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u(x, t) = 4e-t '''Ã'_______________________________________________________________________V n_Ó n_4 _
’ n=1 Xn (Xn-1)(1 + cos2v^n) n=1 Xn (Xn-1)(1 + cos^v^n)-
Recall from Example 1 in Section 11.1 that the eigenvalues Xn are very nearly proportional to n2. In the first series on the right side of Eq. (51) the trigonometric factors are all bounded as n ^ to; therefore, this series converges similarly to the series
X-2 or n-4. Hence at most two or three terms are required to obtain an excellent
approximation to this part of the solution. The second series contains the additional factor e~Xnt, so its convergence is even more rapid for t > 0; all terms after the first are almost surely negligible.
Further Discussion. Eigenfunction expansions can be used to solve a much greater variety of problems than the preceding discussion and examples may suggest. The following brief remarks are intended to indicate to some extent the scope of this method:
Chapter 11. Boundary Value Problems and Sturm-Liouville Theory
1. Nonhomogeneous Time-Independent Boundary Conditions. Suppose that the boundary conditions (26) in the heat conduction problem are replaced by
u(0, t) = T1, u(1, t) = T2. (52)
Then we can proceed much as in Section 10.6 to reduce the problem to one with homogeneous boundary conditions; that is, we subtract from u a function v that is chosen to satisfy Eqs. (52). Then the difference w = u - v satisfies a problem with homogeneous boundary conditions, but with modified forcing term and initial condition. This problem can be solved by the procedure described in this section.
2. Nonhomogeneous Time-Dependent Boundary Conditions. Suppose that the boundary conditions (26) are replaced by
u(0, t) = T(), u(1, t) = T2(t). (53)
If T1 and T2 are differentiable functions, one can proceed just as in paragraph
1. In the same way it is possible to deal with more general boundary conditions involving both u and ux. If T1 and T2 are not differentiable, however, then the method fails. A further refinement of the method of eigenfunction expansions can be used to cope with this type of problem, but the procedure is more complicated and we will not discuss it.
3. Use of Functions Other Than Eigenfunctions. One potential difficulty in using eigenfunction expansions is that the normalized eigenfunctions of the corresponding homogeneous problem must be found. For a differential equation with variable coefficients this may be difficult, if not impossible. In such a case it is sometimes possible to use other functions, such as eigenfunctions of a simpler problem, that satisfy the same boundary conditions. For instance, if the boundary conditions are
u(0, t) = 0, u(1, t) = 0, (54)
then it may be convenient to replace the functions Ôï(x) in Eq. (30) by sin nnx.
These functions at least satisfy the correct boundary conditions, although in general they are not solutions of the corresponding homogeneous differential equation. Next we expand the nonhomogeneous term F(x, t) in a series of the form (34), again with Ôï (x) replaced by sin nn x, and then substitute for both u and F in Eq. (25). Upon collecting the coefficients of sin nnx for each n, we have an infinite set of linear first order differential equations from which to determine b1(t), b2(t),.... The essential difference between this case and the one considered earlier is that now the equations for the functions bn(t) are coupled. Thus they cannot be solved one by one, as before, but must be dealt with simultaneously. In practice, the infinite system is replaced by an approximating finite system, from which approximations to a finite number of coefficients are calculated. Despite its complexity, this procedure has proved very useful in solving certain types of difficult problems.
4. Higher Order Problems. Boundary value problems for equations of higher than second order can often be solved by eigenfunction expansions. In some cases the procedure parallels almost exactly that for second order problems. However, a variety of complications can also arise, and we will not discuss such problems in this book.
Finally, we emphasize that the discussion in this section has been purely formal.
Separate and sometimes elaborate arguments must be used to establish convergence of
11.3 Nonhomogeneous Boundary Value Problems
eigenfunction expansions, or to justify some of the steps used, such as term-by-term differentiation of eigenfunction series.
There are also other altogether different methods for solving nonhomogeneous boundary value problems. One of these leads to a solution expressed as a definite integral rather than as an infinite series. This approach involves certain functions known as Green’s functions, and for ordinary differential equations is the subject of Problems 28 through 36.