Let's break down the problem: ### Initial Condition: - The rod is 7 meters long. - The initial temperature distribution \( u(x, 0) \) is: - \( 43.72^\circ \) for \( 0 \leq x < 0.70 \) - \( 87.72^\circ \) for \( 0.70 \leq x \leq 7 \) ### Boundary Conditions: - \( u(0, t) = 0 \) - \( u(7, t) = 0 \) ### Required: Find the temperature \( u(x, t) \) at the given values of \( x \), \( t \), and \( \kappa \). --- This is a heat equation problem with a piecewise initial condition and Dirichlet boundary conditions. #### The general solution for the heat equation with these boundary conditions is: \[ u(x, t) = \sum_{n=1}^{\infty} B_n \sin\left(\frac{n\pi x}{L}\right) e^{-\kappa \left(\frac{n\pi}{L}\right)^2 t} \] where \( L = 7 \) and \( B_n \) are the Fourier sine coefficients determined by the initial condition. #### The coefficients \( B_n \) are: \[ B_n = \frac{2}{L} \int_0^L u(x, 0) \sin\left(\frac{n\pi x}{L}\right) dx \] Given the piecewise initial condition: \[ B_n = \frac{2}{7} \left[ \int_0^{0.70} 43.72 \sin\left(\frac{n\pi x}{7}\right) dx + \int_{0.70}^{7} 87.72 \sin\left(\frac{n\pi x}{7}\right) dx \right] \] --- ### Now, let's compute the temperature at the given points: #### For each row in the table: - Plug \( x \), \( t \), \( \kappa \) into the formula above. --- #### Let's walk through the first row as an example: - \( x = 0.91 \) - \( t = 1.14 \) s - \( \kappa = 0.08 \) 1. **Calculate \( B_n \):** - For practical purposes, let's consider only the first term (\( n=1 \)), which is usually sufficient for an estimate: \[ B_1 = \frac{2}{7} \left[ \int_0^{0.70} 43.72 \sin\left(\frac{\pi x}{7}\right) dx + \int_{0.70}^{7} 87.72 \sin\left(\frac{\pi x}{7}\right) dx \right] \] Let's denote: - \( I_1 = \int_0^{0.70} 43.72 \sin\left(\frac{\pi x}{7}\right) dx \) - \( I_2 = \int_{0.70}^{7} 87.72 \sin\left(\frac{\pi x}{7}\right) dx \) Compute these integrals (using substitution, or a calculator): \[ I_1 = 43.72 \left[ -\frac{7}{\pi} \cos\left(\frac{\pi x}{7}\right) \right]_0^{0.70} \] \[ = 43.72 \left[ -\frac{7}{\pi} \left( \cos\left(\frac{\pi \cdot 0.70}{7}\right) - \cos(0) \right) \right] \] Similarly, \[ I_2 = 87.72 \left[ -\frac{7}{\pi} \cos\left(\frac{\pi x}{7}\right) \right]_{0.70}^{7} \] \[ = 87.72 \left[ -\frac{7}{\pi} \left( \cos(\pi) - \cos\left(\frac{\pi \cdot 0.70}{7}\right) \right) \right] \] Now: - \( \cos(0) = 1 \) - \( \cos(\pi) = -1 \) - \( \cos\left(\frac{\pi \cdot 0.70}{7}\right) = \cos(0.1\pi) \approx 0.9511 \) So, \[ I_1 = 43.72 \left[ -\frac{7}{\pi} (0.9511 - 1) \right] = 43.72 \left[ -\frac{7}{\pi} (-0.0489) \right] \] \[ = 43.72 \times \frac{7}{\pi} \times 0.0489 \approx 43.72 \times 2.228 \times 0.0489 \approx 43.72 \times 0.109 \approx 4.77 \] \[ I_2 = 87.72 \left[ -\frac{7}{\pi} (-1 - 0.9511) \right] = 87.72 \left[ -\frac{7}{\pi} (-1.9511) \right] \] \[ = 87.72 \times \frac{7}{\pi} \times 1.9511 \approx 87.72 \times 2.228 \times 1.9511 \approx 87.72 \times 4.345 \approx 381.4 \] \[ B_1 = \frac{2}{7}(4.77 + 381.4) = \frac{2}{7}(386.17) \approx 110.62 \] 2. **Plug into the solution:** \[ u(x, t) \approx B_1 \sin\left(\frac{\pi x}{7}\right) e^{-0.08 \left(\frac{\pi}{7}\right)^2 \times 1.14} \] \[ \frac{\pi}{7} \approx 0.4488 \] \[ \left(\frac{\pi}{7}\right)^2 \approx 0.2014 \] \[ 0.08 \times 0.2014 \times 1.14 \approx 0.01836 \] \[ e^{-0.01836} \approx 0.9818 \] \[ \sin\left(\frac{\pi \times 0.91}{7}\right) = \sin(0.4089) \approx 0.397 \] \[ u(0.91, 1.14) \approx 110.62 \times 0.397 \times 0.9818 \approx 110.62 \times 0.3896 \approx 43.06 \] --- ### **Summary Table (First Row Example):** | x-coordinate | Time | Conductivity | Temperature | |--------------|-------|--------------|--------------| | x = 0.91 | 1.14s | κ = 0.08 | 43.06 °C | --- Would you like to see the calculations for the remaining rows, or would you like help setting this up in Python or Excel to automate the calculations?

