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Chapter 10: Problem 4

Let (10.138) be the wave equation \(u_{t t}-a^{2} u_{x x}=0\) for the interval \(0

Short Answer

Expert verified
In summary, the exact solution of the wave equation \(u_{t t}-a^{2} u_{x x}=0\) is given by: $$u(x, t) = (B_1\cos(axk) + B_2\sin(axk))(C_1\cos(akt) + C_2\sin(akt)).$$ The approximating system \((10.139)\) has characteristic values and functions given by: $$\lambda_{n}=\frac{2 a(N+1)}{\pi} \sin \frac{n \pi}{2(N+1)},$$ $$A_{n}\left(x_{\sigma}\right)=\sin \left(n x_{\sigma}\right).$$ As we can see, for each fixed \(n\), \(\lambda_{n}\) approaches \(an\) as \(N\) goes to infinity, which is consistent with the frequency term in the exact solution. This demonstrates the convergence of the approximating system to the exact solution as the number of points in the discretization (\(N\)) increases.

Step by step solution

01

Find the exact solution of the wave equation

To find the exact solution for the wave equation \(u_{t t}-a^{2} u_{x x}=0\), we will use the method of separation of variables. Assume \(u(x, t) = X(x) T(t)\), where \(X(x)\) and \(T(t)\) are functions of \(x\) and \(t\) respectively. Substitute this ansatz into the wave equation: $$X''(x)T(t) - a^2X(x)T''(t) = 0. $$ Divide the equation by \(a^2XT\) to obtain: $$\frac{X''(x)}{a^2X(x)} = \frac{T''(t)}{T(t)}.$$ Since the left-hand side is a function of \(x\) only and the right-hand side is a function of \(t\) only, both sides must be equal to the same constant, which we denote as \(-k^2\): $$\frac{X''(x)}{a^2X(x)} = \frac{T''(t)}{T(t)}=-k^2.$$
02

Solve the separated equations for X(x) and T(t)

Now we must solve both separated equations individually: (1) For the function \(X(x)\), the equation becomes: $$X''(x) + a^2k^2 X(x) = 0.$$ The general form of the solution is: $$X(x) = B_1\cos(axk) + B_2\sin(axk).$$ (2) For the function \(T(t)\), the equation becomes: $$T''(t) + a^2k^2 T(t) = 0.$$ The general form of the solution is: $$T(t) = C_1\cos(akt) + C_2\sin(akt).$$
03

Write the general form of the solution for u(x, t)

The general form of the solution for \(u(x, t)\) is the product of \(X(x)\) and \(T(t)\): $$u(x, t) = (B_1\cos(axk) + B_2\sin(axk))(C_1\cos(akt) + C_2\sin(akt)).$$
04

Compare the approximating system to the exact solution

Now we compare the approximating system, which is given by: $$\lambda_{n}=\frac{2 a(N+1)}{\pi} \sin \frac{n \pi}{2(N+1)},$$ $$A_{n}\left(x_{\sigma}\right)=\sin \left(n x_{\sigma}\right),$$ to the general form of the exact solution for the wave equation. As \(N \rightarrow \infty\), we have: $$\lim_{N \rightarrow \infty}\left(\frac{2 a(N+1)}{\pi} \sin \frac{n \pi}{2(N+1)}\right) = an.$$ So for each fixed \(n\), \(\lambda_{n} \rightarrow an\) as \(N \rightarrow \infty\).

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Key Concepts

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

Approximation Methods in Calculus
A crucial part of solving equations in calculus, particularly when exact solutions are difficult or impossible to obtain, involves approximation methods. These methods provide ways to get as close as possible to the true value of a function or expression. For complex systems, particular boundary conditions, or non-linearity, approximation becomes inevitable.

One approximation strategy referenced in our exercise is using a system that approximates characteristic values for large values of N. As \( N \rightarrow \infty \), the values within this system converge to the actual, continuous characteristic values of the wave equation. In calculus and its applications, such approximations come under various forms, including numerical methods like finite element analysis and finite difference methods. Utilizing these approximation methods can simplify complex problems and provide tangible solutions, especially when dealing with real-world physical systems where exact solutions are often unattainable due to various constraints. In educational contexts, mastering these approximation techniques is essential for students aiming to work with real-life applications of mathematics.

