Question

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

Short answer

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

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 \(-k^2\): $$\frac{X''(x)}{a^2X(x)} = \frac{T''(t)}{T(t)}=-k^2.$$

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: $$T(t) = C_1\cos(akt) + C_2\sin(akt).$$

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 \(T(t)\): $$u(x, t) = (B_1\cos(axk) + B_2\sin(axk))(C_1\cos(akt) + C_2\sin(akt)).$$

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 \(N \rightarrow \infty\).

Key concepts

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.