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Question 1: Finite Difference Method

Problem Statement Recap:

Given the partial differential equation (PDE): $u_t(x,t) = u_{xx}(x,t) + e^{-(x-1)^2}, \quad 0 \leq x \leq 1, \, t > 0$ with boundary conditions $u(0,t) = u(1,t) = 0$, and the initial condition $u(x,0) = 2x(1-x)$.

Objective:

Use the finite difference method to discretize and solve the PDE.

Proof (Sketch):

1. Discretization:

   • Spatial domain discretized with points $x_i = i\Delta x$ where $i = 0, 1, \ldots, N$ and $\Delta x = \frac{1}{N}$.
   • Time discretized as $t_n = n\Delta t$ where $n = 0, 1, 2, \ldots$.

2. Finite Difference Approximation:

   Use central difference for the second derivative in space and forward difference in time: $u_{xx}(x_i,t_n) \approx \frac{u_{i+1}^n - 2u_i^n + u_{i-1}^n}{(\Delta x)^2}$ $u_t(x_i,t_n) \approx \frac{u_i^{n+1} - u_i^n}{\Delta t}$

3. Discretized Equation:

   Substituting the approximations into the original PDE gives: [ \frac{u_i^{n+1} - u_i^n}{\Delta t} = \frac{u_{