Question

Use Exercise 17 to solve the initial-boundaryvalue problem. In some of these exercises Theorem 11.3.5(c) or Exercise 11.3.42(b) will simplify the computation of the coefficients in the mixed Fourier cosine series. $$ \begin{array}{l} u_{t t}=4 u_{x x}, \quad 00 \\ u_{x}(0, t)=0, \quad u(1, t)=0, \quad t>0, \\ u(x, 0)=0, \quad u_{t}(x, 0)=x^{2}(1-x), \quad 0 \leq x \leq 1 \end{array} $$

Short answer

The solution to the given initial-boundary value problem is: $$u(x,t) = \sum_{n=1}^{\infty} \left(\frac{4}{n^3\pi}\right) \cos(n\pi x)e^{-(n\pi)^2 t}$$

Step-by-step solution

Compute the Fourier cosine series coefficients for initial conditions

We need to find the Fourier cosine series coefficients for the initial conditions \(u(x,0) = 0\) and \(u_t(x,0) = x^2(1-x)\). The Fourier cosine series of a function \(f(x)\) is given by: $$f(x) = \sum_{n=0}^{\infty} A_n \cos(n\pi x)$$ Since \(u(x,0) = 0\), the coefficients \(A_n\) are all zero, so we have: $$u(x,0) = \sum_{n=0}^{\infty} A_n \cos(n\pi x) = 0$$ Now, we need to find the Fourier cosine series coefficients for \(u_t\), which can be represented as: $$u_t(x,0) = x^2(1-x) = \sum_{n=0}^{\infty} B_n \cos(n\pi x)$$ Now, we need to find \(B_n\). To do this, we multiply both sides of the equation by \(\cos(m\pi x)\) and then integrate: $$\int_0^1 u_t(x, 0) \cos(m\pi x) \;\text{d}x = \int_0^1 x^2(1-x)\cos(m\pi x) \;\text{d}x = \sum_{n=0}^{\infty} B_n \int_0^1 \cos(n\pi x)\cos(m\pi x) \;\text{d}x$$ Using the orthogonality property of the cosine function, we can solve for \(B_n\): $$B_n = \frac{2}{\pi}\int_0^1 x^2(1-x)\cos(n\pi x) \;\text{d}x$$

Compute the \(B_n\) coefficients

Now that we have a formula for \(B_n\), we will compute the \(B_n\) coefficients. First, we'll integrate \(x^2(1-x)\cos(n\pi x)\): $$B_n = \frac{2}{\pi}\int_0^1 x^2(1-x)\cos(n\pi x) \;\text{d}x$$ This integral can be computed using integration by parts. After going through the integration by parts steps, we get: $$B_n = \frac{2}{n^3\pi^3}[2n^2\pi^2 - 3(\pi n^2 - 2)\sin(\pi n) - 3(\pi n - \sin(\pi n))\sin(\pi n)]$$ Since \(\sin(\pi n) = 0\) for all integer values of \(n\), we have: $$B_n = \frac{4}{n^3\pi}$$

Write the solution

Now that we have the \(B_n\) coefficients, we can write the solution of the initial-boundary value problem using the mixed Fourier cosine series: $$u(x,t) = \sum_{n=0}^{\infty} B_n \cos(n\pi x)e^{-(n\pi)^2 t} = \sum_{n=0}^{\infty} \left(\frac{4}{n^3\pi}\right) \cos(n\pi x)e^{-(n\pi)^2 t}$$ This is the solution to the initial-boundary value problem we were given.

Key concepts

Fourier Cosine Series

The Fourier cosine series is a way to represent a function as a sum of cosine functions multiplied by coefficients. The series is particularly useful for functions defined on a symmetric interval around zero and can be used to solve boundary value problems. When dealing with initial-boundary value problems such as the exercise given, the Fourier cosine series helps in expressing the initial conditions of the unknown function in terms of cosine functions.

For a given function, the coefficients of the series are found by utilizing the orthogonality property of the cosine functions over the interval. These coefficients, which we call Fourier coefficients, encapsulate the frequency content of the original function. With them, you can reconstruct the function's shape, essentially showcasing the function as a superposition of harmonic waves. This property was exercised in determining the Fourier cosine series for the initial velocity, represented by the second initial condition in the problem.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to multiple variables. These types of equations are at the core of mathematical physics and many other applied sciences because they describe various phenomena such as heat, sound, diffusion, electrostatics, and quantum mechanics.

The PDE in the given exercise is a second-order equation in both time and space, representing a physical system where the rate of change of the wave's acceleration in time (represented as \(u_{tt}\)) is proportional to the rate of change of the curvature of the wave in space (represented as \(4u_{xx}\)). This specific PDE is known as the one-dimensional wave equation and is vital in fields such as acoustics and materials science.

Boundary Value Problems

Boundary value problems are problems where a differential equation is to be solved in a domain subject to specific boundary conditions. These problems are common when modeling physical situations where a solution is confined within certain limits, such as the vibrations of a string fastened at two points, or the temperature distribution along a rod with fixed temperatures at its ends.

In the given exercise, the boundary conditions are specified at the spatial borders x=0 and x=1 for all times t. The condition \(u_x(0, t) = 0\) implies that there's no spatial gradient at the end x=0 (it's an insulated or symmetry boundary), and \(u(1, t) = 0\) fixes the value of the solution at the end x=1, which is a typical Dirichlet boundary condition. These conditions greatly influence the form of the solution and are essential in determining the Fourier coefficients for the series expansion.

Separation of Variables

The method of separation of variables is a technique to solve partial differential equations, such as the one in this exercise, by assuming that the solution is a product of functions of one variable each. This process splits a PDE into easier-to-solve ordinary differential equations (ODEs).

Though this method wasn't directly demonstrated in the provided solution, it is essential to understand that separation of variables would be the next logical step had there not been an initial condition that allowed us to skip directly to solving for the Fourier coefficients. Assuming separation of variables, one could find a general solution for the PDE, and then use the initial and boundary conditions to find a specific solution, as was effectively done with the Fourier cosine series expansion.