1. Boyce/DiPrima 9th ed, Ch 11.2: Sturm-Liouville Boundary Value Problems Elementary Differential Equations and Boundary Value Problems, 9th edition, by William E. Boyce and Richard C. DiPrima, ©2009 by John Wiley & Sons, Inc.
We now consider two-point boundary value problems of the type obtained in Section 11.1 by separating the variables in a heat conduction problem for a bar of variable material properties and with a source term proportional to temperature.
This kind of problem also occurs in many other applications.
These boundary value problems are commonly known as Sturm-Liouville problems.
They consist of a differential equation of the form
together with the boundary conditions

2. Differential Operator Form
Consider the linear homogeneous differential operator L:
The differential equation can then be written as
We assume that the functions p, p', q and r are continuous on 0  x  1, and further, that p(x) > 0 and r(x) > 0 on [0,1].
The boundary conditions are said to be separated, since each one only involves one of the boundary points:

3. Lagrange’s Identity (1 of 3)
We next establish Lagrange’s identity, which is basic to the study of linear boundary value problems.
Let u and v be functions having continuous second derivatives on the interval 0  x  1. Then
Integrating the first term on the right twice by parts, we obtain
We thus obtain Lagrange’s identity:

4. Lagrange’s Identity Using Boundary Conditions (2 of 3)
Suppose that u and v satisfy the boundary conditions
If neither 2 nor 2 is zero, then
The result also holds if either 2 or 2 is zero.
Thus Lagrange’s identity reduces to

5. Lagrange’s Identity: Inner Product Form (3 of 3)
From the previous slide, we have Lagrange’s identity
In Section 10.2 we introduced the inner product (u,v) with
Thus Lagrange’s identity can be rewritten as
provided the previously mentioned assumptions are satisfied.

6. Complex Inner Products
The inner product of two complex-valued functions on [0,1] is defined as
where is the complex conjugate of v.
Lagrange’s identity, as shown in the previous slides,
is still valid in this case since p(x), q(x), 1, 1, 2, and 2 are all real quantities.

7. Sturm-Liouville Problems, Lagrange’s Identity
Consider again the Sturm-Liouville boundary value problem
This problem always has eigenvalues and eigenvectors.
In Theorems – of this section, we will state several of their important properties.
Each property is illustrated by the Sturm-Liouville problem whose eigenvalues are n = n2 2 and corresponding eigenfunctions n(x) = sin n x.
These results rely on Lagrange’s identity.

8. Theorem 11.2.1 Consider the Sturm-Liouville boundary value problem
All of the eigenvalues of this problem are real.
We give an outline of the proof on the next slide.
It can also be shown that all of the eigenfunctions are real.

9. Theorem 11.2.1: Proof Outline
Suppose that  is a (possibly complex) eigenvalue with corresponding eigenvector .
Let  = u + iv, (x) = U(x) + iV(x), where u, v, U, V are real.
Then it follows from Lagrange’s identity that or
Since r(x) is real, we have
Recalling r(x) > 0 on [0,1], it follows that  is real.

10. Theorem 11.2.2 Consider the Sturm-Liouville boundary value problem
If 1(x) and 2(x) are two eigenfunctions corresponding to the eigenvalues 1 and 2, respectively, and if 1  2, then
Thus 1(x) and 2(x) are orthogonal with respect to the weight function r(x).

11. Theorem 11.2.2: Proof Outline
Let 1, 2, 1(x) and 2(x) satisfy the hypotheses of theorem.
Then or
Since r(x), 2 and 2(x) are all real, we have
Since 1  2, we have

12. Theorem 11.2.3 Consider the Sturm-Liouville boundary value problem
The eigenvalues are all simple. That is, to each eigenvalue there corresponds one and only one linearly independent eigenfunction.
Further, the eigenvalues form an infinite sequence and can be ordered according to increasing magnitude so that
Moreover,

13. Illustration of Sturm-Liouville Properties
The properties stated in Theorems – are again illustrated by the Sturm-Liouville problem whose eigenvalues n = n2 2 are real, distinct and increasing, such that n   as n  .
Also, the corresponding real eigenfunctions n(x) = sin n x are orthogonal to each other with respect to r(x) = 1 on [0, 1].

