A real-valued function $u ( x , t )$ is defined in $0 \leq x \leq 1$ and $t \geq 0$. Here, $x$ and $t$ are independent. Suppose solving the following partial differential equation:
$$\frac { \partial u } { \partial t } = \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } ,$$
under the following conditions:
$$\begin{aligned} \text { Boundary condition : } & u ( 0 , t ) = u ( 1 , t ) = 0 , \\ \text { Initial condition : } & u ( x , 0 ) = x - x ^ { 2 } . \end{aligned}$$
Since the constant function $u ( x , t ) = 0$ is obviously a solution of the partial differential equation, consider the other solutions. Answer the following questions.
(1) Calculate the following expression, where $n$ and $m$ are positive integers.
$$\int _ { 0 } ^ { 1 } \sin ( n \pi x ) \sin ( m \pi x ) \mathrm { d } x$$
(2) Suppose $u ( x , t ) = \xi ( x ) \tau ( t )$, where $\xi ( x )$ is a function only of $x$ and $\tau ( t )$ is a function only of $t$. Express the ordinary differential equations for $\xi$ and $\tau$ using an arbitrary constant $C$. You may use that $f ( x )$ and $g ( t )$ are constant functions when $f ( x )$ and $g ( t )$ satisfy $f ( x ) = g ( t )$ for arbitrary $x$ and $t$.
(3) Solve the ordinary differential equations in question (2). Next, show that a solution of partial differential equation $(*)$ which satisfies the boundary condition is given by the following $u _ { n } ( x , t )$, and express $\alpha$ and $\beta$ using a positive integer $n$.
$$u _ { n } ( x , t ) = e ^ { \alpha t } \sin ( \beta x )$$
(4) The solution of partial differential equation $(*)$ which satisfies the boundary and initial conditions is represented by the linear combination of $u _ { n } ( x , t )$ as shown below. Obtain $c _ { n }$. You may use the result of question (1).
$$u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } u _ { n } ( x , t )$$