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. $$ \begin{aligned} \text { } u_{t t} &=u_{x x}, \quad 0< x < 1, & & t>0 \\ u_{x}(0, t) &=0, \quad u_{x}(1, t)=0, & & t>0 \\ u(x, 0) &=x^{2}\left(3 x^{2}-8 x+6\right), & & u_{t}(x, 0)=0, \quad 0 \leq x \leq 1 \end{aligned} $$

Short answer

Answer: The final solution to the given IBVP for the function u(x, t) is $$u(x, t) = 32\sum_{n=0}^{\infty} \frac{n^2\pi^2 -6}{n^4\pi^4}e^{-n^2\pi^2 t}\cos(n\pi x)$$

Step-by-step solution

The first step is to apply the Fourier cosine series on the given PDE. According to Exercise 49, the solution for the function u(x, t) can be written as: $$u(x, t) = \sum_{n=0}^{\infty} A_n e^{-\lambda_n^2 t}\cos(\lambda_n x)$$ where \(\lambda_n = n\pi\), and \(A_n\) is the Fourier cosine coefficients given by: $$A_n = \frac{2}{1}\int_{0}^{1} u(x, 0)\cos(n\pi x) dx$$ #Step 2: Calculating \(A_n\)#

Now, we will calculate the coefficients \(A_n\) using the given initial condition and the provided formula: $$A_n = 2\int_{0}^{1} x^2(3x^2 - 8x + 6)\cos(n\pi x) dx$$ #Step 3: Integrating the function#

The next step is to integrate the function to find the value of \(A_n\). Here, we can use integration by parts or a software tool to compute the integral. The result is: $$A_n = \frac{32(n^2\pi^2 -6)}{n^4\pi^4}$$ #Step 4: Putting it all together#

Now that we have the value of \(A_n\), we can plug it back into the formula for u(x, t) to obtain the final solution: $$u(x, t) = 32\sum_{n=0}^{\infty} \frac{n^2\pi^2 -6}{n^4\pi^4}e^{-n^2\pi^2 t}\cos(n\pi x)$$ And that's the solution to the given IBVP.

Key concepts

Partial Differential Equations

Partial Differential Equations (PDEs) involve multivariable functions and their partial derivatives. They are crucial in modeling various physical phenomena such as sound, heat, diffusion, and fluid flow. In the given exercise, the equation \( u_{tt} = u_{xx} \) represents a wave equation. Here, the subscripts denote partial derivatives, where \( u_{tt} \) represents the second partial derivative of \( u \) with respect to time \( t \), and \( u_{xx} \) is the second partial derivative with respect to space \( x \).
This form of PDE is known for modeling the propagation of waves. The wave equation in this context implies that the acceleration of the wave (given by \( u_{tt} \)) is equal to its curvature (represented by \( u_{xx} \)). Solving PDEs like this one requires methods that often include separation of variables and transform techniques, making them a rich area of study in applied mathematics.

Boundary Value Problems

Boundary Value Problems (BVPs) are used to find solutions to differential equations that must satisfy specific conditions at the boundaries of the domain. In this exercise, we are given boundary conditions: \( u_x(0, t) = 0 \) and \( u_x(1, t) = 0 \) for \( t > 0 \). These conditions indicate that the spatial derivative is zero at both ends of the interval \( [0, 1] \).
This condition is a type of Neumann boundary condition, suggesting that there is no heat flux entering or leaving the ends of the spatial domain. Establishing and solving BVPs allow us to understand how a physical system behaves within its spatial constraints. By fulfilling these conditions, we ensure the system's modeled behavior accurately represents reality within those limits.
Solving BVPs usually involves finding a function that satisfies both the differential equation and the given boundary constraints, which can be solved analytically or numerically depending on the complexity.

Fourier Cosine Series

The Fourier Cosine Series is a powerful tool used to express a function as a sum of cosines. It is particularly useful in solving PDEs with even and symmetric boundary conditions, as it naturally satisfies certain boundary constraints like the Neumann conditions used here.
In the given scenario, the solution \( u(x, t) \) is expressed as an infinite sum of cosine functions multiplied by time-dependent factors. This representation is:

• \( u(x, t) = \sum_{n=0}^{\infty} A_n e^{-\lambda_n^2 t}\cos(\lambda_n x) \)

Where \( \lambda_n = n\pi \) and \( A_n \) are the coefficients that depend on the specific function \( u(x, 0) \). The series converges to the solution as \( n \rightarrow \infty \).
The Fourier Cosine Series is particularly effective because it transforms a complex problem into simpler ones by decomposing it into basic trigonometric components, which can be manipulated and solved more easily.

Initial Conditions

Initial conditions in the context of PDEs provide necessary data to obtain specific solutions and ensure unique solutions to the differential equations given the system's constraints. For this exercise, the initial conditions are given as \( u(x, 0) = x^2(3x^2 - 8x + 6) \) and \( u_t(x, 0) = 0 \).
These conditions specify the initial shape and velocity of the wave at time \( t = 0 \). Specifically, \( u(x, 0) \) describes the initial displacement of the wave, while \( u_t(x, 0) = 0 \) indicates that the initial velocity is zero across the domain.
Initial conditions help determine the coefficients in the Fourier series solution. Without these, the solution would be under-specified, producing many possible solutions. By incorporating the initial conditions, the problem becomes well-defined, allowing us to calculate the Fourier coefficients \( A_n \), which in turn, forms the specific solution that respects the initial configuration of the wave in the system.