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Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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The normalized eigenfunctions of the given problem are
\[2 sin Fklx
(x) = ---------/ ; n = 1,2,.... (29)
n (1 + cos2 ,/Tn )1/2
We now turn to the question of expressing a given function f as a series of eigenfunctions of the Sturm-Liouville problem (1), (2). We have already seen examples of such expansions in Sections 10.2 to 10.4. For example, it was shown there that if f is continuous and has a piecewise continuous derivative on 0 < x < 1, and satisfies the boundary conditions f (0) = f (1) = 0, then f can be expanded in a Fourier sine series of the form
TO
f(x) = ? bn sin nn x. (30)
n= 1
The functions sin nn x, n = 1, 2,, are precisely the eigenfunctions of the boundary value problem (10). The coefficients bn are given by
bn = 2 f f (x) sin nn x dx (31)
n0
and the series (30) converges for each x in0 < x < 1. In a similar way f can be expanded in a Fourier cosine series using the eigenfunctions cos nnx, n = 0, 1, 2,..., of the boundary value problem y" + ky = 0, y'(0) = 0, /(1) = 0.
Now suppose that a given function f, satisfying suitable conditions, can be expanded in an infinite series of eigenfunctions of the more general Sturm-Liouville problem
(1), (2). If this can be done, then we have
TO
f(x) = E Cn(x), (32)
n= 1
where the functions (x) satisfy Eqs. (1), (2), and also the orthogonality condition (22). To compute the coefficients in the series (32) we multiply Eq. (32) by r(x)(x), where m is a fixed positive integer, and integrate from x = 0 to x = 1. Assuming that the series can be integrated term by term we obtain
/1 TO /1 TO
/ r(x) f (x)(x) dx =^2 Cn (x)(x)(x) dx =Yh CnSmn. (33)
J0 n=1 J0 n=1
Hence, using the definition of Smn, we have
Cm =f r(x) f (x^m(x) dx = ( f, m), m = 1, 2,.... (34)
636
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
Theorem 11.2.4
EXAMPLE
3
The coefficients in the series (32) have thus been formally determined. Equation (34) has the same structure as the Euler-Fourier formulas for the coefficients in a Fourier series, and the eigenfunction series (32) also has convergence properties similar to those of Fourier series. The following theorem is analogous to Theorem 10.3.1.
Let 1, 2,... , ,... be the normalized eigenfunctions ofthe Sturm-Liouville problem (1), (2):
[p(x)/]'- q(x)y + kr(x)y = 0,
31 y(0) + a2y (0) = 0, bl y (1) + V(1) = 0.
Let f and f be piecewise continuous on 0 < x < 1. Then the series (32) whose coefficients cm are given by Eq. (34) converges to [ f (x+) + f (x-)]/2 at each point in the open interval 0 < x < 1.
If f satisfies further conditions, then a stronger conclusion can be established. Suppose that, in addition to the hypotheses of Theorem 11.2.4, the function f is continuous on 0 < x < 1. If a2 = 0 in the first of Eqs. (2) [so that (0) = 0], then assume that f (0) = 0. Similarly, if b2 = 0 in the second of Eqs. (2), assume that f (1) = 0. Otherwise no boundary conditions need be prescribed for f. Then the series (32) converges to f (x) at each point in the closed interval 0 < x < 1.
Expand the function
f(x) = x, 0 < x < 1 (35)
in terms of the normalized eigenfunctions (x) of the problem (25).
In Example 2 we found the normalized eigenfunctions to be
(x) = kn sin yjknx> (36)
where kn is given by Eq. (28) and kn satisfies Eq. (26). To find the expansion for f in terms of the eigenfunctions we write
TO
f(x) = E Cn (x), (37)
n=1
where the coefficients are given by Eq. (34). Thus
C" = L f{X)*n(x) dx = kn / xsin^x dx.
Integrating by parts, we obtain
_t /sinv^n cosv^M _ , 2sinv^n
c = kn ,/r) = n kn
n n
11.2 Sturm-Liouville Boundary Value Problems
637
where we have used Eq. (26) in the last step. Upon substituting for kn from Eq. (28) we obtain
2V2sin
Thus
f (x) = 4J2
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