Let $\left(a_{k}\right)_{k \in \mathbb{N}}$ be a sequence of complex terms such that the series with general term $k^{2} a_{k}$ is absolutely convergent. For $(t, x) \in \mathbb{R}^{2}$ we then denote $$\Phi_{0}(t, x) = \sum_{k=0}^{+\infty} a_{k} e^{-ik^{2}t + ikx}$$
(a) Show that $\Phi_{0}$ is well defined on $\mathbb{R}^{2}$.
(b) Show that for all $(t, x) \in \mathbb{R}^{2}$ we have $\Phi_{0}(., x) \in C^{1}(\mathbb{R}, \mathbb{C})$ and $\Phi_{0}(t, .) \in C^{2}(\mathbb{R})$. Calculate $\frac{\partial \Phi_{0}}{\partial t}(t, x)$ and $\frac{\partial^{2} \Phi_{0}}{\partial x^{2}}(t, x)$.
(c) Let $(c_{k})$ be a sequence of complex terms such that the series with general term $k^{2} c_{k}$ is absolutely convergent. For $x \in \mathbb{R}$, we set $$f_{0}(x) = \sum_{k=0}^{+\infty} c_{k} e^{ikx}$$ Construct a function $\Psi_{0}$ defined on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ which satisfies

• For all $(t, x) \in \mathbb{R}_{+}^{*}$, $\Psi_{0}(., x) \in C^{1}(\mathbb{R}_{+}^{*}, \mathbb{C})$ and $\Psi_{0}(t, .) \in C^{2}(\mathbb{R}, \mathbb{C})$ and $\Psi_{0}$ is a solution of equation $(F_{0})$, that is $$\forall (t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}, \quad i\frac{\partial \Psi_{0}}{\partial t}(t, x) + \frac{\partial^{2} \Psi_{0}}{\partial x^{2}}(t, x) + \frac{1}{2t}\Psi_{0}(t, x) = 0$$
• For all $t > 0$, $\Psi_{0}(t, .)$ is periodic.
• For all $x \in \mathbb{R}$, $\Psi_{0}(1, x) = f_{0}(x)$.

Problem 5
I. A function $f ( x )$ is continuous and defined on the interval $0 \leq x \leq \pi$. If $f ( x )$ is extended to the interval $- \pi \leq x \leq \pi$ as an odd function, it can be expanded in the following Fourier sine series:
$$\begin{aligned} & f ( x ) \doteq \sum _ { n = 1 } ^ { \infty } \left( b _ { n } \sin n x \right) \\ & b _ { n } = \frac { 2 } { \pi } \int _ { 0 } ^ { \pi } f ( x ) \sin n x \, d x \quad ( n = 1,2,3 , \cdots ) \end{aligned}$$
Here, $f ( 0 ) = f ( \pi ) = 0$.

1. Find the Fourier sine series for the following function $f ( x )$: $$f ( x ) = x ( \pi - x ) \quad ( 0 \leq x \leq \pi )$$
2. Derive the following equation using the result obtained in Question I.1, $$\frac { 1 } { 1 ^ { 3 } } - \frac { 1 } { 3 ^ { 3 } } + \frac { 1 } { 5 ^ { 3 } } - \frac { 1 } { 7 ^ { 3 } } + \cdots = \frac { \pi ^ { 3 } } { 32 }$$

II. A two-variable function $f ( x , y )$ is continuous and defined in the region $0 \leq x \leq \pi$ and $0 \leq y \leq \pi$. Using a similar method to Question I, $f ( x , y )$ can be expanded in the following double Fourier sine series:
$$\begin{aligned} & f ( x , y ) = \sum _ { m = 1 } ^ { \infty } \sum _ { n = 1 } ^ { \infty } \left( B _ { m n } \sin m x \sin n y \right) \\ & B _ { m n } = \frac { 4 } { \pi ^ { 2 } } \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { \pi } f ( x , y ) \sin m x \sin n y \, d x \, d y \quad ( m , n = 1,2,3 , \cdots ) \end{aligned}$$
Here, $f ( 0 , y ) = f ( \pi , y ) = f ( x , 0 ) = f ( x , \pi ) = 0$.

1. Find the double Fourier sine series for the following function $f ( x , y )$: $$f ( x , y ) = x ( \pi - x ) \sin y \quad ( 0 \leq x \leq \pi , 0 \leq y \leq \pi )$$
2. Function $u ( x , y , t )$ is defined in the region $0 \leq x \leq \pi , 0 \leq y \leq \pi$ and $t \geq 0$. Obtain the solution for the following partial differential equation of $u ( x , y , t )$ by the method of separation of variables: $$\frac { \partial u } { \partial t } = c ^ { 2 } \left( \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u } { \partial y ^ { 2 } } \right)$$ where $c$ is a positive constant and the following boundary and initial conditions apply: $$\begin{aligned} & u ( 0 , y , t ) = u ( \pi , y , t ) = u ( x , 0 , t ) = u ( x , \pi , t ) = 0 \\ & u ( x , y , 0 ) = x ( \pi - x ) \sin y \end{aligned}$$

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).
$$u ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } u _ { n } ( x , t )$$