9.2 The Green’s function and the Fredholm alternative
Theorem 9.2.1. Suppose the homogeneous problem \(Ly = 0\), \(B_ay = B_by = 0\) has only the zero solution. Let \(u_1\) be a non-zero solution of \(Lu = 0\) with \(B_au_1 = 0\), and \(u_2\) one with \(B_bu_2 = 0\). Then \(C = p(u_1'u_2 - u_1u_2')\) is a non-zero constant, and for every continuous \(f\) the problem \(Ly = f\), \(B_ay = B_by = 0\) has the unique solution \[ y(t) = \int _a^bG(t, s)f(s)\,ds, \qquad G(t, s) = \frac 1C\begin {cases} u_1(s)u_2(t), & s \leq t,\\ u_1(t)u_2(s), & s \geq t .\end {cases} \] The Green’s function \(G\) is continuous, symmetric, \(G(t, s) = G(s, t)\), and its \(t\)-derivative jumps by \(-1/p(s)\) across \(t = s\).
Proof. By Lemma 9.1.1 with \(Lu_1 = Lu_2 = 0\), \((p(u_1u_2' - u_2u_1'))' = 0\), so \(C\) is constant. If \(C = 0\), \(u_1\) and \(u_2\) are dependent, so \(u_1\) satisfies both boundary conditions and is a non-zero solution of the homogeneous problem, which is excluded. Now write \(y(t) = \frac 1C\bigl [u_2(t)\int _a^tu_1f\,ds + u_1(t)\int _t^bu_2f\,ds\bigr ]\). Differentiating, the terms from the variable limits cancel, and \(y' = \frac 1C\bigl [u_2'\int _a^tu_1f + u_1'\int _t^bu_2f\bigr ]\). Differentiating \(py'\) once more, \[ (py')' = \frac 1C\Bigl [(pu_2')'\int _a^tu_1f + (pu_1')'\int _t^bu_2f + p(u_2'u_1 - u_1'u_2)f\Bigr ] = \frac 1C\Bigl [qu_2\int _a^tu_1f + qu_1\int _t^bu_2f\Bigr ] - f , \] using \((pu_i')' = qu_i\) and \(p(u_1u_2' - u_1'u_2) = -C\). So \((py')' = qy - f\), that is \(Ly = f\). At \(t = a\), \(y = \frac {u_1(a)}{C}\int _a^bu_2f\) and \(y' = \frac {u_1'(a)}{C}\int _a^bu_2f\), a multiple of \((u_1(a), u_1'(a))\), so \(B_ay = 0\); similarly at \(b\). Uniqueness: the difference of two solutions solves the homogeneous problem. Symmetry and the jump in \(\partial G/\partial t\), which is \(\frac 1C(u_1(s)u_2'(s) - u_1'(s)u_2(s)) = -1/p(s)\), are visible from the formula. □
The Green’s function is the response at \(t\) to a unit point source at \(s\): a string under unit tension with a unit point load at \(s\) is deflected into the tent-shaped \(G(\cdot , s)\), and a general load \(f\) produces the superposition \(\int G(t, s)f(s)\,ds\). For \(-y'' = f\) with \(y(0) = y(1) = 0\), \(u_1 = t\), \(u_2 = 1 - t\), \(C = 1\), and \(G(t, s) = \min (t, s)\bigl (1 - \max (t, s)\bigr )\).
When the homogeneous problem does have non-zero solutions, the situation is that of a singular matrix.
Theorem 9.2.2 (Fredholm alternative). Either the homogeneous problem \(L\phi = 0\), \(B_a\phi = B_b\phi = 0\) has only the zero solution, and then \(Ly = f\) with the boundary conditions has a unique solution for every continuous \(f\); or it has a non-zero solution \(\phi \), unique up to a constant multiple, and then \(Ly = f\) has a solution if and only if \(\int _a^bf\phi \,dt = 0\), in which case the solution is unique up to adding a multiple of \(\phi \).
Proof. The first alternative is Theorem 9.2.1. In the second, \(\phi \) is unique up to a multiple because two solutions satisfying the condition at \(a\) are dependent (proof of Lemma 9.1.1). If \(Ly = f\) has a solution, then by Lemma 9.1.1, \(\int f\phi = \langle Ly, \phi \rangle = \langle y, L\phi \rangle = 0\). Conversely, if \(\int f\phi = 0\), one constructs a solution with a modified Green’s function, which inverts \(L\) on the functions orthogonal to \(\phi \); we omit the details. □
Example 9.2.3. For \(y'' + y = f\), \(y(0) = y(\pi ) = 0\), the homogeneous problem has \(\phi = \sin t\). So the problem is solvable if and only if \(\int _0^\pi f(t)\sin t\,dt = 0\). For \(f = \cos t\) the condition holds, and \(y = \frac t2\sin t + c\sin t\) solves it; for \(f = 1\) it fails (\(\int _0^\pi \sin t = 2\)), and no solution exists: the string is being forced at its own natural frequency.
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