Advanced Real Analysis by Anthony W. Knapp

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By Anthony W. Knapp

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4: t (a) Show directly by expanding out u ∗ (t) = a G 0 (t, s) f (s) ds that u ∗ satisfies L(u ∗ ) = f . (b) Calculate G 0 (t, s) and G 1 (t, s) explicitly for the case that L(u) = u + u when the conditions (SL2) are that u(0) = 0 and u(π/2) = 0. I. Introduction to Boundary-Value Problems 32 4. This problem discusses the starting point for Sturm’s original theory. Suppose that p(t), p (t), g1 (t), and g2 (t) are real-valued and continuous on [a, b] and that p(t) > 0 and g2 (t) > g1 (t) everywhere on [a, b].

Sturm–Liouville Theory The name “Sturm–Liouville theory” refers to the analysis of certain kinds of “eigenvalue” problems for linear ordinary differential equations, particularly equations of the second order. In this section we shall concentrate on one theorem of this kind, which was stated explicitly in Section 2 and was used as a tool for verifying that the method of separation of variables succeeded, for some examples, in solving a boundary-value problem for one of the standard partial differential equations.

2. Separation of Variables 17 for solutions that are independent of θ. If f (ϕ, θ ) is independent of θ , we determine cm by the formula f (ϕ, θ ) ∼ ∞ cm Pm (cos ϕ). m=0 The coefficients can be determined because the polynomials Pm are orthogonal under integration over [−1, 1]. To see this fact, we first rewrite the equation for P as ((1 − t 2 )P ) + m(m + 1)P = 0. 3, but the coefficient 1 − t 2 vanishes at the endpoints t = ±1. 3; see Problem 2 at the end of the chapter. A part of the completeness question is easily settled by observing that Pm is of degree m and that therefore the linear span of {P0 , P1 , .

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