14. Orthonormal Eigenfunctions
We now assume that the eigenvalues of the Sturm-Liouville problem are ordered as indicated in Theorem
Associated with the eigenvalue n is a corresponding real eigenfunction n(x) determined up to a multiplicative constant.
It is often convenient to choose the arbitrary constant multiplying each eigenfunction so as to satisfy the condition
The eigenfunctions are said to be normalized, and form an orthonormal set with respect to the weight function r, since they are orthogonal and normalized.

15. Kronecker Delta
For orthonormal eigenfunctions, it is useful to combine the orthogonal and normalization relations into one symbol.
To this end, the Kronecker delta mn is helpful:
Thus, for our orthonormal eigenfunctions, we have

16. Example 1: Orthonormal Eigenfunctions
Consider the Sturm-Liouville problem
In this case the weight function is r(x) = 1. The eigenvalues and eigenfunctions are n = n2 2 and yn(x) = sin n x.
To find the orthonormal eigenfunctions, we choose kn so that
Now
and hence the orthonormal eigenfunctions are

17. Example 2: Boundary Value Problem (1 of 3)
Consider the Sturm-Liouville problem
In this case is r(x) = 1. The eigenvalues n satisfy
and the corresponding eigenfunctions are
To find the orthonormal eigenfunctions, we choose kn so that or

18. Example 2: Orthonormal Eigenfunctions (2 of 2)
We have
where in the last step we have used
Thus the orthonormal eigenfunctions are

19. Eigenfunction Expansions (1 of 2)
We now investigate expressing a given function f as a series of eigenfunctions of the Sturm-Liouville boundary value problem
For example, if f is continuous and has a piecewise continuous derivative on [0, 1], and satisfies f (0) = f (1) = 0, then f can be expanded in a Fourier sine series of the form
which converges for each x in [0, 1].
This expansion of f is given in terms of the eigenfunctions of

20. Eigenfunction Expansions (2 of 2)
Suppose that a given function f, satisfying suitable conditions, can be expressed in a series of orthonormal eigenfunctions n(x) of the Sturm-Liouville boundary value problem.
Then
To obtain cm, we assume the series can be integrated term by term, after multiplying both sides by m(x)r(x) and integrating:
In inner product form, we have

21. Theorem 11.2.4 Consider the Sturm-Liouville boundary value problem
Let 1, 2, …, n,… be the normalized eigenfunctions for this problem, and let f and f ' be piecewise continuous on 0  x 1.
Then the series
converges to [ f (x+) + f (x-)]/2 at each point x in the open interval 0 < x < 1.

22. Theorem : Discussion
If f satisfies further conditions, then a stronger conclusion can be established. Consider again the Sturm-Liouville problem
Suppose f is continuous and f ' piecewise continuous on[0,1].
If 2 = 0, then assume that f(0) = 0. If 2 = 0, then assume that f(1) = 0. Otherwise no boundary conditions need to be prescribed for f. Then the series
converges to f (x) at each point in the closed interval [0, 1].

23. Example 3: Function f (1 of 3)
Consider the function
Recall from Example 2, for the Sturm-Liouville problem
the orthonormal eigenfunctions are
Then by Theorem , we have

24. Example 3: Coefficients (2 of 3)
Thus
Integrating by parts, we obtain
where in the last step we have used
Next, recall that

25. Example 3: Eigenfunction Expansion (3 of 3)
We have
It follows that
Thus

26. Sturm-Liouville Problems and Algebraic Eigenvalue Problems
Sturm-Liouville boundary value problems are of great importance in their own right, but they can also be viewed as belonging to a much more extensive class of problems that have many of the same properties.
For example, there are many similarities between Sturm-Liouville problems and the algebraic system Ax = x, where the n x n matrix A is real symmetric or Hermitian.
Comparing the results mentioned in Section 7.3 with those of this section, in both cases the eigenvalues are real and the eigenfunctions or eigenvectors form an orthogonal set, and can be used as a basis for expressing an essentially arbitrary function or vector, respectively, as a sum.

