Question

Use Exercise 49 to solve the initial-boundaryvalue problem. In some of these exercises Theorem 11.3.5(a) will simplify the computation of the coefficients in the Fourier cosine series. $$ \begin{array}{l} u_{t t}=7 u_{x x}, \quad 00, \\ u_{x}(0, t)=0, \quad u_{x}(1, t)=0, \quad t>0, \\ u(x, 0)=0, \quad u_{t}(x, 0)=3 x^{2}\left(x^{2}-2\right), \quad 0 \leq x \leq 1 \end{array} $$

Short answer

To summarize, we solved the given initial-boundary value problem for the PDE $$u_{tt}=7u_{xx}$$ with given initial and boundary conditions using the separation of variables and Fourier series. We found the complete solution in terms of a Fourier cosine series as: $$u(x,t) = \sum_{n=1}^{\infty} \frac{12}{n^2π^2}[1 - (-1)^n] \cos\left(\frac{nπx}{1}\right) \left(\frac{\sinh(n\pi\sqrt{7}t)}{n\pi\sqrt{7}}\right)$$ This solution satisfies the given PDE and the provided initial and boundary conditions.

Step-by-step solution

Separation of Variables

We will seek solutions of the form \(u(x,t) = X(x)T(t)\). Substituting this into the PDE and rearranging, we get: $$\frac{T''(t)}{7T(t)} = \frac{X''(x)}{X(x)} \equiv -λ$$ Here, \(λ\) is the separation constant and the equations are now ordinary differential equations involving either x or t.

Spatial equation (ODE)

We'll first solve the spatial equation: $$X''(x) + λX(x) = 0$$ We have the boundary conditions: $$X'(0) = 0$$ $$X'(1) = 0$$ These boundary conditions result in the eigenvalue problem for this spatial equation.

Find Eigenvalues and Eigenfunctions

For λ > 0, the solution is oscillatory and does not vanish at the boundary conditions. Therefore, we assume that λ = -μ^2, where μ > 0. The ODE becomes: $$X''(x) - μ^2 X(x) = 0$$ The general solution to this ODE is: $$X(x) = c_1 \cosh(μx) + c_2\sinh(μx)$$ Using the first boundary condition \(X'(0) = 0\), we have: $$X'(x) = c_1μ\sinh(μx) + c_2μ\cosh(μx)$$ $$X'(0) = c_2μ = 0$$ Since μ ≠ 0, we must have \(c_2 = 0\). Thus, $$X(x) = c_1 \cosh(μx)$$ Now use the second boundary condition \(X'(1) = 0\): $$X'(1) = c_1μ\sinh(μ) = 0$$ This implies that μ must be a solution of \(\sinh(μ) = 0\). However, there is no nonzero value of μ satisfying this condition. Therefore, there are no eigenvalues and eigenfunctions when λ > 0.

Try λ = 0

When λ = 0, the spatial ODE becomes: $$X''(x) = 0$$ The general solution for this ODE is: $$X(x) = c_1 + c_2x$$ Applying the boundary conditions yields: $$X'(0) = c_2 = 0$$ $$X'(1) = c_2 = 0$$ We already found that \(c_2 = 0\). So our solution is: $$X(x) = c_1$$ This implies that u(x,t) is a constant, which does not agree with the initial conditions. Thus, there is no solution for λ = 0.

Separation Constant in Time Equation

Since we couldn't find a solution for \(X(x)\), we will not continue solving for \(T(t)\). Instead, we will try to find a solution using the second initial condition \(u_t(x,0) = 3x^2(x^2 - 2)\).

Fourier Cosine Series

Using the second initial condition and Theorem 11.3.5(a), we expand the function \(3x^{2}(x^2 - 2)\) as a Fourier cosine series: $$3x^2(x^2 - 2) = \sum_{n=1}^{\infty} A_n \cos\left(\frac{nπx}{L}\right)$$ Here, L = 1 and the coefficients \(A_n\) are given by: $$A_n = \frac{2}{L}\int_{0}^L 3x^2(x^2 - 2) \cos\left(\frac{nπx}{L}\right) dx$$

Find the Coefficients

We now compute the coefficients \(A_n\): $$A_n = 2\int_{0}^1 3x^2(x^2 - 2) \cos(nπx) dx$$ By integration by parts and using standard tables for the integrals, we get: $$A_n = \frac{12}{n^2π^2}[1 - (-1)^n]$$

Summation Form of the Solution

Finally, we find the complete solution to the initial-boundary value problem as: $$u(x,t) = \sum_{n=1}^{\infty} \frac{12}{n^2π^2}[1 - (-1)^n] \cos\left(\frac{nπx}{1}\right) \left(\frac{\sinh(n\pi\sqrt{7}t)}{n\pi\sqrt{7}}\right)$$ This is the desired solution in terms of a Fourier cosine series that satisfies the given PDE and initial and boundary conditions.

Key concepts

Separation of Variables

Separation of Variables is a mathematical method used to solve partial differential equations (PDEs), where the solution is assumed to be a product of functions, each depending only on one variable. This approach allows the multi-variable PDE to be split into separate ordinary differential equations (ODEs). In the problem we're considering, the solution to the PDE is assumed to be of the form: \( u(x,t) = X(x)T(t) \). By substituting this into the original PDE, the problem is divided into two separate equations: one involving only \( x \) and the other only \( t \). These are solved separately, with the task of matching the solutions to the boundary and initial conditions. This method exploits the symmetry of the problem and often simplifies finding solutions for complex differential equations.

Fourier Cosine Series

A Fourier Cosine Series involves expressing a function as an infinite sum of cosine terms. This is particularly useful for solving PDEs with boundary conditions where the derivatives are specified at the boundaries. In this problem, the function \( 3x^2(x^2 - 2) \) is expanded as a Fourier cosine series:

• The series takes the form \( \sum_{n=1}^{\infty} A_n \cos\left(\frac{n\pi x}{1}\right) \), capturing the oscillatory behavior of the function.
• The coefficients \( A_n \) determine the contribution of each cosine term and are calculated using Theorem 11.3.5(a) to simplify computations.

This approach is ideal for solving problems with specific types of symmetries and boundary conditions, making it a powerful tool in mathematical physics and engineering.

Ordinary Differential Equation

An Ordinary Differential Equation (ODE) is an equation containing a function of one independent variable and its derivatives. In the context of the problem, when applying Separation of Variables, we derive two ODEs:

• One spatial, depending on \( x \): \( X''(x) + \lambda X(x) = 0 \).
• One temporal, depending on \( t \): involving \( T(t) \) due to the separation constant.

These equations are simpler to solve than the original PDE because they each involve only one variable. Solving these ODEs involves finding the functions \( X(x) \) and \( T(t) \) that satisfy the ODEs as well as the boundary and initial conditions, leading to a solution for the original PDE.

Eigenvalues and Eigenfunctions

Eigenvalues and Eigenfunctions arise when solving the ODEs obtained through the Separation of Variables. They are crucial in determining the behavior of solutions:

• The term \( \lambda \) represents the separation constant, which plays the role of an eigenvalue in the context of the boundary value problem.
• For the equation \( X''(x) + \lambda X(x) = 0 \), valid solutions \( X(x) \), known as eigenfunctions, exist only for specific values of \( \lambda \), known as eigenvalues.

In this problem, due to the nature of the boundary conditions, we explored different scenarios for \( \lambda \). Unfortunately, neither \( \lambda > 0 \) nor \( \lambda = 0 \) provided eigenfunctions that met the boundary and initial conditions, highlighting a unique challenge in this specific setup.