[latex]
\[
\textbf{Introduction to Hyperbolic Partial Differential Equations}
\]
\[
\text{Hyperbolic partial differential equations (PDEs) describe phenomena characterized by wave-like behavior.}
\]
\[
\text{These equations are fundamental in modeling systems such as sound waves, light waves, and water waves.}
\]
\[
\textbf{General Form:}
\]
\[
\text{A second-order PDE is considered hyperbolic if its principal part resembles the wave equation:}
\]
\[
\frac{\partial^2 u}{\partial t^2} – c^2 \nabla^2 u = 0, \quad \text{where } c > 0 \text{ is the wave speed.}
\]
\[
\textbf{Characteristics:}
\]
\[
\text{Hyperbolic PDEs have real and distinct characteristic lines along which information propagates.}
\]
\[
\text{This property ensures that the solution depends only on initial or boundary data in a finite region.}
\]
\[
\textbf{Applications:}
\]
\[
\begin{aligned}
1. & \quad \text{Wave propagation in acoustics, electromagnetics, and fluid dynamics.} \\
2. & \quad \text{Vibrations of strings, membranes, and elastic solids.} \\
3. & \quad \text{Traffic flow and other transport phenomena.}
\end{aligned}
\]
\[
\textbf{Example: The Wave Equation}
\]
\[
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad \text{where } u(x,t) \text{ represents the displacement of a wave.}
\]
\[
\text{Hyperbolic PDEs are essential for understanding dynamic systems governed by causality and finite propagation speed.}
\]
\[
\textbf{Exercise: Hyperbolic Partial Differential Equations}
\]
\[
\textbf{Q1: Solve the wave equation:}
\]
\[
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad 0 < x < L, \; t > 0
\]
\[
\text{with initial conditions:}
\]
\[
u(x,0) = f(x), \quad \frac{\partial u}{\partial t}(x,0) = g(x), \; \text{and boundary conditions: } u(0,t) = u(L,t) = 0.
\]
\[
\textbf{Solution:}
\]
\[
\text{Step 1: Assume a solution of the form: } u(x,t) = X(x)T(t).
\]
\[
\text{Step 2: Substituting } u(x,t) \text{ into the wave equation gives:}
\]
\[
\frac{1}{c^2} \frac{T”(t)}{T(t)} = \frac{X”(x)}{X(x)} = -\lambda.
\]
\[
\text{Step 3: Solve the spatial equation: } X”(x) + \lambda X(x) = 0, \text{ with } X(0) = X(L) = 0.
\]
\[
\text{Solution to spatial equation: } X_n(x) = \sin\left(\frac{n\pi x}{L}\right), \quad \lambda_n = \left(\frac{n\pi}{L}\right)^2, \; n = 1, 2, 3, \ldots
\]
\[
\text{Step 4: Solve the temporal equation: } T”(t) + \lambda_n c^2 T(t) = 0.
\]
\[
\text{Solution to temporal equation: } T_n(t) = A_n \cos\left(\frac{n\pi c t}{L}\right) + B_n \sin\left(\frac{n\pi c t}{L}\right).
\]
\[
\text{Step 5: Combine solutions: } u(x,t) = \sum_{n=1}^\infty \left[ A_n \cos\left(\frac{n\pi c t}{L}\right) + B_n \sin\left(\frac{n\pi c t}{L}\right) \right] \sin\left(\frac{n\pi x}{L}\right).
\]
\[
\text{Step 6: Use initial conditions to determine coefficients:}
\]
\[
A_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx, \quad B_n = \frac{2}{n\pi c} \int_0^L g(x) \sin\left(\frac{n\pi x}{L}\right) dx.
\]
\[
\text{Final solution:}
\]
\[
u(x,t) = \sum_{n=1}^\infty \left[ \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx \cos\left(\frac{n\pi c t}{L}\right)
+ \frac{2}{n\pi c} \int_0^L g(x) \sin\left(\frac{n\pi x}{L}\right) dx \sin\left(\frac{n\pi c t}{L}\right) \right] \sin\left(\frac{n\pi x}{L}\right).
\]