Numerical Analysis 9th. Edition by Richard L. Burden, J. Douglas Faires | 9780538733519 | 538733519 | Holooly

Numerical Analysis

Numerical Analysis

172 SOLVED PROBLEMS

Question: 9.5.2

Apply one iteration of the QR method to the matrix that was given in Example 1: A = [ 3 1 0 1 3 1 0 1 3 ]. ...

Verified Answer:

Let

A^{(1)}=A

be the given matrix...

Question: 12.3.1

Approximate the solution to the hyperbolic problem ∂²u/∂t² (x, t) − 4 ∂²u/∂x² (x, t) = 0, 0 < x < 1, 0 < t, with boundary conditions u(0, t) = u(1, t) = 0, for 0 < t, and initial conditions u(x, 0) = sin(πx), 0 ≤ x ≤ 1, and ∂u/∂t (x, 0) = 0, 0 ≤ x ≤ 1, using h = 0.1 and k = 0.05. ...

Verified Answer:

Choosing h = 0.1 and k = 0.05 gives λ = 1, m = 10,...

Question: 12.2.3

Use the Crank-Nicolson method with h = 0.1 and k = 0.01 to approximate the solution to the problem ∂u/∂t (x, t) − ∂²u/∂x² (x, t) = 0, 0 < x < 1 0 < t, subject to the conditions u(0, t) = u(1, t) = 0, 0 < t, and u(x, 0) = sin(πx), 0 ≤ x ≤ 1. ...

Verified Answer:

Choosing h = 0.1 and k = 0.01 gives m = 10, N = 50...

Question: 12.2.2

Use the Backward-Difference method (Algorithm 12.2) with h = 0.1 and k = 0.01 to approximate the solution to the heat equation ∂u/∂t (x, t) − ∂²u/∂x² (x, t) = 0, 0 < x < 1, 0 < t, subject to the constraints u(0, t) = u(1, t) = 0, 0 < t, u(x, 0) = sin πx, 0 ≤ x ≤ 1. ...

Verified Answer:

This problem was considered in Example 1 where we ...

Question: 12.2.1

Use steps sizes (a) h = 0.1 and k = 0.0005 and (b) h = 0.1 and k = 0.01 to approximate the solution to the heat equation ∂u/∂t (x, t) − ∂²u/∂x² (x, t) = 0, 0 < x < 1, 0 ≤ t, with boundary conditions u(0, t) = u(1, t) = 0, 0 < t, and initial conditions u(x, 0) = sin(πx), 0 ≤ x ≤ 1. ...

Verified Answer:

(a) Forward-Difference method with h = 0.1, k = 0....

Question: 12.1.2

Use the Poisson finite-difference method with n = 6 , m = 5, and a tolerance of 10^−10 to approximate the solution to ∂²u/∂x² (x, y) + ∂²u/∂y² (x, y) = xe^y , 0 < x < 2, 0 < y < 1, with the boundary conditions u(0, y) = 0, u(2, y) = 2e^y , 0 ≤ y ≤ 1, u(x, 0) = x, u(x, 1) = ex, 0 ≤ x ≤ 2, ...

Verified Answer:

Using Algorithm 12.1 with a maximum number of iter...

Question: 12.1.1

Determine the steady-state heat distribution in a thin square metal plate with dimensions 0.5 m by 0.5 m using n = m = 4. Two adjacent boundaries are held at 0°C, and the heat on the other boundaries increases linearly from 0°C at one corner to 100°C where the sides meet. ...

Verified Answer:

Place the sides with the zero boundary conditions ...

Question: 11.4.1

Apply Algorithm 11.4, with h = 0.1, to the nonlinear boundary-value problem y” = 1/8 (32 + 2x³ − yy’), for 1 ≤ x ≤ 3, with y(1) = 17 and y(3) = 43/3 , and compare the results to those obtained in Example 1 of Section 11.2. ...

Verified Answer:

The stopping procedure used in Algorithm 11.4 was ...

Question: 11.3.2

Apply Richardson’s extrapolation to approximate the solution to the boundary-value problem y” = −2/x y’ + 2/x² y + sin(ln x)/x² , for 1 ≤ x ≤ 2, with y(1) = 1 and y(2) = 2, using h = 0.1, 0.05, and 0.025. ...

Verified Answer:

The results are listed in Table 11.4. The first ex...

Question: 11.3.1

Use Algorithm 11.3 with N = 9 to approximate the solution to the linear boundary-value problem y” = −2/x y’+ 2/x² y + sin(ln x)/x² , for 1 ≤ x ≤ 2, with y(1) = 1 and y(2) = 2, and compare the results to those obtained using the Shooting method in Example 2 of Section 11.1. ...

Verified Answer:

For this example, we will use N = 9, so h = 0.1, a...

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