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| | [No title] (Site not responding. Last check: 2007-10-18) |
 | | Throughout, we let $u(x)$ be a function in $L^2[a,b]$ and define \eq Lu \equiv a_2(x) u''(x) +a_1(x) u'(x) + a_0(x) u(x) \quad, \quad x \in (a,b) \endeq and $B = [b_{ij}] \in \reals^{2\times 4}$.We use the overbar notation $\bar{u}$ to denote \eq \bar{u} = (u(a),u'(a),u(b),u'(b))^T \in \reals^4 \endeq for any function $u(x)$. |
 | | However, the Greens function $g_0(x,t)$ solving \eq L_0 g_0 = \delta (x-t) \quad, u(0)=u(1)=0 \endeq is easily found as \eq g_0(x,t) = \left\{ \begin{array}{ll} t(x-1) & 1 \ge x > t \ge 0\\ x(t-1) & 1 \ge t > x \ge 0 \end{array} \right. |
 | | Since the Greens function $g_0(x,t)$ exists, we know that \eq L^*v = g \quad, \quad B^*\bar{v} = 0 \endeq is equivalent to the integral equation \eq v(x)+ \int_a^b a_0(x) g_0(t,x) v(t) dt = G(x) = \int_a^b g(t) g_0(t,x) dt \endeq The result follows for $g=0$. |
| www.math.montana.edu /~pernarow/M560/2000/2ndorderBVP.texOLD (1251 words) |
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