Certainly! Here's a step-by-step guide to set up this problem in Excel, allowing you to compute \( u(x,t) \) at various points efficiently: --- ### **Step 1: Organize your data** Create a table with the following columns: | A | B | C | D | E | F | G | H | |---|---|---|---|---|---|---|---| | x | t | κ | N_terms | B_n (for n=1 to N_terms) | sin(nπx/7) | e^{-κ(nπ/7)^2 t} | u(x,t) | --- ### **Step 2: Input your known data** - **Column A:** x-coordinates (e.g., 0.91) - **Column B:** t (time in seconds) - **Column C:** κ (thermal diffusivity) Example: | A2 | B2 | C2 | |-----|-----|-----| | 0.91 | 1.14 | 0.08 | --- ### **Step 3: Set parameters** - In a separate cell, define \( L = 7 \) (say, cell `J1`) - Decide on the number of terms \( N \) for the Fourier series (say, 10). Put this in cell `J2`. ```excel J1: 7 J2: 10 ``` --- ### **Step 4: Compute Fourier coefficients \( B_n \)** Since \( B_n \) depends on the initial piecewise distribution, you can precompute \( B_n \) for each \( n \). **Method:** - For \( n=1 \) to \( N \), compute: \[ B_n = \frac{2}{L} \left[ \int_0^{0.70} u_0(x) \sin\left(\frac{n\pi x}{L}\right) dx + \int_{0.70}^{L} u_0(x) \sin\left(\frac{n\pi x}{L}\right) dx \right] \] - These integrals are straightforward to evaluate analytically as shown earlier, or numerically via Excel. **Implementation:** - In Excel, create a column for \( n \) (say, starting at `K2`): fill from 1 to N. - For each \( n \), compute \( B_n \): ```excel B_n = (2 / L) * [43.72 * integral over 0 to 0.7 + 87.72 * integral over 0.7 to 7] ``` - The integrals: \[ \int a \sin(bx) dx = -\frac{a}{b} \cos(bx) \] - So, for each \( n \): ```excel b = n * PI() / L ``` - Compute: ```excel I1 = 43.72 * [ - (7 / (n * PI())) * (cos(b * 0.7) - cos(0)) ] I2 = 87.72 * [ - (7 / (n * PI())) * (cos(b * 7) - cos(b * 0.7)) ] ``` - Then, ```excel B_n = (2 / L) * (I1 + I2) ``` **Example in Excel:** Suppose for n in `K2`: ```excel n: 1 b: =K2 * PI() / $J$1 I1: =43.72 * ( - (7 / (K2 * PI())) * (COS(b * 0.7) - COS(0)) ) I2: =87.72 * ( - (7 / (K2 * PI())) * (COS(b * 7) - COS(b * 0.7)) ) B_n: =(2 / $J$1) * (I1 + I2) ``` Drag down for all n. --- ### **Step 5: Compute the series sum for \( u(x,t) \)** - For each \( n \), compute: \[ \sin\left(\frac{n\pi x}{L}\right) \] and \[ e^{-\kappa \left(\frac{n\pi}{L}\right)^2 t} \] - In Excel, for each \( n \): ```excel sin_part: =SIN( K2 * PI() * A2 / $J$1 ) exp_part: =EXP( - $C$2 * (K2 * PI() / $J$1)^2 * B2 ) ``` - Then, the contribution of each term: ```excel Term_n: = B_n * sin_part * exp_part ``` - Sum over all \( n \): In a cell (say, `L2`), sum all `Term_n` from `K2` to `K_(N+1)`: ```excel =SUM( L2: L_(N+1) ) ``` --- ### **Step 6: Automate for multiple points** - Repeat steps 4 and 5 for each row of your data. - To automate, you can set up the calculations in a table and copy formulas accordingly. --- ### **Summary of key formulas:** | Step | Formula | Explanation | |--------|---------|--------------| | \( b \) | `=K2 * PI() / $J$1` | \( n \pi / L \) | | \( I_1 \) | `=43.72 * ( - (7 / (K2 * PI())) * (COS(b * 0.7) - COS(0)) )` | First integral | | \( I_2 \) | `=87.72 * ( - (7 / (K2 * PI())) * (COS(b * 7) - COS(b * 0.7)) )` | Second integral | | \( B_n \) | `=(2 / $J$1) * (I_1 + I_2)` | Fourier coefficient | | \( \sin(n\pi x / 7) \) | `=SIN( K2 * PI() * A2 / $J$1 )` | Spatial sine component | | \( e^{-\kappa (n\pi/7)^2 t} \) | `=EXP( - C2 * (K2 * PI() / $J$1)^2 * B2 )` | Exponential decay | --- ### **Final tip:** - Use a **helper column** for each \( n \)'s contribution. - Sum all contributions for the final \( u(x,t) \). --- ### **Would you like a sample Excel file?** I can prepare a downloadable template with these formulas pre-entered for you. --- **Let me know if you'd like the actual file, or if you'd like further customization!**