Question

Find an appropriate but simple trial function and use it to estimate the lowest eigenvalue \(\lambda_{0}\) of Stokes' equation $$ \frac{d^{2} y}{d x^{2}}+\lambda x y=0, \quad y(0)=y(\pi)=0 $$ Explain why your estimate must be strictly greater than \(\lambda_{0}\) -

Short answer

Use sinusoidal trial function, compute second derivative, integrate and explain upper bound reason.

Step-by-step solution

- Choose Trial Function

Select a simple trial function that satisfies the boundary conditions. A common choice is a sinusoidal function like \[ y(x) = \frac{x(\frac{\text{π}}{2} - x)}{\text{π}} \] because it satisfies both \( y(0) = 0 \) and \( y(\text{π}) = 0 \).

- Plug Trial Function into Stokes' Equation

To use the trial function in the differential equation, compute the second derivative: \[ \frac{d^2}{dx^2} \bigg( \frac{x(\frac{π}{2} - x)}{π} \bigg). \] The first derivative \[ \frac{d}{dx} \bigg( \frac{x(\frac{π}{2} - x)}{π} \bigg) = \frac{1}{π} \bigg( \frac{π}{2} -2x \bigg), \] and the second derivative \[ \frac{d^2}{dx^2} \bigg( \frac{x(\frac{π}{2} - x)}{π} \bigg) = - \frac{2}{π}. \]

- Integrate to Find Estimate

Using the trial function in the Stokes' equation, evaluate the integral to find an estimate for the eigenvalue. \[ \frac{\frac{d^2} {dx^2} y(x) + λ x y}{π}. \]

- Explanation

The estimate found from the trial function must be strictly greater than the lowest eigenvalue \(\lambda_0\) because variational methods, which are being employed here, provide upper bound estimates for the true eigenvalues. They tend to overestimate the lowest eigenvalue.

Key concepts

Trial Function

In solving differential equations, a **trial function** is an initial guess at the form of the solution. It must satisfy the given boundary conditions of the problem. Here, we use the trial function:

• \[ y(x) = \frac{x(\frac{π}{2} - x)}{π} \]

This function is chosen because it naturally satisfies the boundary conditions. Specifically, it equals zero at both ends, at \( x = 0 \) and \( x = π \). This simplifies the process of fitting the function within the constraints.
This step is crucial because an appropriately chosen trial function can make finding an approximate solution more manageable.

Consequently, we use trial functions in variational methods and to simplify the complex differential equations. It is a vital first step in estimating eigenvalues accurately.

Boundary Conditions

**Boundary conditions** are crucial in solving differential equations. They define the values or behaviors that a solution must exhibit at the boundaries of the domain. For the given Stokes' equation:

• \[ \frac{d^2 y}{d x^2} + \text{λ} x y = 0 \]

the boundary conditions are:

• \[ y(0) = 0 \]
• \[ y(π) = 0 \]

This means our solution function, \( y(x) \), must be zero at the boundaries \( x = 0 \) and \( x = π \). Boundary conditions are like anchor points, restraining the solution to fit within specified limits. Properly applying these conditions guarantees that we are looking at valid solutions to the differential equation. They help ensure any solutions derived do not deviate from the physical or geometrical interpretations that the problem's context implies.

Differential Equation

A **differential equation** involves an unknown function and its derivatives. In this problem, our differential equation is:

• \[ \frac{d^2 y}{d x^2} + \text{λ} x y = 0 \]

To solve it using the chosen trial function, we first find the second derivative of our trial function:

• \[ \frac{d^2}{dx^2} \bigg( \frac{x(\frac{π}{2} - x)}{π} \bigg) = -\frac{2}{π}. \]

We then use this second derivative in the original differential equation to simplify the problem for finding an estimate of the eigenvalue, \( \text{λ} \). This transformation turns the problem into one of fitting our trial function and satisfying the equation over all given domains. Properly handling this step allows us to progress toward finding eigenvalues and understanding the behavior of the solution in context.

Variational Methods

**Variational Methods** are mathematical techniques used to approximate solutions for complex problems. Here, we use such a method to estimate the lowest eigenvalue \( \text{λ}_0 \) of Stokes' equation. The chosen trial function is evaluated to find an upper bound for the eigenvalue.

The variational method applied ensures that this estimate is greater than or equal to the actual lowest eigenvalue. It operates on the principle of minimizing or maximizing functionals, which are expressions involving integrals of functions. By testing different trial functions, one can derive close approximations to true solutions. Given expectations from variational principles, any such estimate tends to be higher than the exact value, hence providing a safe upper bound.

Variational methods are powerful tools in physics and engineering, as they simplify otherwise intractable problems, offering useful approximations to complex physical phenomena.