Give step-by-step solution with explanation and final answer:Question 1. Consider the operator L = d?/dz? on the interval [0,7] with boundary conditions u(0) =
0,/(r) = 0. Using the standard second o

Question

Give step-by-step solution with explanation and final answer:Question 1. Consider the operator L = d?/dz? on the interval [0,7] with boundary conditions u(0) =
0,/(r) = 0. Using the standard second order centered difference expression for the second derivative,
derive a matrix approximation D to L. Namely, divide the interval into N subintervals of size Az = m/N.
To a function u(x) in the domain of L, let u be the vector whose N values are u(nAz), n = 1,2,..., N (note
that we do not include the value at z = 0 why?). Waite function that returns a (N) x (N) dimensional
matrix D such that Du contains the estimated values of the second derivative using the second order
centered difference formula at the points nAz, n = 1,2,..., N. Hint: D is a matrix whose only nonzero
elements lie on the main diagonal and on the diagonal just below and just above the main diagonal. To
automate the generation of the matrix, use the matlab function diag. Give a formula for the eigenvalues
of D.
Submit to Matlab grader and Gradescope (40 points)
Question 2. Consider the one dimensional heat equation
ou tu du(r, 1) 1 if0

VIPSolutions · Accepted Answer

Let's break down **Question 1** step by step:

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### **Question 1:** Finite Difference Matrix for $ L = \frac{d^2}{dx^2} $

#### **Step 1: Discretize the Interval**
- Interval: $ [0, \pi] $
- Divide into $ N $ subintervals of size $ \Delta x = \pi / N $.
- Points: $ x_n = n\Delta x $, $ n = 1, 2, ..., N $ (do **not** include $ x=0 $ or $ x=\pi $; these are boundary points).

#### **Step 2: Centered Second Difference Approximation**

For $ u''(x_n) $:
$$
u''(x_n) \approx \frac{u(x_{n-1}) - 2u(x_n) + u(x_{n+1})}{\Delta x^2}
$$

#### **Step 3: Construct the Matrix $ D $**
- $ D $ is an $ N \times N $ tridiagonal matrix.
- Main diagonal: $ -2/\Delta x^2 $
- Super- and sub-diagonal: $ 1/\Delta x^2 $

So,
$$
D = \frac{1}{\Delta x^2}
\begin{bmatrix}
-2 & 1 & 0 & \cdots & 0 \
1 & -2 & 1 & \ddots & \vdots \
0 & 1 & -2 & \ddots & 0 \
\vdots & \ddots & \ddots & \ddots & 1 \
0 & \cdots & 0 & 1 & -2
\end{bmatrix}
$$

In **Matlab**:
```matlab
N = ...;      % number of interior points
dx = pi/N;
D = diag(-2*ones(N,1)) + diag(ones(N-1,1),1) + diag(ones(N-1,1),-1);
D = D/(dx^2);
```

#### **Step 4: Formula for Eigenvalues of $ D $**
The eigenvalues of this matrix are:
$$
\lambda_j = -\frac{4}{\Delta x^2} \sin^2\left(\frac{j\pi}{2(N+1)}\right) \quad \text{for } j = 1, 2, ..., N
$$

#### **Summary Table**

| Step                    | Expression/Formula                                                                                 |
|-------------------------|---------------------------------------------------------------------------------------------------|
| Discretization          | $ x_n = n\Delta x, \quad n=1,\ldots,N $                                                         |
| Centered difference     | $ u''(x_n) \approx \frac{u(x_{n-1}) - 2u(x_n) + u(x_{n+1})}{\Delta x^2} $                       |
| Matrix $ D $          | Tridiagonal: main = $-2/\Delta x^2$, off-diagonals = $1/\Delta x^2$                           |
| Matlab code             | See above                                                                                         |
| Eigenvalues of $ D $  | $ \lambda_j = -\frac{4}{\Delta x^2} \sin^2\left(\frac{j\pi}{2(N+1)}\right),\; j=1,\ldots,N $    |

---

### **Final Answers**
#### **a) Matrix $ D $ formula**:
$$
D = \frac{1}{\Delta x^2}
\begin{bmatrix}
-2 & 1 & 0 & \cdots & 0 \
1 & -2 & 1 & \ddots & \vdots \
0 & 1 & -2 & \ddots & 0 \
\vdots & \ddots & \ddots & \ddots & 1 \
0 & \cdots & 0 & 1 & -2
\end{bmatrix}
$$

#### **b) Eigenvalues of $ D $**:
$$
\boxed{
\lambda_j = -\frac{4}{\Delta x^2} \sin^2\left(\frac{j\pi}{2(N+1)}\right),\quad j=1,\ldots,N
}
$$

---

If you need the Matlab function for matrix generation or more details, let me know!

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