Question

Justify defining the formal solution of the initial-boundary value problem $$ \begin{array}{c} u_{t t}=a^{2} u_{x x}, \quad 00, \\ u_{x}(0, t)=0, \quad u_{x}(L, t)=0, \quad t>0, \\ u(x, 0)=f(x), \quad u_{t}(x, 0)=g(x), \quad 0 \leq x \leq L. \end{array} $$ to be $$ u(x, t)=\alpha_{0}+\beta_{0} t+\sum_{n=1}^{\infty}\left(\alpha_{n} \cos \frac{n \pi a t}{L}+\frac{L \beta_{n}}{n \pi a} \sin \frac{n \pi a t}{L}\right) \cos \frac{n \pi x}{L} $$ where $$ C_{f}(x)=\alpha_{0}+\sum_{n=1}^{\infty} \alpha_{n} \cos \frac{n \pi x}{L} \quad \text { and } \quad C_{g}(x)=\beta_{0}+\sum_{n=1}^{\infty} \beta_{n} \cos \frac{n \pi x}{L} $$ are the Fourier cosine series of \(f\) and \(g\) on \([0, L] ;\) that is, $$ \begin{array}{c} \alpha_{0}=\frac{1}{L} \int_{0}^{L} f(x) d x, \quad \beta_{0}=\frac{1}{L} \int_{0}^{L} g(x) d x \\ \alpha_{n}=\frac{2}{L} \int_{0}^{L} f(x) \cos \frac{n \pi x}{L} d x, \quad \text { and } \quad \beta_{n}=\frac{2}{L} \int_{0}^{L} g(x) \cos \frac{n \pi x}{L} d x, \quad n=1,2,3, \ldots . \end{array} $$ are the Fourier cosine series of \(f\) and \(g\) on \([0, L] ;\) that is, $$ \begin{array}{c} \alpha_{0}=\frac{1}{L} \int_{0}^{L} f(x) d x, \quad \beta_{0}=\frac{1}{L} \int_{0}^{L} g(x) d x \\ \alpha_{n}=\frac{2}{L} \int_{0}^{L} f(x) \cos \frac{n \pi x}{L} d x, \quad \text { and } \quad \beta_{n}=\frac{2}{L} \int_{0}^{L} g(x) \cos \frac{n \pi x}{L} d x, \quad n=1,2,3, \ldots \end{array} $$

Short answer

Question: Verify that the given formal solution satisfies the initial-boundary value problem with the second-order partial differential equation (PDE), boundary conditions, and initial conditions. Answer: The given formal solution satisfies the initial-boundary value problem as it satisfies the PDE, boundary conditions, and initial conditions. This was verified by substituting the given solution into the conditions and showing that they are satisfied.

Step-by-step solution

Verify the PDE

Start by computing the second-order time and space derivatives for the function \(u(x,t)\), with respect to \(t\) and \(x\), respectively, and substituting them back into the PDE \(u_{tt} = a^2u_{xx}\). Compute second-order time derivative: $$ \begin{align} u_t &= \beta_0 + \sum_{n=1}^{\infty}\left(-\frac{n\pi a}{L}\alpha_n\sin\frac{n\pi at}{L} + \frac{n\pi a\beta_n}{L}\cos\frac{n\pi at}{L}\right)\cos\frac{n\pi x}{L}\\ u_{tt} &= -\sum_{n=1}^{\infty}\left(\frac{n^2\pi^2 a^2}{L^2}\alpha_n\cos\frac{n\pi at}{L} - \frac{n^2\pi^2 a^2\beta_n}{L^2}\sin\frac{n\pi at}{L}\right)\cos\frac{n\pi x}{L} \end{align} $$ Compute second-order space derivative: $$ \begin{align} u_x &= \sum_{n=1}^{\infty}\left(-\frac{n\pi}{L}\alpha_n\cos\frac{n\pi at}{L} - \frac{n\pi \beta_n}{L}\sin\frac{n\pi at}{L}\right)\sin\frac{n\pi x}{L}\\ u_{xx} &= -\sum_{n=1}^{\infty}\left(\frac{n^2\pi^2}{L^2}\alpha_n\cos\frac{n\pi at}{L} + \frac{n^2\pi^2\beta_n}{L^2}\sin\frac{n\pi at}{L}\right)\cos\frac{n\pi x}{L} \end{align} $$ Now substitute the computed derivatives back into the PDE: $$u_{tt} = a^2u_{xx} \implies -\sum_{n=1}^{\infty}\left(\frac{n^2\pi^2 a^2}{L^2}\alpha_n\cos\frac{n\pi at}{L} - \frac{n^2\pi^2 a^2\beta_n}{L^2}\sin\frac{n\pi at}{L}\right)\cos\frac{n\pi x}{L} = a^2\left(-\sum_{n=1}^{\infty}\left(\frac{n^2\pi^2}{L^2}\alpha_n\cos\frac{n\pi at}{L} + \frac{n^2\pi^2\beta_n}{L^2}\sin\frac{n\pi at}{L}\right)\cos\frac{n\pi x}{L}\right)$$ Since the sums on both sides of the equation are the same, the PDE is satisfied.