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Most popular questions from this chapter

Let \(u(x, t)\) have continuous derivatives through the second order in \(x\) and \(t\) for \(t \geq 0\) and \(0 \leq x \leq \pi\) and let \(u(0, t)=u(\pi, t)=0\). Prove that if \(u(x, t)\) satisfies the heat equation (10.91) for \(t>0,0

Let \(h_{1}=h_{2}=\cdots=h_{N}=h, m_{\sigma}=0\) and \(F_{a}(t) \equiv 0\) for \(\sigma=1, \ldots, N_{0}=u_{N+1}=\) 0 in (10.51), so that one has case (c), with equal friction coefficients. Show that the substitution \(u_{\sigma}=A(\sigma) e^{\lambda /}\) leads to the difference equation with boundary conditions: he. $$ \begin{aligned} \Delta^{2} A(\sigma)+p^{2} A(\sigma) &=0, & & p^{2}=-h \lambda / k^{2} \\ A(0) &=0, & & A(N+1)=0 . \end{aligned} $$ Use the result of Problem 5(d) to obtain the "modes of decay": $$ \begin{aligned} u_{\sigma}(t) &=\sin \left(\frac{n \pi}{N+1} \sigma\right) e^{-a_{n} t} \\ a_{n} &=\frac{2 k^{2}}{h}\left(1-\cos \frac{n \pi}{N+1}\right) . \end{aligned} $$ Show that \(0

Find the solution of the partial differential equation $$ \frac{\partial u}{\partial t}-\frac{\partial^{2} u}{\partial x^{2}}=x^{2} \cos t-2 \sin t, \quad 00 \text {, } $$ satisfying boundary conditions \(u(0, t)=0, u(\pi, t)=\pi^{2} \sin t\), and initial conditions \(u(x, 0)=\pi x-x^{2}\).

Show that the solution of the homogeneous problem \((F=0)\) corresponding to (10.121), with boundary conditions \(w(0, t)=0, w(\pi, t)=0\), is given by $$ \begin{aligned} &w=\sum_{n=1}^{\infty} \sin n x\left[\alpha_{n} \sin n a t+\beta_{n} \cos n a t\right] \\ &z=\sum_{n=1}^{\infty} \sin n x\left[n a \alpha_{n} \cos n a t-n a \beta_{n} \sin n a t\right] . \end{aligned} $$

(Difference equations) We consider functions \(f(\sigma)\) of an integer variable \(\sigma: \sigma=0\), \(\pm 1, \pm 2, \ldots\) Let \(f(\sigma)\) be defined for \(\sigma=m, \sigma=m+1, \ldots, \sigma=n\). Then the first difference \(\Delta_{+} f(\sigma)\) is the function \(f(\sigma+1)-f(\sigma)(m \leq \sigma4\), distinct numbers \(a_{1}\) and \(a_{2}\) can be chosen so that the functions \(c_{1} a_{1}^{\sigma}+c_{2} a_{2}^{\sigma}\) are solutions and that when \(p^{2}=4\), the functions \(c_{1}(-1)^{\sigma}+c_{2} \sigma(-1)^{\sigma}\) are solutions. c) Show that if \(f(0)\) and \(f(1)\) are given, then the difference equation of part (b) successively determines \(f(2) . f(3)\)..... Hence for such "initial conditions" there is one and only one solution. Determine the constants \(c_{1}, c_{2}\) for each of the cases of part (b) to match the given initial conditions. The fact that this is possible ensures that each expression gives the general solution for the corresponding value of \(p\). d) Show that the only solutions of the difference equation of part (b) which satisfy the boundary conditions: \(f(0)=0, f(N+1)=0\) are constant multiples of the \(N\) functions $$ \phi_{n}(\sigma)=\sin \left(\frac{n \pi}{N+1} \sigma\right) \quad(n=1, \ldots, N) $$ and that \(\phi_{n}(\sigma)\) is a solution only when $$ p^{2}=2\left(1-\cos \frac{n \pi}{N+1}\right) $$
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