Can You Solve an ODE Using the Rayleigh-Ritz Method? - A Short Tutorial

[member 428835]

Hi PF!

I'm trying to understand the Rayleigh-Ritz variational technique to estimate eigenvalues of an ODE. Could someone familiar with this technique either give me a good link to a worked example or write a BVP here and see if I can solve it under your guidance?

Thanks!

 

[jvicens]

Hi there,

The Rayleigh-Ritz variational technique is a powerful method for estimating eigenvalues of an ODE. It involves finding an approximate solution to the ODE by minimizing the functional associated with it. This method is commonly used in engineering and physics to solve problems with complex boundary conditions.

To help you understand this technique better, let's work through an example together. Consider the following boundary value problem:

y'' + 2y' + 5y = λy
y(0) = 0
y(π/2) = 0

To solve this problem using the Rayleigh-Ritz variational technique, we first need to choose a trial function that satisfies the boundary conditions. Let's choose the trial function to be:

y(x) = a sin(x) + b cos(x)

where a and b are unknown constants. Now, we can plug this trial function into the ODE and solve for the eigenvalue λ.

Substituting y(x) into the ODE, we get:

(a cos(x) - b sin(x))'' + 2(a cos(x) - b sin(x))' + 5(a sin(x) + b cos(x)) = λ(a sin(x) + b cos(x))

Simplifying, we get:

-4a sin(x) - 4b cos(x) + 5a sin(x) + 5b cos(x) = λ(a sin(x) + b cos(x))

Now, we use the boundary conditions to solve for the constants a and b. Plugging in x = 0, we get:

b = 0

Plugging in x = π/2, we get:

a = 0

Therefore, our trial function satisfies the boundary conditions, and the eigenvalue λ is equal to 5.

I hope this example helps you understand the Rayleigh-Ritz variational technique better. Please let me know if you have any further questions or need clarification. Best of luck with your studies!