Let's break down the problem and solve it step by step:

Question 1: Solve the wave equation

The given equation is:

$\frac{\partial^2 u}{\partial t^2} = -4 \frac{\partial^2 u}{\partial x^2}$

with the domain:

$0 \leq x \leq 2, \quad 0 \leq t \leq 1$

1.1. General Solution to the Wave Equation

This is a standard form of the wave equation, which has the general solution of the form:

$u(x, t) = f(x - 2t) + g(x + 2t)$

where $f$ and $g$ are arbitrary functions that will be determined from the initial and boundary conditions.

1.2. Apply Initial Conditions

Typically, initial conditions would be given for $u(x, 0)$ and $\frac{\partial u}{\partial t}(x, 0)$. However, since the exact initial conditions were not provided, we assume the general case with functions:

• $u(x, 0) = f(x) + g(x)$
• $\frac{\partial u}{\partial t}(x, 0) = -2f'(x) + 2g'(x)$

If initial conditions were given, we would use these to solve for $f(x)$ and $g(x)$.

1.3. Boundary Conditions

In most problems of this type, boundary conditions are also provided, which would typically involve the behavior of the solution at $x = 0$ and $x = 2$. If given, you would apply these conditions to solve for the specific form of $f$ and $g$.

Question 2: Derive the explicit finite difference scheme for the heat equation

The heat equation is given by:

$\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$

2.1. Discretizing the Heat Equation

To derive an explicit finite difference scheme, we discretize both time and space.

Let the grid points in space be $x_i = i\Delta x$ where $i = 0, 1, 2, \dots, N$, and the time steps be $t^n = n\Delta t$ where $n = 0, 1, 2, \dots, M$.

We approximate the derivatives as follows:

• Time derivative: $\frac{\partial u}{\partial t} \approx \frac{u_i^{n+1} - u_i^n}{\Delta t}$
• Spatial second derivative: $\frac{\partial^2 u}{\partial x^2} \approx \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\Delta x)^2}$

Substituting these into the heat equation:

$\frac{u_i^{n+1} - u_i^n}{\Delta t} = \alpha \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\Delta x)^2}$

Rearranging:

$u_i^{n+1} = u_i^n + \frac{\alpha \Delta t}{(\Delta x)^2} (u_{i+1}^n - 2u_i^n + u_{i-1}^n)$

2.2. Stability Condition

For the explicit finite difference scheme to be stable, the following condition (known as the CFL condition) must be satisfied:

$\frac{\alpha \Delta t}{(\Delta x)^2} \leq \frac{1}{2}$

This ensures that the numerical solution does not grow unbounded over time.

Summary of Steps:

1. For the wave equation, solve it in terms of $f(x-2t)$ and $g(x+2t)$ using the initial and boundary conditions.
2. For the heat equation, derive the explicit finite difference scheme and enforce the stability condition $\frac{\alpha \Delta t}{(\Delta x)^2} \leq \frac{1}{2}$.

Let me know if you need further clarification or specific details on any of the steps. I can also help with solving using specific initial conditions if you provide them!

Related Questions:

1. How do you solve the wave equation with specific initial and boundary conditions?
2. How does the stability condition for the heat equation affect the time step and space step?
3. What are the key differences between the heat equation and the wave equation in terms of their numerical schemes?
4. How would you derive the implicit finite difference scheme for the heat equation?
5. What role does the CFL condition play in the stability of the explicit finite difference scheme?

Tip:

When solving PDEs numerically, always verify the stability of your scheme first; unstable schemes can lead to physically meaningless results.