

Example 1 :

     The boundary - value problem 

             d²y/dx² + λy = 0,   ..............(4)

        y(0) = 0 , y'(π) = 0 ......................(5)

 is a Sturm - Liouville Problem , since , equation (4) may be written as 

d/dx [1. dy/dx ] + [0 + λ.1] y = 0 

which is of the form (1) with p(x) =1 ,        q(x) =0 and γ(x) =1 and the boundary condition (5) is the form (2) with a1 = 1 ,     a2 = 0 , b1= 0 , b2= 1.

    It is easy to see that the identically  vanishing function φ i.e. φ(x) = 0 for all x in [a,b] is always a solution of a Sturm - Liouville problem . But this trivial solution is not very useful . So we seek for nontrivial solutions of the problems . That is , we seek for functions , identically zero , which satisfy both the differential equation(1) and the boundary conditions (2) . In the next example we shall see that the existence of nontrivial solutions depends upon the value of the parameter λ . 

Example 2 :

        Find nontrivial solutions of the Sturm - Liouville problem

                 d²y/dx² + λy = 0 ,    ..........(4)

  y(0) =0, y'(π) =0      ...............(5)

Solution :

      We shall discuss the problem under three separate cases : λ=0 , λ<0 and λ>0.

Case (i) :

       λ=0 . In this case the differential equation(4)  becomes 

            d²y/dx² = 0,

and hence the general solution is 

      y(x) = c1 x + c2 ...............(6)

    Using y(0) = 0 in (6) we obtain c2= 0.  The condition y'(π) = 0 yields                            (since v'(x) = c1)c1 =0 . Thus the solution of the given problem is y(x) =0 for all values of x . So when λ=0, the only solution of the given Sturm - Liouville problem is the trivial solution . 

Case (ii) :

      λ<0 . In this case the auxiliary equation of (4) is m² + λ =0 which has roots ±√(-λ). Since λ<0 , -λ is positive so that √(-λ) is real . Hence the general solution is 

    y(x) = c1 e^√(-λ)x + c2 e^-√(-λ )x ........(7)

Application of y(0) = 0 in (7) yields 

    c1 + c2 = 0 .................(8)

Since  y'(x) = √(-λ)[c1 e^√(-λ)x - c2 e^-√(-λ )x]

 the condition y'(π) = 0 gives 

 √(-λ)[c1 e^√(-λ)π - c2 e^-√(-λ)π]=0 .........(9)

   Since √(-λ), e^√(-λ)π and e^-√(-λ) are all positive (8) and (9) yield c1=c2=0 . Thus when λ<0 , the only solution of the given problem is trivial solution y(x) =0 for all values of x . 

Case (iii) :

     λ>0 . Since λ>0 the roots ±√(-λ) of the auxiliary equation are conjugate complex numbers ±√(λ)i and the general solution is of the form 

   y(x) = c1 sin√(λ)x + c2 cos√(λ )x ...........(10)

  Using the condition y(0)  in (10) we obtain 

   0 = c1 sin 0 + c2 cos 0 

and hence c2 = 0 . Since 

 y'(x) = √(λ) [c1 cos √(λ)x - c2 sin √(λ)x]

 from the condition y'(π) =0 we obtain 

 0 = √(λ)[c1 cos √(λ)π - c2 sin √(λ)π]

  Which reduces to c1 cos√(λ) π = 0, .........(11)

   since c2 =0 and √(λ) ≠ 0 .

If c1 = 0 in (11) then we obtain from (10) the unwanted trivial solution y(x) . So for a nontrivial solution of the problem we must take 

     cos √(λ)π = 0 

 which gives √(λ)π = (2n - 1)π/2 

 i.e. λ = (2n - 1)²/4 , where n=1,2,3...

 Thus for a nontrivial solution the parameter λ must have one of the values of the sequence 

            {(2n-1)²/4} , n=1,2,3.........

and the nontrivial solution , as obtained from (10) , corresponding to 

     λ = (2n-1)²/4 are 

   y= ci sin (2n-1)/2 x, n=1,2,3... .........(13)

    where ci' s are nonzero constants. 

    The values of λ in (12) are called eigen values and the corresponding solution in (13) are called eigen functions . Thus the following definition is given 

Definition :

    The values of the parameter λ in (1) for which there exist nontrivial solutions of the Sturm - Liouville problem (1) and (2) are called the eigen values or characteristic values of the problem and the corresponding nontrivial solutions are called eigen functions or characteristic functions of the problem . 

    It is observed from the above example that the eigen values are uniquely determined by the problem , but a one - parameter family of eigen functions since ci is arbitrary for each n correspond to each eigen value and any two eigen functions corresponding to definite eigen value are non zero constant multiples of each other . Further it is seen that the eigen values form an increasing sequence of positive numbers that approach ∞ and the nth eigen function sin (2n-1)/2 x has exactly n-1 zeroes inside the interval [0,π] . [The zeros of                    sin (2n-1 / 2)x are given by (2n-1 / 2) x =K π i.e. x= K π/(n-1/2) , K is any integer . So the zeros which lie in 0<x<π are those for which K = 1,2,.....n-1]

      Let us check the above observation is true or not for Sturm - Liouville problem using the following theorem .

 Theorem :

        If p(x)>0 and γ(x)>0 for all x on [a,b] the Sturm - Liouville problem consisting of the equation(1) and the boundary condition (2) has an infinite number of eigen values λn which can be arranged in a monotonic increasing sequence λ1<λ2<..... with no finite limit . Further corresponding to each eigen value there exists a one- parameter family of eigen functions such that  any two eigen functions corresponding to the same eigen value λn has exactly (n-1) zeros in the open interval a<x<b.