Let's break down **Part (a)** into two clear steps with explanations. --- ## **Step 1: Set Up the Problem and Discretize the PDE** **Explanation:** You are given the heat equation: \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \] with the domain \( \leq x \leq 1\) and \(t \geq \), and the following boundary/initial conditions: - \(u_x(, t) = \) (Neumann, insulated left boundary) - \(u_x(1, t) = \) (Neumann, insulated right boundary) - \(u(x, ) = P(x)\) (initial condition), where \(P(x) = 4x^3 - 3x^4 + [\sin(2\pi x)]^6\) To solve numerically, use the **finite difference method**: - **Discretize \(x\)**: Divide \(x\) into \(N\) points, \(x_j = j\Delta x\), \(j = , 1, ..., N\), \(\Delta x = 1/N\). - **Discretize \(t\)**: Choose a time step \(\Delta t\). - **Approximate derivatives**: \[ \frac{\partial u}{\partial t} \approx \frac{u_j^{n+1} - u_j^n}{\Delta t} \] \[ \frac{\partial^2 u}{\partial x^2} \approx \frac{u_{j+1}^n - 2u_j^n + u_{j-1}^n}{(\Delta x)^2} \] - **Update formula (explicit Euler)**: \[ u_j^{n+1} = u_j^n + \Delta t \frac{u_{j+1}^n - 2u_j^n + u_{j-1}^n}{(\Delta x)^2} \] - **Boundary conditions** (Neumann): - At \(x=\): \(\frac{\partial u}{\partial x}|_{x=} \approx \frac{u_1^n - u_^n}{\Delta x} = \implies u_1^n = u_^n\) - At \(x=1\): \(\frac{\partial u}{\partial x}|_{x=1} \approx \frac{u_N^n - u_{N-1}^n}{\Delta x} = \implies u_N^n = u_{N-1}^n\) --- ## **Step 2: Implement and Plot the Solution** **Explanation:** - **Initial condition:** Set \(u_j^ = P(x_j)\) for all \(j\). - **Iterate in time:** For each time step \(n\), update \(u_j^{n+1}\) using the formula above for all \(j=1\) to \(N-1\). - **Enforce boundary conditions:** Update \(u_^{n+1} = u_1^{n+1}\), \(u_N^{n+1} = u_{N-1}^{n+1}\) after every time step. - **Extract and plot:** At desired times (\(t = , .004, .016, .2\)), plot \(u(x, t)\) vs \(x\). **Final Answer (Summary for Part (a)):** > 1. **Discretize the PDE** using finite differences with Neumann boundary conditions and specified initial profile \(P(x)\). > 2. **Iterate the explicit update formula in time,** enforce boundary conditions, and plot \(u(x, t)\) at \(t = , .004, .016, .2\) on the same graph. --- If you need a sample Python/Matlab code for this, let me know!

Let's solve **Parts (a)**, **(b)**, and **(c)** of the problem step by step. --- ## **Part (a)**: Solve the PDE with Given Boundary Conditions ### **Step 1: Set Up the Problem and Discretize the PDE** **Explanation:** We start with the heat equation: \[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \] with the boundary conditions: - \(\frac{\partial u}{\partial x}(0, t) = 0\) (insulated left boundary) - \(\frac{\partial u}{\partial x}(1, t) = 0\) (insulated right boundary) - Initial condition: \(u(x, 0) = P(x)\) where \[ P(x) = 4x^3 - 3x^4 + [\sin(2\pi x)]^6 \] **Discretization:** - Divide \(x\) into \(N\) points: \(x_j = j\Delta x\) where \(\Delta x = \frac{1}{N}\). - Choose a time step \(\Delta t\). - Approximate derivatives: \[ \frac{\partial u}{\partial t} \approx \frac{u_j^{n+1} - u_j^n}{\Delta t} \] \[ \frac{\partial^2 u}{\partial x^2} \approx \frac{u_{j+1}^n - 2u_j^n + u_{j-1}^n}{(\Delta x)^2} \] **Update Formula:** \[ u_j^{n+1} = u_j^n + \Delta t \frac{u_{j+1}^n - 2u_j^n + u_{j-1}^n}{(\Delta x)^2} \] ### **Step 2: Implement and Plot the Solution** **Explanation:** - **Initial Condition:** Set \(u_j^0 = P(x_j)\) for all \(j\). - **Iterate in Time:** For each time step \(n\), update \(u_j^{n+1}\) using the formula. - **Boundary Conditions:** - For \(j=0\) (left boundary): \(u_1^n = u_0^n\) - For \(j=N\) (right boundary): \(u_N^n = u_{N-1}^n\) **Plotting:** Extract and plot \(u(x, t)\) at \(t = 0\), \(0.004\), \(0.016\), and \(0.2\). **Final Answer for Part (a):** > 1. **Discretize the PDE** using finite differences and apply Neumann boundary conditions. > 2. **Iterate and plot** \(u(x, t)\) for specified time values. --- ## **Part (b)**: Repeat with Modified Boundary Condition ### **Step 1: Set Up the Problem** **Explanation:** This time, change the first boundary condition to: - \(\frac{\partial u}{\partial x}(0, t) = 0\) (Neumann condition) The second boundary condition remains: - \(\frac{\partial u}{\partial x}(1, t) = 0\) - Initial condition: \(u(x, 0) = P(x)\) ### **Step 2: Implement and Plot the Solution** **Explanation:** - **Initial Condition:** Set \(u_j^0 = P(x_j)\) for all \(j\). - **Iterate in Time:** Update \(u_j^{n+1}\) using the same finite difference formula. - **Boundary Conditions:** - For \(j=0\): \(\frac{\partial u}{\partial x}(0, t)\) leads to: \[ u_1^n = u_0^n \] - For \(j=N\): Same as before, \(u_N^n = u_{N-1}^n\). **Plotting:** Extract and plot \(u(x, t)\) at \(t = 0\), \(0.004\), \(0.016\), and \(0.2\). **Final Answer for Part (b):** > 1. **Use the modified boundary condition** for the left endpoint while keeping the right endpoint condition the same. > 2. **Iterate and plot** \(u(x, t)\) for specified time values. --- ## **Part (c)**: Plot Temperature at Right Endpoint ### **Step 1: Extract Temperature at Right Endpoint** **Explanation:** From the results obtained in Part (b), extract the temperature at the right endpoint: \[ u(1, t) \text{ for } 0 < t < 0.1 \] ### **Step 2: Create the Plot** **Explanation:** - Choose a time increment \(\Delta t = 0.001\). - Calculate \(u(1, t)\) at each time step and store the values for plotting. **Final Answer for Part (c):** > 1. **Extract temperatures** at the right endpoint \(u(1, t)\) over the specified range. > 2. **Plot the results** to visualize the temperature evolution. --- This structured approach solves each part of the problem methodically. If you would like to see the implementation in Python or another programming language, please let me know!