Chapter 6: Problem 28

In Example 6.20, we found a bound on the lowest eigenvalue for the n eigenvalue problem. a. Verify the computation in the example. b. Apply the method using $$ y(x)=\left\\{\begin{array}{cl} x, & 0

Short Answer

Expert verified

Due to the abstract nature of the problem and without specific numerical values, the explicit answer cannot be provided. Nevertheless, the result will be an upper bound for the lowest eigenvalue of the given eigenvalue problem, derived from the Rayleigh quotient step.

Step by step solution

Verify the computation in the example

To start off, ensure that you understand the given example by going through each calculation step. Double-check the results for accuracy. Without the specific example provided, however, it cannot be explicitly detailed here.

Apply the method using $y(x)$

Next, employ the eigenvalue problem using your defined function: \[y(x)=\left\{\begin{array}{cl} x, & 0<x<\frac{1}{2} \ 1-x, & \frac{1}{2}<x<1\end{array}\right.\]. Compute the integral value and compare it with the eigenvalue $\lambda_{1}$, which you may have found in the example or elsewhere.

Use the Rayleigh quotient

Lastly, apply the Rayleigh quotient to obtain an upper bound for the lowest eigenvalue of the problem. The Rayleigh quotient is given by: \[R(x) = \frac{\phi^{\prime} (x)}{\phi(x)}\], for $\phi(x) \neq 0$. Obtain the derivative of your function, substitute it into the Rayleigh quotient, and determine the upper bound for your eigenvalue problem: \[\phi^{\prime\prime} + (\lambda - x^{2})\phi = 0, \phi(0) = 0, \phi^{\prime}(1) = 0\]