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11.2 Sturm–Liouville Boundary Value Problems 629

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11.2 Sturm–Liouville Boundary Value Problems 629 11.2 Sturm–Liouville Boundary Value Problems 629 where E is Young’s modulus and ρ is the mass per unit volume. If the end x 0isfixed, then the boundary condition there is = u(0, t) 0, t > 0. (ii) = Suppose that the end x L is rigidly attached to a mass m but is otherwise unrestrained. We can obtain the boundary= condition here by writing Newton’s law for the mass. From the theory of elasticity it can be shown that the force exerted by the bar on the mass is given by EAu (L, t). Hence the boundary condition is − x EAu (L, t) mu (L, t) 0, t > 0. (iii) x + tt = (a) Assume that u(x, t) X (x)T (t), and show that X (x) and T (t) satisfy the differential equations = X ′′ λX 0, (iv) + = T ′′ λ(E/ρ)T 0. (v) + = (b) Show that the boundary conditions are X (0) 0, X ′(L) γλLX(L) 0, (vi) = − = where γ m/ρ AL is a dimensionless parameter that gives the ratio of the end mass to the mass of the= rod. Hint: Use the differential equation for T (t) in simplifying the boundary condition at x L. (c) Determine the form of the eigenfunctions and the equation satisfied by the real eigen-= values of Eqs. (iv) and (vi). Find the first two eigenvalues λ and λ if γ 0.5. 1 2 = 11.2 Sturm–Liouville Boundary Value Problems 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 the temperature. This kind of problem also occurs in many other applications. These boundary value problems are commonly associated with the names of Sturm and Liouville.1 They consist of a differential equation of the form [p(x)y′]′ q(x)y λr(x)y 0(1) − + = on the interval 0 < x < 1, together with the boundary conditions a y(0) a y′(0) 0, b y(1) b y′(1) 0(2) 1 + 2 = 1 + 2 = 1Charles-Franc¸ois Sturm (1803–1855) and Joseph Liouville (1809–1882), in a series of papers in 1836 and 1837, set forth many properties of the class of boundary value problems associated with their names, including the results stated in Theorems 11.2.1 to 11.2.4. Sturm is also famous for a theorem on the number of real zeros of a polynomial, and in addition, did extensive work in physics and mechanics. Besides his own research in analysis, algebra, and number theory, Liouville was the founder, and for 39 years the editor, of the influential Journal de mathe´matiques pures et applique´es. One of his most important results was the proof (in 1844) of the existence of transcendental numbers. 630 Chapter 11. Boundary Value Problems and Sturm–Liouville Theory at the endpoints. It is often convenient to introduce the linear homogeneous differential operator L defined by L[y] [p(x)y′]′ q(x)y. (3) =− + Then the differential equation (1) can be written as L[y] λr(x)y. (4) = We assume that the functions p, p′, q,andr are continuous on the interval 0 x 1 and, further, that p(x)>0andr(x)>0 at all points in 0 x 1. These assumptions≤ ≤ are necessary to render the theory as simple as possible w≤hile≤retaining considerable generality. It turns out that these conditions are satisfied in many significant problems in mathematical physics. For example, the equation y′′ λy 0, which arose repeatedly in the preceding chapter, is of the form (1) with p(+x) =1, q(x) 0, and r(x) 1. The boundary conditions (2) are said to be separated; t=hat is, each=involves only=one of the boundary points. These are the most general separated boundary conditions that are possible for a second order differential equation. Before proceeding to establish some of the properties of the Sturm–Liouville problem (1), (2), it is necessary to derive an identity, known as Lagrange’s identity,whichis 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. Then2 ≤ ≤ 1 1 L[u]v dx [ (pu′)′v quv] dx. = − + 0 0 Integrating the first term on the right side twice by parts, we obtain 1 1 1 1 L[u]v dx p(x)u′(x)v(x) p(x)u(x)v′(x) [ u(pv′)′ uqv] dx =− + + − + 0 