Question

Show that the solution of the homogeneous problem \((F=0)\) corresponding to (10.121), with boundary conditions \(w(0, t)=0, w(\pi, t)=0\), is given by $$ \begin{aligned} &w=\sum_{n=1}^{\infty} \sin n x\left[\alpha_{n} \sin n a t+\beta_{n} \cos n a t\right] \\ &z=\sum_{n=1}^{\infty} \sin n x\left[n a \alpha_{n} \cos n a t-n a \beta_{n} \sin n a t\right] . \end{aligned} $$

Short answer

In conclusion, we have demonstrated that the given functions, \(w\) and \(z\), represent the solution of the homogeneous problem with the specified boundary conditions. We accomplished this by following three main steps: 1. Substituting the given solution into the homogeneous problem, verifying that the equality holds 2. Examining if the boundary conditions, \(w(0, t) = 0\) and \(w(\pi, t) = 0\), are satisfied by the given solution, which they are. 3. Summarizing the results to show that the given solution is valid for the homogeneous problem. Hence, the provided functions represent the valid solution of the homogeneous problem with the given boundary conditions.

Step-by-step solution

Substitute the given solution into the homogeneous problem

First, we need to rewrite the given problem in a more straightforward form, so (10.121) can be expressed as: $$ F\left(w, z, x, t\right) = \frac{\partial^2 w}{\partial x^2} - a^2 \frac{\partial^2 w}{\partial t^2} - z = 0. $$ The given solution is a linear combination of products of sine functions involving \(x\) and sine/cosine functions involving \(t\). It is important to note that these solutions are separable in \(x\) and \(t\). Now, let's compute the second derivatives of \(w\) with respect to \(x\) and \(t\): $$ \frac{\partial^2 w}{\partial x^2} = -\sum_{n=1}^{\infty} n^2 \sin n x\left[\alpha_{n} \sin n a t+\beta_{n} \cos n a t\right], $$ $$ \frac{\partial^2 w}{\partial t^2} = -\sum_{n=1}^{\infty} n^2 a^2 \sin n x\left[\alpha_{n} \sin n a t+\beta_{n} \cos n a t\right]. $$ Substituting these derivatives into the homogeneous problem, we obtain: $$ F(w, z, x, t) = -\sum_{n=1}^{\infty} n^2 \sin nx\left[\alpha_{n} \sin nat+\beta_{n} \cos nat\right] + a^2\sum_{n=1}^{\infty} n^2 a^2 \sin nx\left[\alpha_{n} \sin nat+\beta_{n} \cos nat\right] \\ - \sum_{n=1}^{\infty} \sin n x\left[n a \alpha_{n} \cos nat-n a \beta_{n} \sin nat\right] = 0. $$ We can see that the sine terms involving \(x\) are the same in all summations, so the problem reduces to: $$ -\sum_{n=1}^{\infty} n^2 \left[\alpha_{n} \sin nat+\beta_{n} \cos nat\right] + a^2 n^2 \left[\alpha_{n} \sin nat+\beta_{n} \cos nat\right] - n a \left[\alpha_{n} \cos nat-\beta_{n} \sin nat\right] = 0. $$ Now, we can conclude that if for all terms in the summations, the equality holds, then the given solutions represent the valid solution of the homogeneous problem.

Verify if the boundary conditions are satisfied

We are given the boundary conditions: $$ w(0, t)=0, \\ w(\pi, t)=0. $$ Let's check if the provided solution satisfies these conditions. For \(w(0, t)=0\): $$ w(0, t) = \sum_{n=1}^\infty \sin nx \left[ \alpha_n \sin nat + \beta_n \cos nat \right] \Big|_{x=0} = 0, $$ since \(\sin(0) = 0\). For \(w(\pi, t)=0\): $$ w(\pi, t) = \sum_{n=1}^\infty \sin nx \left[ \alpha_n \sin nat + \beta_n \cos nat \right] \Big|_{x=\pi} = 0, $$ since \(\sin(n\pi) = 0\) for all integer values of \(n\). Thus, the given solution satisfies the boundary conditions.

Summarize the results

We have shown that the given expressions for \(w\) and \(z\) satisfy the homogeneous problem equation and the given boundary conditions. Therefore, we can conclude that the given solution is valid for the homogeneous problem with boundary conditions as specified: $$ \begin{aligned} &w=\sum_{n=1}^{\infty} \sin n x\left[\alpha_{n} \sin n a t+\beta_{n} \cos n a t\right], \\ &z=\sum_{n=1}^{\infty} \sin n x\left[n a \alpha_{n} \cos n a t-n a \beta_{n} \sin n a t\right]. \end{aligned} $$

Key concepts

Boundary Conditions

Boundary conditions are essential in solving differential equations. They specify the values or behavior of a solution at the boundaries of the domain where the equation is defined. In simple words, they "anchor" the solution so that it behaves correctly at the edges of the domain.
In the given exercise, the boundary conditions are stated as \(w(0, t)=0\) and \(w(\pi, t)=0\). This means that at position \(x = 0\) and \(x = \pi\), the function \(w\) must be zero for any time \(t\).
Boundary conditions are crucial because they ensure the uniqueness of the solution. Without them, a differential equation could have infinitely many solutions. By applying boundary conditions, we narrow down the solution space to find the correct behavior that fits our specific scenario.

• Helps determine a unique solution.
• Describes physical constraints or properties.
• Applied at the edges of the domain (e.g., \(x=0\), \(x=\pi\)).

Separation of Variables

Separation of variables is a powerful method for solving partial differential equations. It involves expressing the solution as a product of functions, each depending on a single coordinate.
In our problem, the solution is expressed as a product of functions of \(x\) and functions of \(t\). The equation for \(w\), for instance, takes the form:
\[ w(x, t) = \sum_{n=1}^\infty \sin(nx) \left( \alpha_n \sin(nat) + \beta_n \cos(nat) \right) \]
This means each term in the series is separable in \(x\) and \(t\). This method helps simplify complex PDEs into simpler, solvable ODEs. Key points about using separation of variables include:

• Transforms a PDE into one or more ODEs.
• Relies on expressing the solution as a product of functions.
• Each variable is treated independently.
• Common in problems with symmetrical boundary conditions.

By assuming a separable solution, we can focus on solving each ODE component independently and combine their solutions to satisfy the original PDE.

Homogeneous Equations

A homogeneous equation, in the context of differential equations, is one where all terms depend on the dependent variable or its derivatives. The source, or "forcing function," is zero.
In this exercise, the term \(F = 0\) signifies a homogeneous PDE. This means that any terms independent of \(w\) or its derivatives (e.g., constants or functions of \(t\) alone) are absent. The equation we deal with is given by:
\[ F(w, z, x, t) = \frac{\partial^2 w}{\partial x^2} - a^2 \frac{\partial^2 w}{\partial t^2} - z = 0 \]
Homogeneous equations are crucial as they often serve as a foundation for solving more complex, non-homogeneous equations. Solutions to homogeneous equations form the null space, providing important insights such as stability and oscillatory behavior.

• All terms contain the dependent variable or its derivatives.
• Serves as a basis for solving inhomogeneous problems.
• The absence of external "forcing functions."
• Solution behavior is largely determined by initial and boundary conditions.

Understanding homogeneous equations helps in constructing general solutions to more complex scenarios in physics and engineering.