Verify the boundary conditions

Evaluate the spatial derivatives \(u_x\) at the boundary conditions \(x = 0\) and \(x = L\). At \(x = 0\): $$ u_x(0, t) = \sum_{n=1}^{\infty}\left(-\frac{n\pi}{L}\alpha_n\cos\frac{n\pi at}{L} - \frac{n\pi \beta_n}{L}\sin\frac{n\pi at}{L}\right)\sin 0 = 0 $$ At \(x = L\): $$ u_x(L, t) = \sum_{n=1}^{\infty}\left(-\frac{n\pi}{L}\alpha_n\cos\frac{n\pi at}{L} - \frac{n\pi \beta_n}{L}\sin\frac{n\pi at}{L}\right)\sin n\pi = 0 $$ The boundary conditions are satisfied.

Verify the initial conditions

Evaluate the function \(u(x,t)\) and its time derivative \(u_t(x,t)\) at the initial condition \(t=0\). At \(t=0\): $$ u(x, 0) = \alpha_0 + \beta_0 \cdot 0 + \sum_{n=1}^{\infty}\left(\alpha_n \cos 0 + \frac{L \beta_n}{n\pi a}\sin0\right)\cos\frac{n \pi x}{L} = \alpha_0 + \sum_{n=1}^{\infty} \alpha_n \cos\frac{n\pi x}{L} = C_f(x) = f(x) $$ At \(t=0\): $$ u_t(x, 0) = \beta_0 + \sum_{n=1}^{\infty}\left(-\frac{n\pi a}{L}\alpha_n\sin0 + \frac{n\pi a\beta_n}{L}\cos0\right)\cos\frac{n\pi x}{L} = \beta_0 + \sum_{n=1}^{\infty} \beta_n \cos\frac{n\pi x}{L} = C_g(x) = g(x) $$ The initial conditions are satisfied. Since the PDE, boundary conditions, and initial conditions are satisfied, the given formal solution is justified.

Key concepts

Fourier Cosine Series

Understanding the Fourier cosine series is fundamental in solving certain types of partial differential equations (PDEs), especially initial-boundary value problems. It represents a function as a sum of cosine terms, each multiplied by a coefficient that dictates the amplitude at various frequencies.

The Fourier cosine series is particularly useful when dealing with functions defined on a finite interval and subject to even boundary conditions, such as those encountered in the textbook problem we are looking at. The coefficients ( ) and ( ) in our textbook example are evaluated using definite integrals that project the initial functions and onto the cosine basis functions. This process extracts the frequency components of the initial conditions, allowing the PDE to be solved in a series form.

It's important to realize that the effectiveness of representing a function with a Fourier series relies on the function being periodic. Since our functions of interest are defined on a finite interval and are not necessarily periodic, we focus on the cosine series to maintain the even symmetry, thus implicitly extending our function as an even function beyond the interval.

Partial Differential Equations

Partial differential equations (PDEs) involve functions of multiple variables and their partial derivatives. They are essential in expressing phenomena such as diffusion, heat conduction, wave propagation, and fluid dynamics.

In the context of our textbook problem, the PDE is the one-dimensional wave equation, where (00), and describes wave propagation along a string of length with wave speed . This second-order linear PDE is a fundamental equation in physics and engineering.

Solving a PDE analytically can be challenging and typically requires simplifying assumptions or techniques like separation of variables, which, when combined with the Fourier series approach, leads to a solution that can be made as accurate as necessary by considering enough terms in the series. The verification of the PDE solution involves confirming that both the form of the function and its derivatives satisfy the original PDE, as demonstrated in the step-by-step solution.

Boundary Conditions

Boundary conditions are constraints necessary for uniquely determining the solution to a PDE. They define the behavior of a physical quantity at the boundaries of the domain in which the PDE is posed.

In our problem, we have Neumann boundary conditions given by (at and (at for a string fixed at both ends. This implies that the derivative of the displacement with respect to (which is related to the slope of the string) is zero at both ends, corresponding to a horizontal tangent at these points. The fulfillment of these conditions ensures the physical realism of the solution, as verified in the step-by-step solution.

Without these conditions, the wave equation would have infinitely many solutions, so boundary conditions serve to 'anchor' the string's ends in place, reflecting the real-world constraints of a physical system.

Initial Value Problem

An initial value problem is an aspect of solving PDEs that includes conditions that specify the state of the system at the beginning of the time variable considered. It is crucial to predict the future behavior of the system accurately.

In the given exercise, the initial value problem is defined by ( at ) and ( at 0 < x < L), which represents the initial displacement and initial velocity distribution along the string, respectively. Solving the initial value problem involves verifying that the proposed solution satisfies these initial conditions at = 0, thus ensuring that the solution not only respects the dynamics dictated by the PDE but also starts from the correct configuration, as highlighted in the step-by-step solution.

When students embrace the relevance of initial conditions, they can better understand the broader implications of solutions to PDEs, allowing them to model and predict physical systems with greater confidence and precision.