EXERCISES 3.4
3.4.1. 
The integration-by-parts formula
       $$\int_a_b u\,dv = \left. uv\right|_{x}^{b} - \int_a_b v\,du$$
is known to be valid for functions $u(x)$ and $v(x)$, which are continuous and
have continuous first derivatives. However, we will assume that $u, v, du/dx$,
and $dv/dx$ are continuous only for $a < x < c$ and $c < x < b$; we assume
that all quantities may have a jump discontinuity at $x = c$.

$*$(a) Derive an expression for $\int_a^b u\,dv$ in terms of $\int_a^b v\,du$.

(b) Show that this reduces to the integration-by-parts formula if $u$ and $v$
are continuous across $x =c$. It is not necessary for $du/dx$ and $dv/dx$
to be continuous at $x = c$.

3.4.2. 
Suppose that $f (x)$ and $df /dx$ are piecewise smooth. Prove that the Fourier
series of $f (x)$ can be differentiated term by term if the Fourier series of $f (x)$
is continuous.

3.4.3. 
Suppose that $f (x)$ is continuous [except for a jump discontinuity at $x = x_0,
f(x_0-) = \alpha$ and $f(x_0+)=\beta$] and $df/dx$ is piecewise smooth. 

$*$(a) Determine the Fourier sine series of $df /dx$ in terms of the Fourier cosine
series coefficients of $f (x)$.

(b) Determine the Fourier cosine series of $df /dx$ in terms of the Fourier
sine series coefficients of $f(x)$.

3.4.4. 
Suppose that $f (x)$ and $df /dx$ are piecewise smooth.  

(a) Prove that the Fourier sine series of a continuous function $f (x)$ can
only be differentiated term by term if $f (0) = 0$ and $f (L) = 0$.  

(b) Prove that the Fourier cosine series of a continuous function $f (x)$ can
be differentiated term by term.

3.4.5.
Determine the Fourier cosine series of $\sin(\pi x/L)$, using Fourier sine series formulas
developed in the previous section.

3.4.6. 
There are some things wrong in the following demonstration. Find the
mistakes and correct them.

In this exercise we attempt to obtain the Fourier cosine coefficients of 
$e^x$:
$$e^x = A_0 + \sum_{n=1}^\infty A_n \cos(n\pi x/L).$$
Differentiating yields
$$e^x =  -
\sum_{n=1}^\infty A_n \frac{n\pi}{L} \sin(n\pi x/L),$$
the Fourier sine series of $e^x$. Differentiating      again yields
$$ e^x = - \sum_{n=1}^\infty \left(\frac{n\pi}{L}\right)^2 A_n \cos(n\pi x/L).
Next, we match coefficients   
in the 
first and third series, because they are both cosine series.
Thus
$$ A_0=0,\quad A_n=0,\quad \mbox{Obviously wrong!}$$
By correcting the mistakes, you should be able to obtain $A_0$ and $A_n$
without using the typical technique, that is, 
$A_n=(2/L)\int_0^L e^x\cos(n\pi x/L)dx$.

3.4.7. 
Prove that the Fourier series of a continuous function $u(x, t)$ can be differentiated
term by term with respect to the parameter $t$ if $\partial u/\partial t$ is piecewise
smooth.

3.4.8. 
Consider
$$u_t = k u_{xx}$$
subject to
$u_x=0$ at $x=0$ and $x=L$, and
$u(x,0) = f(x)$.
Solve in the following way. Look for the solution as a Fourier cosine series.
Assume that $u$ and $u_x$ are continuous and $u_{xx}$ and $u_t$ are piecewise smooth.
Justify all differentiations of infinite series.

$*$3.4.9 
Consider the heat equation with a known source $q(x, t)$:
 $$u_t = k u_{xx} + q(x,t)$$
with $u(0, t) = 0$ and $u(L, t) = 0$.
Assume that $q(x, t)$ (for each $t > 0$) is a piecewise smooth function of $x$.
Also assume that $u$ and $u_x$
are continuous functions of $x$ (for $t > 0$) and
$u_{xx}$ and $u_t$
are piecewise smooth. Thus,
      $$u(x,t)=\sum_{n=1}^\infty b_n(t) \sin(n\pi x/L).$$
What ordinary differential equation does $b_n(t)$  satisfy? Do not solve this
differential equation.

3.4.10. 
Modify the previous exercise if instead $u_x=0$ at $x=0$ and $x=L$.

3.4.11. 
Consider the nonhomogeneous heat equation (with a steady heat source):
$$u_t = k u_{xx} + g(x).$$
Solve this equation with the initial condition
$u(x,0) = f(x)$
and the boundary conditions
$u(O,t) = 0$ and $u(L, t) = 0$.
Assume that a continuous solution exists (with continuous derivatives).
[Hints: Expand the solution as a Fourier sine series (i.e., use the method
of eigenfunction expansion). Expand $g(x)$ as a Fourier sine series. Solve
for the Fourier sine series of the solution. Justify all differentiations with
respect to $x$.]

*3.4.12. 
Solve the following nonhomogeneous problem:
$$u_t = k u_{xx} + e^{-t} + e^{-2t}\cos(3\pi x/L) \quad \mbox{[assume that $2\ne k(3\pi/L)^2$]}
$$
subject to
$u_x=0$ at $x=0$ and $x=L$, and $u(x,0)=f(x)$.
Use the following method. Look for the solution as a Fourier cosine series.
Justify all differentiations of infinite series (assume appropriate continuity).

3.4.13. 
Consider  
$$u_t = k u_{xx}$$
subject to
$u(0, t) = A(t)$, $u(L, t) = 0$, and $u(x, 0) = g(x)$.
Assume that $u(x,t)$ has a 
Fourier  sine series. 
Determine a differential equation
for the Fourier coefficients (assume appropriate continuity).