This problem focuses on solving a 1st order hyperbolic partial differential equation (PDE) using both analytical and numerical methods. The equation is:

$3u_x + 4u_t = \frac{11}{x + 2t + 1}$

with the initial condition $u(x, 0) = \ln(x + 1)$ over the interval $0 \leq x \leq 1$. For the numerical methods, a spatial step size $h = 0.2$ and time step size $k = 0.1$ are provided.

Part (a)

You are asked to use the method of characteristics to solve the PDE analytically. This method involves solving a system of ordinary differential equations (ODEs) derived from the PDE to trace characteristic curves along which the solution is constant.

Steps:

1. The characteristic equations can be derived from the PDE by solving for $\frac{dx}{dt}$ and integrating both sides.
2. Then, the solution $u(x, t)$ is found by integrating along the characteristic curves, using the boundary condition $u(x, 0) = \ln(x + 1)$.
3. Finally, compute the solution at $u(x, 2)$ for the given boundary values, rounding to 4 decimal places.

Part (b)

You are asked to determine the stability of three numerical methods: FTFS (Forward Time Forward Space), FTBS (Forward Time Backward Space), and the Lax-Wendroff method. This involves calculating the Courant-Friedrichs-Lewy (CFL) condition or ratio to check whether the methods are stable or unstable for the given step sizes $h$ and $k$.

Stability criteria:

1. Compute the Courant number $C = \frac{k}{h} \cdot \frac{a}{b}$, where $a$ and $b$ are derived from the PDE.
2. Compare the CFL number against stability bounds for each method (e.g., FTFS is stable if $C \leq 1$, etc.).

Part (c)

For this part, you are required to use the FTBS method and Maple (a computational tool) to compute the numerical solution for $u(x, 2)$ up to 4 decimal places. The task involves discretizing the PDE using the finite difference method (FTBS scheme) and solving the resulting system numerically.

Steps:

1. Write down the finite difference scheme for the FTBS method.
2. Apply the scheme iteratively for the spatial and temporal discretization with $h = 0.2$ and $k = 0.1$.
3. Use Maple to assist in solving the system and compute $u(x, 2)$.

Do you want me to work through the detailed solution for any of these parts or explain any concepts further?

Here are 5 related questions that build on this problem:

1. How do you derive the characteristic equations for first-order PDEs like this one?
2. What is the Courant-Friedrichs-Lewy (CFL) condition, and why is it important for stability?
3. How does the Lax-Wendroff method differ from FTFS and FTBS in terms of stability?
4. Can you explain how to implement the FTBS method numerically step-by-step?
5. What are the advantages and disadvantages of using numerical methods for solving PDEs compared to analytical solutions?

Tip: In numerical methods for PDEs, always check the stability of your method before implementing it to avoid divergent solutions.