Advanced Calculus fi..

71 0 Advanced Calculus, Fifth Edition '"1"Problem 7 following Section 10.4. In the notation used here the characteristic valuesand functions found were2a(N + 1) sin nnI, =A, (x,) = sin (nx,),n 2(N + 1) 'whereu = 0, 1, ..., N + l,n = 1 ,..., N;comparewiththeexactsolutionsofthewaveequation. Show that for each fixed n, A, -+ an as N + CQ.5. Study the behavior of the solutions of the initial value problem:as I increases from 0 to CQ; note in particular the appearance of values of 1 for which thecondition u(1) = 0 is satisfied. It can be shown that the same qualitative picture holds for .the general Sturm-Liouville problem of Section 10.13.6. Show that the function 4 defined by (10.145) has precisely one critical point at which +4 takes on its absolute minimum. [Hint: Show that by proper choice of the constants,cul, . . . , (YN, the substitutiontransforms 4 into an expressionThis shows that 4 + CQ as w: + . . . + W; + 00, SO that 6 has at least one critical pointthat gives its absolute minimum. The equations in ul ,..., u~ for the critical point aresimultaneous linear equations. The equations have a unique solution, if the correspondinghomogeneous equations (all F, zero) have a unique solution; in the homogeneouscase ul = u2 = ... = UN = 0 is one critical point; if u;, ..., U; were a second one, thena@/au, would be 0 for all u when ul = u;t,. .., uN = u&t and -oo c t c CQ. Thiscontradicts the fact that 4 + oo as w: + -.. + w: -+ CQ. The uniqueness of the criticalpoint can also be established by using difference equations, as in Problem 5 followingSection 10.4.17. Prove that the function Q, defined by (10.147) attains its minimum value, among smoothfunctions u(x) satisfying the boundary conditions u(0) = u(L) = 0, when u is the solutionof the equation -K2u"(x) = F(x). [Hint: Take L = n for convenience. Then expressthe integral in terms of Fourier sine coefficients of u(x), ul(x), and F(x), using Theorem14 of Section 7.13. This gives a separate minimum problem for each n, which is solvedprecisely when - K2u" = F(x).]8. a) Determine the function u(x) that minimizesb) Use the Rayleigh-Ritz procedure to solve the problem of part (a), using as trial functionsthe functions9. Verify that the Green's function for (10.157), as described in the text, is the followingfunction when L = 1: