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

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:

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:

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

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.

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.

Sturm-Liouville Problems, Lagrange’s Identity Consider again the Sturm-Liouville boundary value problem This problem always has eigenvalues and eigenvectors. In Theorems 11.2.1 – 11.2.4 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.

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.

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.

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).

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

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,

Illustration of Sturm-Liouville Properties The properties stated in Theorems 11.2.1 – 11.2.3 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].

Orthonormal Eigenfunctions We now assume that the eigenvalues of the Sturm-Liouville problem are ordered as indicated in Theorem 11.2.3. 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.

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

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

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

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

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

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

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.

Theorem 11.2.4: 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].

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 11.2.4, we have

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.