Problem 5 Solve the initial-boundaryvalue ... [FREE SOLUTION]

Chapter 12: Problem 5

Solve the initial-boundaryvalue problem. In some of these exercises, Theorem 11.3.5(b) or Exercise 11.3.35 will simplify the computation of the coefficients in the Fourier sine series. $$ \begin{array}{ll} u_{t t}=7 u_{x x}, & 00, \\ u(0, t)=0, & u(1, t)=0, \quad t>0, \\ u(x, 0)=0 & u_{t}(x, 0)=x^{2}(1-x), \quad 0 \leq x \leq 1 \end{array} $$

Short Answer

Expert verified

Answer: The solution to the given initial-boundary value problem is: $$ u(x,t) = \sum_{n=1}^\infty \frac{4}{(n\pi)^2\sqrt{7}}\left[\pi n(2-\pi^2 n^2)\right]\sin\left(\sqrt{7}n\pi t\right)\sin(n\pi x) $$

Step by step solution

Determine the required Fourier sine series for initial conditions

First, we need to find the coefficients of the Fourier sine series for the given initial condition $u(x, 0)=0$ and $u_{t}(x, 0)=x^{2}(1-x)$: $$ u(x, 0)=\sum_{n=1}^{\infty}B_{n} \sin{\left(n \pi x\right)} = 0 $$ Since $u(x, 0)=0$, we don't need to compute the coefficients $B_n$. Now, we calculate the coefficients for $u_t(x,0)$: $$ u_{t}(x, 0)=x^{2}(1-x)=\sum_{n=1}^{\infty}C_{n}\sin{\left(n\pi x\right)} $$ To find the coefficients $C_n$, we will use the following formula: $$ C_{n}=\frac{2}{\operatorname{length}}\int_{0}^{\mathrm{length}}f(x) \sin \frac{n \pi}{\mathrm{length}}dx $$ Here, length = 1. Therefore, we have: $$ C_{n}=2\int_0^1 x^2(1-x)\sin(n\pi x)dx $$

Calculate the coefficients $C_n$

To find $C_n$, we need to evaluate the integral: $$ C_{n}=2\int_0^1 x^2(1-x)\sin(n\pi x)dx $$ Using integration by parts twice, we find $C_n$: $$ C_n = \frac{2}{(n\pi)^3}\left[\pi n(2-\pi^2 n^2)\right] $$

Solve the PDE using separation of variables

As given, the PDE is: $$ u_{tt} = 7u_{xx} $$ Using separation of variables, let $u(x,t)= X(x) T(t)$. Then, the above equation becomes: $$ \frac{X''(x)}{X(x)} = \frac{1}{7}\frac{T''(t)}{T(t)} = -\lambda $$ Now, solve the spatial ODE with the given boundary conditions: $$ X''(x) + \lambda X(x) = 0, \quad X(0) = 0, \quad X(1) = 0 $$ We obtain the following eigenvalues and eigenfunctions: $$ \lambda_n = (n\pi)^2, \quad X_n(x) = \sin(n\pi x) $$ Then the solution of the PDE is of the form: $$ u(x,t) = \sum_{n=1}^\infty T_n(t)\sin(n\pi x) $$

Solve the time ODE

Now, we plug the eigenvalues into the time ODE and solve it: $$ T_n''(t) + 7(n\pi)^2 T_n(t) = 0 $$ Since this ODE is a simple harmonic oscillator, we have: $$ T_n(t) = a_n \cos\left(\sqrt{7}n\pi t\right) + b_n \sin\left(\sqrt{7}n\pi t\right) $$

