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?

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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.