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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_{

Explore the application of the finite difference method in solving partial differential equations (PDEs). Step-by-step solution using central and forward difference approximations. Suitable for advanced undergraduate students and researchers in mathematics and engineering.

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