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. $$ \begin{array}{l|l} \text u_{t t}=7 u_{x x}, \quad 0< x < 1, \quad t>0, \\ u_{x}(0, t)=0, \quad u(1, t)=0, \quad t>0, \\ u(x, 0)=0, \quad u_{t}(x, 0)=4 x^{3}+3 x^{2}-7, \quad 0 \leq x \leq 1 \end{array} $$

Short answer

The general form of the solution for the given initial-boundary value problem is: $$ u(x, t) = \sum_{n=1}^{\infty}(B_n \sin(\sqrt{7}\lambda_n t))(\cos(\lambda_n x)) $$ where \(\lambda_n = n\pi\) and the coefficients \(B_n\) are given by: $$ B_n = \frac{2}{n\pi}\left[ \int_0^1 4x^3\cos(n\pi x) dx + \int_0^1 3x^2\cos(n\pi x) dx \right] $$

Step-by-step solution

Identify the general solution format

Given the PDE \(u_{tt} = 7u_{xx}\) and the boundary conditions, we know that the general solution will be of the form: $$ u(x, t) = \sum_{n=1}^{\infty} (\cos(\lambda_n x))\cdot T_n(t) $$ where \(\lambda_n = n \pi\) and \(T_n(t)\) are the time-dependent Fourier modes.

Determine the T_n(t) function using the PDE

Substitute the general solution format into the PDE, and obtain the ordinary differential equation (ODE) for \(T_n(t)\), which is: $$ T_n^{''}(t) = 7\lambda_n^2 T_n(t) $$ The solution to this ODE is: $$ T_n(t) = A_n \cos(\sqrt{7}\lambda_n t) + B_n \sin(\sqrt{7}\lambda_n t) $$

Apply initial conditions to determine coefficients

First, let's apply the initial displacement condition \(u(x, 0) = 0\): $$ 0 = A_n \cos(0) \Rightarrow A_n = 0 $$ Next, apply the initial velocity condition \(u_{t}(x, 0) = 4x^3+3x^2-7\): $$ 0 = B_n \sqrt{7}\lambda_n \sin(0)\Rightarrow B_n = \frac{2}{n\pi}\int_0^1(4x^3+3x^2-7)\cos(n\pi x) \, dx $$

Compute the Fourier coefficients

Calculate the coefficients \(B_n\) using the integration above: $$ B_n = \frac{2}{n\pi}\left[ \int_0^1 4x^3\cos(n\pi x) dx + \int_0^1 3x^2\cos(n\pi x) dx - \int_0^1 7\cos(n\pi x) dx \right] $$ Notice that the integral for the last term will be equal to zero due to the cosine function's symmetry. Therefore, we can simplify the expression: $$ B_n = \frac{2}{n\pi}\left[ \int_0^1 4x^3\cos(n\pi x) dx + \int_0^1 3x^2\cos(n\pi x) dx \right] $$ Continue by integrating and computing the expression for \(B_n\).

Write the final solution

Using the coefficients \(A_n=0\) and the computed \(B_n\), we can now write the solution for the initial-boundary value problem as follows: $$ u(x, t) = \sum_{n=1}^{\infty}(B_n \sin(\sqrt{7}\lambda_n t))(\cos(\lambda_n x)) $$ We have now found the solution for the given initial-boundary value problem, where all the required conditions are satisfied.

Key concepts

Fourier series

A Fourier series is a way to represent a function as the sum of simple sine and cosine waves. By breaking down a complex wave pattern into these simpler waves, we can analyze it more easily. For the given initial-boundary value problem, the function is expressed in terms of a mixed Fourier cosine series. This entails using cosine functions due to the boundary conditions provided, which fit a half-range expansion over the interval. These expansions are instrumental when dealing with problems that have specific boundary conditions such as those given at 0 and 1 in this exercise.

• Sine and Cosine Components: These are fundamental to Fourier series. They allow the transformation of a signal or function into a frequency domain.
• Orthogonality: A property of sine and cosine that simplifies integration, making the calculation of coefficients easier.
• Periodicity: Fourier series assumes the function repeats over a fixed interval.

This breakdown might simplify the complex behaviors observed in waves and other oscillatory phenomena, which directly applies to solving the provided exercise.

Partial differential equations

Partial differential equations (PDEs) are equations that involve multiple independent variables, their partial derivatives, and an unknown function. The given PDE in the exercise, \(u_{tt} = 7u_{xx}\), suggests that we are dealing with a wave equation. These types of PDEs describe phenomena involving waves or signals over time.

• Second Order: The PDE is second-order due to the second derivatives \(u_{tt}\) and \(u_{xx}\).
• Separation of Variables: A method often used to solve PDEs by breaking them into simpler ODEs, as shown in the original steps where we differentiate between \(x\) and \(t\).
• Application: These equations are crucial in fields like physics and engineering, where they model vibrations, heat conduction, fluid dynamics, etc.

Understanding PDEs is key to solving complex systems where change across space and time is modeled.

Boundary conditions

Boundary conditions are crucial in solving differential equations as they specify the values of the solutions at the boundaries of the domain. In our exercise, the boundary conditions were \(u_x(0,t) = 0\) and \(u(1,t) = 0\). These conditions help in determining the specific form of the solution among the infinite possibilities a differential equation might have.

• Dirichlet Boundary Condition: Specifies the value a solution must take on the boundary, such as \(u(1, t) = 0\).
• Neumann Boundary Condition: Specifies the value of the derivative of the solution on the boundary, such as \(u_x(0, t) = 0\).
• Initial Conditions: These provide the needed information to solve for constants in time-dependent solutions.

Applying these conditions ensures that the solved PDE accurately models the real-world scenario we intend to represent.

Ordinary differential equations

Ordinary differential equations (ODEs) involve functions of only one variable and their derivatives. In our solution process, we derived an ODE, \(T_n''(t) = 7\lambda_n^2 T_n(t)\), for obtaining the time-dependent factors \(T_n(t)\) of the wave equation solution.

• Second Order Homogeneous ODEs: The ODE obtained is homogeneous and second order, meaning it has constant coefficients and can be solved directly using standard methods.
• Characteristic Equations: A method for solving linear ODEs that transforms them into algebraic problems. Here, it would yield terms involving \(\cos\) and \(\sin\).
• Solutions: Typically involve exponential, sinusoidal, or polynomial expressions.

Solving ODEs is a critical step in tackling more intricate PDEs as seen in this exercise.

Fourier coefficients

Fourier coefficients play a pivotal role in representing the function as a Fourier series, providing the weights or amplitudes for each of the sine and cosine terms. In our problem, determining the coefficients \(A_n\) and \(B_n\) involved using the initial conditions to ensure the series accurately models the problem's behavior.

• Integral Calculations: Involve integrating the product of the original function with sine or cosine terms over the given domain.
• Specific Values: Obtained as shown, where \(B_n\) is calculated based on the type of series (cosine series) and the problem's specific conditions.
• Zero Coefficients: Sometimes coefficients such as \(A_n\) may be zero due to symmetry or specific initial or boundary conditions.

These coefficients are crucial to accurately reconstruct the function from its components and form the basis for our full solution to the exercise.