0 0 0 1 1 p(x)[u′(x)v(x) u(x)v′(x)] uL[v] dx. =− − + 0 0 Hence, on transposing the integral on the right side, we have 1 1 L[u]v uL[v] dx p(x)[u′(x)v(x) u(x)v′(x)] , (5) { − } =− − 0 0 which is Lagrange’s identity. Now let us suppose that the functions u and v in Eq. (5) also satisfy the boundary conditions (2). Then, if we assume that a2 0andb2 0, the right side of Eq. (5) becomes = = 1 p(x)[u′(x)v(x) u(x)v′(x)] − − 0 p(1)[u′(1)v(1) u(1)v′(1)] p(0)[u′(0)v(0) u(0)v′(0)] =− − + − b b a a p(1) 1 u(1)v(1) 1 u(1)v(1) p(0) 1 u(0)v(0) 1 u(0)v(0) =− −b + b + −a + a 2 2 2 2 0. = 1 1 2For brevity we sometimes use the notation fdxrather than f (x) dx in this chapter. 0 0 11.2 Sturm–Liouville Boundary Value Problems 631 The same result holds if either a2 or b2 is zero; the proof in this case is even simpler, and is left for you. Thus, if the differential operator L is defined by Eq. (3), and if the functions u and v satisfy the boundary conditions (2), Lagrange’s identity reduces to 1 L[u]v uL[v] dx 0. (6) { − } = 0 Let us now write Eq. (6) in a slightly different way. In Eq. (4) of Section 10.2 we introduced the inner product (u,v) of two real-valued functions u and v on a given interval; using the interval 0 x 1, we have ≤ ≤ 1 (u,v) u(x)v(x) dx. (7) = 0 In this notation Eq. (6) becomes (L[u],v) (u, L[v]) 0. (8) − = In proving Theorem 11.2.1 below it is necessary to deal with complex-valued func- tions. By analogy with the definition in Section 7.2 for vectors, we define the inner product of two complex-valued functions on 0 x 1as ≤ ≤ 1 (u,v) u(x)v(x) dx, (9) = 0 where v is the complex conjugate of v. Clearly, Eq. (9) coincides with Eq. (7) if u(x) and v(x) are real. It is important to know that Eq. (8) remains valid under the stated conditions if u and v are complex-valued functions and if the inner product (9) is used. 1 To see this, one can start with the quantity L[u]v dx and retrace the steps leading to 0 Eq. (6), making use of the fact that p(x), q(x), a1, a2, b1,andb2 are all real quantities (see Problem 22). We now consider some of the implications of Eq. (8) for the Sturm–Liouville bound- ary value problem (1), (2). We assume without proof 3 that this problem actually has eigenvalues and eigenfunctions. In Theorems 11.2.1 to 11.2.4 below, we state several of their important, but relatively elementary, properties. Each of these properties is illustrated by the basic Sturm–Liouville problem y′′ λy 0, y(0) 0, y(1) 0, (10) + = = = 2 2 whose eigenvalues are λn n π , with the corresponding eigenfunctions φn(x) sin nπx. = = Theorem 11.2.1 All the eigenvalues of the Sturm–Liouville problem (1), (2) are real. To prove this theorem let us suppose that λ is a (possibly complex) eigenvalue of the problem (1), (2) and that φ is a corresponding eigenfunction, also possibly complex- valued. Let us write λ µ iν and φ(x) U(x) iV(x),whereµ, ν, U(x),and V (x) are real. Then, if we= let+u φ and also=v φ +inEq.(8),wehave = = (L[φ],φ) (φ, L[φ]). (11) = 3The proof may be found, for example, in the references by Sagan (Chapter 5) or Birkhoff and Rota (Chapter 10). 632 Chapter 11. Boundary Value Problems and Sturm–Liouville Theory However, we know that L[φ] λrφ, so Eq. (11) becomes = (λrφ,φ) (φ, λrφ). (12) = Writing out Eq. (12) in full, using the definition (9) of the inner product, we obtain 1 1 λr(x)φ(x)φ(x) dx φ(x)λr(x)φ(x) dx. (13) = 0 0 Since r(x) is real, Eq. (13) reduces to 1 (λ λ) r(x)φ(x)φ(x) dx 0, − = 0 or 1 (λ λ) r(x)[U 2(x) V 2(x)] dx 0. (14) − + = 0 The integrand in Eq. (14) is nonnegative and not identically zero. Since the integrand is also continuous, it follows that the integral is positive. Therefore, the factor λ λ 2iν must be zero. Hence ν 0andλ is real, so the theorem is proved. − = An important consequence= of Theorem 11.2.1 is that in finding eigenvalues and eigenfunctions of a Sturm–Liouville boundary value problem, one need look only for real eigenvalues.
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