Question

For the case where the left end of the bar is kept at zero degrees and the right end is insulated, we saw in Exercise 9 of Section \(9.2\) that the boundary value problem for \(X(x)\) is $$ X^{\prime \prime}-\sigma X=0, \quad 0

Short answer

Question: Show that eigenvalues are solutions of the equation \(\exp (2 \sqrt{\sigma}l) = -1\) and determine the corresponding eigenfunction \(X_n(x)\). Answer: As we have shown during the problem-solving process, when enforcing the boundary conditions for a given boundary value problem, we arrive at the equation $$ \exp(2\sqrt{\sigma}l) = -1. $$ Eigenvalues are solutions of this equation. Additionally, the corresponding eigenfunction, given the relation \(\sqrt{\sigma_n}=\frac{(2n-1)i\pi}{2l}\), can be found as $$ X_n(x) = A_n\left(\exp\left(\frac{(2n-1)i\pi x}{2l}\right) - \exp\left(-\frac{(2n-1)i\pi x}{2l}\right)\right), $$ where \(A_n\) is a constant.

Step-by-step solution

Show that \(\sigma=0\) is not an eigenvalue

Let's take the given boundary value problem $$ X^{\prime \prime}-\sigma X=0, \quad 0<x<l, \quad X(0)=0, \quad X^{\prime}(l)=0. $$ and plug in \(\sigma=0\). We get $$ X^{\prime \prime} = 0. $$ The general solution of this equation is \(X(x) = ax+b\). Using the boundary conditions, we get the following system of equations:{ X(0) = 0 , X' (l) = 0 }. Plug in the general solution and the boundary conditions: $$ \begin{cases} a \cdot 0 + b = 0, \\ a = 0. \end{cases} $$ It follows that \(a=0\) and \(b=0\), so \(X(x) = 0\). Since the trivial solution \(X(x) = 0\) is not an eigenfunction, we conclude that \(\sigma=0\) is not an eigenvalue.

Show that eigenvalues are solutions of the equation \(\exp (2 \sqrt{\sigma}l) = -1\)

Now let's assume \(\sigma > 0\). Then our equation is $$ X^{\prime \prime}-\sigma X=0 $$ which has the following general solution: $$ X(x) = A\exp(\sqrt{\sigma}x) + B\exp(-\sqrt{\sigma}x). $$ Enforcing the boundary conditions we get: $$ \begin{cases} X(0) = A + B = 0 \\ X'(l) = \sqrt{\sigma}Ae^{\sqrt{\sigma} l} - \sqrt{\sigma}Be^{-\sqrt{\sigma} l} = 0 \end{cases} $$ From the first equation, it follows that \(B=-A\). Using this in the second equation and simplifying, we get: $$ \sqrt{\sigma}A(e^{\sqrt{\sigma}l} + e^{-\sqrt{\sigma}l}) = 0. $$ Since \(A\) cannot be 0 (as it would lead to the trivial solution \(X(x) = 0\)), we can divide by \(A\). We get the following equation for better view: $$ e^{2\sqrt{\sigma}l} + 1 = 0. $$ Multiplying both sides by \(e^{\sqrt{\sigma}l}\), we have $$ \exp(2\sqrt{\sigma}l) = -1. $$ Hence, eigenvalues are solutions of this equation.

Use Euler's formula to relate \(\sqrt{\sigma_n}\) to \(\pi\) and \(l\)

Using Euler's formula, we can rewrite the equation \(\exp(2\sqrt{\sigma}l)=-1\) as \[e^{2\sqrt{\sigma}l i}=e^{i \pi (2n-1)}.\] Since the exponents must be equal, it follows that $$ 2\sqrt{\sigma_n} l i = i \pi (2n-1) $$ for \(n=1,2,\ldots\).

Determine a corresponding eigenfunction \(X_n(x)\)

Using the relation found in step 3, we can write \(\sqrt{\sigma_n}=\frac{(2n-1)i\pi}{2l}\). Now we can substitute this value in the general solution $$ X_n(x) = A_n\exp\left(\frac{(2n-1)i\pi x}{2l}\right) + B_n\exp\left(-\frac{(2n-1)i\pi x}{2l}\right). $$ We already know that \(A_n=-B_n\) from the boundary condition \(X_n(0)=0\). So, the eigenfunction is $$ X_n(x) = A_n\left(\exp\left(\frac{(2n-1)i\pi x}{2l}\right) - \exp\left(-\frac{(2n-1)i\pi x}{2l}\right)\right). $$ This eigenfunction \(X(x_n)\) corresponds to the eigenvalue \(\sigma_n\) obtained in the previous step.

