we applied separation of variables successfully to a linear homogeneous PDE: the one-dimensional heat equation with Dirichlet boundary conditions
on the interval (0, e) :
au = k a2u
at ax2 '
x E n = (0, e), t > 0,
u(O, t) = 0,
u(e, t) = 0,
27fx
u(x,O) = f(x) = 3 sin -e-' 

To solve the given one-dimensional heat equation using **separation of variables**, let's break the problem into manageable steps. The equation provided is:

\[
\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}
\]
with the boundary conditions:
\[
u(0, t) = 0, \quad u(e, t) = 0, \quad u(x, 0) = f(x) = 3 \sin \left( \frac{2 \pi x}{e} \right)
\]

### Step 1: Separation of Variables
Assume the solution \( u(x,t) \) can be written as the product of two functions:

\[
u(x,t) = X(x)T(t)
\]

Substitute into the PDE:

\[
X(x)T'(t) = k X''(x)T(t)
\]

Divide both sides by \( X(x)T(t) \):

\[
\frac{T'(t)}{k T(t)} = \frac{X''(x)}{X(x)} = -\lambda
\]

Here, \( \lambda \) is the separation constant.

### Step 2: Solve the Spatial Equation
We now have the spatial ODE:

\[
X''(x) + \lambda X(x) = 0
\]

With the boundary conditions:

\[
X(0) = 0, \quad X(e) = 0
\]

This is a standard Sturm-Liouville problem. The general solution for \( X(x) \) is:

\[
X(x) = A \sin(\sqrt{\lambda} x) + B \cos(\sqrt{\lambda} x)
\]

Apply the boundary conditions:

1. \( X(0) = 0 \Rightarrow B = 0 \), so \( X(x) = A \sin(\sqrt{\lambda} x) \).
2. \( X(e) = 0 \Rightarrow A \sin(\sqrt{\lambda} e) = 0 \).

For non-trivial solutions (\( A \neq 0 \)), this implies:

\[
\sqrt{\lambda} e = n\pi, \quad n = 1, 2, 3, \dots
\]

Thus, \( \lambda_n = \left( \frac{n\pi}{e} \right)^2 \) and the eigenfunctions are:

\[
X_n(x) = \sin\left( \frac{n\pi x}{e} \right)
\]

### Step 3: Solve the Temporal Equation
The temporal equation is:

\[
T'(t) = -\lambda_n k T(t)
\]

The solution is:

\[
T_n(t) = C_n e^{-k \lambda_n t} = C_n e^{-k \left( \frac{n\pi}{e} \right)^2 t}
\]

### Step 4: Form the General Solution
The general solution to the PDE is the sum of all modes:

\[
u(x,t) = \sum_{n=1}^{\infty} C_n \sin\left( \frac{n\pi x}{e} \right) e^{-k \left( \frac{n\pi}{e} \right)^2 t}
\]

### Step 5: Apply the Initial Condition
The initial condition is:

\[
u(x,0) = f(x) = 3 \sin \left( \frac{2 \pi x}{e} \right)
\]

Thus, the Fourier series simplifies to a single term (since \( f(x) \) is already given as a sine function of a specific mode):

\[
u(x,0) = C_2 \sin \left( \frac{2 \pi x}{e} \right)
\]

Comparing this with the given initial condition, we find:

\[
C_2 = 3
\]

### Final Solution
The solution to the PDE is:

\[
u(x,t) = 3 \sin\left( \frac{2\pi x}{e} \right) e^{-k \left( \frac{2\pi}{e} \right)^2 t}
\]

### Key Points Recap:
- **Method:** Separation of variables.
- **Boundary conditions:** Applied to find \( X(x) \).
- **Temporal solution:** Obtained from solving \( T(t) \).
- **Initial condition:** Used to determine the constant \( C_2 \).

Would you like more details or clarifications on any step?

---

### 5 Related Questions:
1. How would the solution change if the boundary conditions were non-homogeneous?
2. What is the physical interpretation of the separation constant \( \lambda \)?
3. How would you extend this solution to a two-dimensional heat equation?
4. Can the method of separation of variables be applied to nonlinear PDEs?
5. How would you compute the solution numerically if \( f(x) \) were more complex?

### Tip:
Always check the orthogonality of the eigenfunctions when applying Fourier series in PDE solutions—it ensures proper coefficients in the series expansion.

We applied separation of variables successfully to a linear homogeneous PDE: the one-dimensional heat equation with Dirichlet boundary conditions on the interval (0, e) : au = k a2u / at ax2, x E n = (0, e), t > 0, u(O, t) = 0, u(e, t) = 0, 27fx u(x,O) = f(x) = 3 sin -e-.

Explore how to solve the one-dimensional heat equation with Dirichlet boundary conditions using separation of variables and Fourier series. The detailed solution includes key concepts such as partial differential equations, the heat equation, and the Sturm-Liouville problem. Suitable for undergraduate-level students.

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