Find the coefficients $a_n$ and $b_n$ using initial conditions

Using the initial condition $u(x,0)=0$ and the fact that $u_t(x,0)=\sum_{n=1}^{\infty}C_{n}\sin{\left(n\pi x\right)}$, we can find the coefficients $a_n$ and $b_n$. We have: $$ a_n = 0, \quad b_n = \frac{C_n}{\sqrt{7}n\pi} $$ So, the solution $u(x,t)$ can be written as: $$ u(x,t) = \sum_{n=1}^\infty \frac{C_n}{\sqrt{7}n\pi}\sin\left(\sqrt{7}n\pi t\right)\sin(n\pi x) $$ Substitute the expression for $C_n$ that we obtained in Step 2: $$ u(x,t) = \sum_{n=1}^\infty \frac{4}{(n\pi)^2\sqrt{7}}\left[\pi n(2-\pi^2 n^2)\right]\sin\left(\sqrt{7}n\pi t\right)\sin(n\pi x) $$ This is the solution to the given initial-boundary value problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial-Boundary Value Problem

An initial-boundary value problem involves finding a solution to a partial differential equation (PDE) that satisfies certain conditions at the boundary and initial time. In our problem, we are dealing with a second-order hyperbolic PDE of the form $u_{tt} = 7u_{xx}$. The challenge is to determine the function $u(x, t)$ that fulfills both the boundary conditions $u(0, t) = 0$ and $u(1, t) = 0$, as well as the initial conditions $u(x, 0) = 0$ and $u_t(x, 0) = x^2(1-x)$. Tying initial conditions with boundary constraints is vital:

Boundary conditions: Define values that the solution must take on the boundaries of the spatial interval.
Initial conditions: Give starting information for evolution over time.

The mathematics involves finding a suitable function that behaves within these constraints across the given domain.

Separation of Variables

Separation of variables is a powerful technique used to solve PDEs by breaking them into simpler, separate ordinary differential equations (ODEs). In the context of our hyperbolic PDE $u_{tt} = 7u_{xx}$, we assume a solution approach by proposing that $u(x, t) = X(x)T(t)$. This transforms the original PDE into two independent ODEs:

One for $X(x)$: $X''(x) + \lambda X(x) = 0$
One for $T(t)$: $T''(t) + 7\lambda T(t) = 0$

Each equation can be solved separately while ensuring both the spatial and temporal functions satisfy the initial and boundary conditions. It's like peeling an onion, going layer by layer to simplify the problem.

Eigenvalues and Eigenfunctions

In solving boundary value problems using separation of variables, we often deal with finding eigenvalues and eigenfunctions. These are key to satisfying boundary conditions. For our spatial ODE: $X''(x) + \lambda X(x) = 0$ with $X(0) = 0$ and $X(1) = 0$, the solution gives rise to eigenvalues $\lambda_n = (n\pi)^2$ and corresponding eigenfunctions $X_n(x) = \sin(n\pi x)$.

Eigenvalues $\lambda_n$: Characterize the frequency modes in the domain.
Eigenfunctions $X_n(x)$: Describe the spatial form of each mode, crucial for expressing solutions in series forms.

By utilizing these, especially in Fourier series expansions, we achieve a solution that fits the requirement for each distinct $n$ mode in the problem.

Harmonic Oscillator

The time-dependent ODE can be identified as a harmonic oscillator, which is a classic system that oscillates at a specific frequency. Our time ODE $T_n''(t) + 7(n\pi)^2 T_n(t) = 0$ leads us to solutions resembling simple harmonic motion. These come in sine and cosine forms:

$T_n(t) = a_n \cos\left(\sqrt{7}n\pi t\right) + b_n \sin\left(\sqrt{7}n\pi t\right)$

This equation describes the temporal evolution influenced by initial conditions. Here, $a_n$ and $b_n$ are coefficients determined by the initial conditions, with $b_n = \frac{C_n}{\sqrt{7}n\pi}$ coming from the Fourier coefficients derived earlier. Understanding this concept helps in appreciating how different modes of vibration contribute to the overall solution.

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Short Answer

Step by step solution

Determine the required Fourier sine series for initial conditions

Calculate the coefficients \(C_n\)

Solve the PDE using separation of variables

Solve the time ODE

Find the coefficients \(a_n\) and \(b_n\) using initial conditions

Key Concepts

Initial-Boundary Value Problem

Separation of Variables

Eigenvalues and Eigenfunctions

Harmonic Oscillator

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