Solve \(T_n^{\prime}(t) - \kappa \sigma_n T_n(t) = 0\) and form the building-block solution

Now we need to solve the differential equation for \(T_n(t)\): $$ T_n^{\prime}(t) - \kappa \sigma_n T_n(t) = 0. $$ This is a first-order linear homogeneous differential equation, and its general solution is $$ T_n(t) = C_n \exp(\kappa \sigma_n t), $$ where \(C_n\) is a constant (possibly different for each \(n\)). Finally, the building-block solution \(u_n(x, t) = X_n(x) T_n(t)\) is given by $$ u_n(x, t) = A_n \left(\exp\left(\frac{(2n-1)i\pi x}{2l}\right) - \exp\left(-\frac{(2n-1)i\pi x}{2l}\right)\right) \cdot C_n \exp(\kappa \sigma_n t). $$

Key concepts

Eigenvalues

Eigenvalues are special numbers associated with a boundary value problem in differential equations, which help in understanding the behavior of the system under given constraints. In this context, an eigenvalue is a value of \(\sigma\) for which non-trivial solutions, called eigenfunctions, exist for the differential equation \(X'' - \sigma X = 0\) with specific boundary conditions such as \(X(0) = 0\) and \(X'(l) = 0\).

Eigenvalues are crucial as they help determine the fundamental properties of systems that can be expressed by differential equations, like vibrations and heat conduction. In practice, to find eigenvalues for this case, we simplify the given equation to reach a form like \(\exp(2\sqrt{\sigma}l) = -1\), which suggests a periodic phenomena tied to the roots derived from Euler's formula, showing that \(\sigma_n\) is complex with a relationship to \(\pi\) and \(l\).

Differential Equations

Differential equations involve unknown functions and their derivatives and are a tool to describe various natural phenomena, much like the temperature distribution in a rod. In boundary value problems, they allow us to model conditions at the boundaries which a solution must satisfy.

For this exercise, the differential equation is \(X'' - \sigma X = 0\), where \(X''\) is the second derivative of the function \(X(x)\) with respect to \(x\). Solving differential equations involves determining \(X(x)\) such that the equation is satisfied under defined boundary conditions. This can lead to different forms of solutions, ranging from simple polynomials to complex exponentials, each describing different possible behaviors depending on the characteristics set by the eigenvalues.

• Second-order: Involves second derivative \(X''\).
• Linear: Linearity implies superposition principle holds.
• Homogeneous: No external forcing term, solutions depend directly on eigenvalues.

Boundary Conditions

Boundary conditions are constraints imposed on the solutions of differential equations, specifying values or derivatives of the solutions at specific points. These conditions are vital to determine the unique solution set for a given differential equation.

In this problem, the boundary conditions are \(X(0) = 0\) and \(X'(l) = 0\), meaning that at the start of the bar, the displacement or temperature is zero, and at the endpoint, the change rate is zero, indicating no heat flux. Such conditions help in pinning the problem down to a specific physical scenario and ensuring well-posedness:

• Dirichlet conditions: Specify the function value \(X(0) = 0\).
• Neumann conditions: Specify the derivative value \(X'(l) = 0\).
• Ensure uniqueness: Unlike initial value problems, boundary value problems need these to prevent multiple solutions.

Eigenfunctions

Eigenfunctions are solutions to the differential equations for specific eigenvalues, effectively describing the modes in which a system can naturally behave without external forcing.

In simpler terms, eigenfunctions are the shapes or forms that a system takes on when it resonates at its natural frequencies, the eigenvalues. For the exercise, if we find an eigenvalue \(\sigma_n\), the associated eigenfunction \(X_n(x)\) helps us construct solutions to the differential equation model inherently satisfying the BVP.

The eigenfunction is a combination of exponential terms due to the nature of the original differential equation, demonstrating harmonic-like properties:

• Involves complex combinations e.g., using Euler's formula.
• Respects boundary conditions \(X_n(0) = 0\) and \(X_n'(l) = 0\).
• Forms building blocks for more complex solutions (superposition).