27. Linear Operator Theory
The most important difference is that a matrix has only a finite number of eigenvalues and eigenvectors, while a Sturm-Liouville system has infinitely many.
It is of fundamental importance in mathematics that these seemingly different problems – the matrix problem Ax = x and the Sturm-Liouville problem,
which arise in different ways, are actually parts of a single underlying theory.
This theory is linear operator theory, and is part of the subject of functional analysis.

28. Self-Adjoint Problems (1 of 4)
Consider the boundary value problem consisting of the differential equation L[y] = r(x)y, where
and n homogeneous boundary conditions at the endpoints.
If Lagrange’s identity is valid for every pair of sufficiently differentiable functions that satisfy the boundary conditions, then the problem is said to be self-adjoint.

29. Self-Adjoint Problems and Structure of Differential Operator L (2 of 4)
Lagrange’s identity involves restrictions on both the differential equation and the boundary conditions.
The differential operator L must be such that the same operator appears in both terms of Lagrange’s identity,
This requires L to be of even order. Further, a second order operator must have the form
and a fourth order operator must have the form
Higher order operators must have an analogous structure.

30. Self-Adjoint Problems and Boundary Conditions (3 of 4)
In addition, the boundary conditions must be such as to eliminate the boundary terms that arise during the integration by parts used in deriving Lagrange’s identity.
For example, in a second order problem this is true for the separated boundary conditions
and also in other cases, one of which is given in Example 4, as we will see.

31. Fourth Order Self-Adjoint Problems and Eigenvalue, Eigenvector Properties (4 of 4)
Suppose we have self-adjoint boundary value problem for L[y] = r(x)y, where L[y] is given by
We assume that p(x), q(x), r(x), s(x) are continuous on [0,1] and that the derivatives p', p'' and q' are also continuous.
Suppose also that p(x) > 0 and r(x) > 0 on [0,1].
Then there is an infinite sequence of real eigenvalues tending to , the eigenfunctions are orthogonal with respect to the weight function r, and an arbitrary function f can be expressed as a series of eigenfunctions.
However, the eigenfunctions may not be simple in these more general problems.

32. Sturm-Liouville Problems and Fourier Series
We have noted previously that Fourier sine (and cosine) series can be obtained using the eigenfunctions of certain Sturm-Liouville problems involving the differential equation
This raises the question of whether we can obtain a full Fourier series, including both sine and cosine terms, by choosing a suitable set of boundary conditions.
The answer is yes, as we will see in the following example.
This example will also serve to illustrate the occurrence of nonseparated boundary conditions.

33. Example 4: Boundary Value Problem (1 of 3)
Consider the boundary value problem
This is not a Sturm-Liouville problem because the boundary conditions are not separated.
The boundary conditions above are called periodic boundary conditions since the require that y and y' assume the same values at x = L as at x = -L.
It is straightforward to show that this problem is self-adjoint.

34. Example 4: Eigenvalues and Eigenfunctions (2 of 3)
Our boundary value problem is
It can be shown that 0 = 0 is an eigenvalue with corresponding eigenfunction 0(x) = 1.
All other eigenvalues are given by
To each of these eigenvalues there corresponds two linearly independent eigenfunctions.
For example, the eigenfunctions corresponding to n are
This shows that the eigenfunctions may not be simple when the boundary conditions are not separated.

35. Example 4: Eigenfunction Expansion (3 of 3)
Further, if we seek to expand a given function f of period 2L in a series of eigenfunctions for this problem, we obtain
which is just the Fourier series of f.