Visualizations & Practice
Interactive heat & wave simulators • 6-question quiz • Summary & reference
Interactive Visualizations
Heat Equation Simulator
Watch an initial temperature profile evolve over time as heat diffuses. Adjust $K$, $L$, the number of Fourier terms, and the initial condition to explore different behaviours.
- High-frequency wiggles in the initial profile die out first.
- Larger $K$ makes the solution smooth out faster.
- With homogeneous Dirichlet BCs, $u \to 0$ as $t \to \infty$.
Wave Equation Simulator
See how a vibrating string (plucked from initial shape $f(x) = \sin(\pi x/L)$) oscillates. The wave equation preserves energy — solutions do not decay.
Practice Quiz
Test your understanding of the key ideas in Chapter 6. Click a choice to check your answer.
Q1: Which equation is the heat equation?
Correct! The heat equation is $u_t = K u_{xx}$ — first-order in time, second-order in space.
Q2: For the heat equation on $[0, L]$ with homogeneous Dirichlet BCs, which three-case outcome holds?
Correct! Cases $\lambda < 0$ and $\lambda = 0$ both force the trivial solution. Only $\lambda > 0$ with $\sin(\alpha L) = 0$ produces the family $\sin(m\pi x/L)$.
Q3: Solve $u_t = 7u_{xx}$ on $[0,\pi]$ with $u(0,t)=u(\pi,t)=0$ and $u(x,0) = 3\sin(2x) - 6\sin(5x)$.
Correct! With $L = \pi$, the decay rate of mode $m$ is $K m^2 = 7 m^2$. So mode 2 decays at rate $28$ and mode 5 at rate $175$.
Q4: For non-homogeneous Dirichlet BCs $u(0,t) = T_1$, $u(L,t) = T_2$, the steady-state is:
Correct! $V$ satisfies $V''(x) = 0$, $V(0) = T_1$, $V(L) = T_2$, giving a linear interpolation.
Q5: With Neumann BCs ($u_x(0,t) = u_x(L,t) = 0$), how does the $\lambda = 0$ case behave?
Correct! For Neumann BCs, $\lambda = 0$ is non-trivial: $P(x) = C_1 x + C_2$ with $P'(0) = P'(L) = 0$ gives $C_1 = 0$ and leaves $C_2$ free. This produces the constant (DC) mode which conserves the spatial average forever.
Q6: The wave equation $u_{tt} = a^2 u_{xx}$ differs from the heat equation because its temporal solution:
Correct! The temporal ODE $Q'' + \omega^2 Q = 0$ has oscillatory solutions $A\cos(\omega t) + B\sin(\omega t)$. Energy is conserved.
0 / 6 correct
Summary of Problem Types
| Problem Type | BCs | Strategy | Eigenfunctions |
|---|---|---|---|
| Type 1: Homogeneous Dirichlet | $u(0,t) = u(L,t) = 0$ | Direct separation of variables | $\sin(m\pi x/L),\; m\geq 1$ |
| Type 2: Non-Homogeneous Dirichlet | $u(0,t) = T_1,\; u(L,t) = T_2$ | $u = V(x) + w$; $V = T_1 + \tfrac{T_2-T_1}{L}x$ | $\sin(m\pi x/L)$ |
| Type 3: Non-Homog. PDE with Source | $u(0,t) = T_1,\; u(L,t) = T_2$ | $u = V(x) + w$; solve $KV'' + p(x) = 0$ | $\sin(m\pi x/L)$ |
| Type 4: Neumann (insulated) | $u_x(0,t) = u_x(L,t) = 0$ | Direct separation; cosine series | $1$ and $\cos(m\pi x/L)$ |
| Type 5: Wave equation | $u(0,t) = u(L,t) = 0$, two ICs | 2nd-order temporal ODE; both $A_m$ and $B_m$ | $\sin(m\pi x/L)$ |
Quick Reference Card
Heat Equation
$$\frac{\partial u}{\partial t} = K\frac{\partial^2 u}{\partial x^2}$$
Decay factor: $e^{-K(m\pi/L)^2 t}$
Wave Equation
$$\frac{\partial^2 u}{\partial t^2} = a^2\frac{\partial^2 u}{\partial x^2}$$
Oscillation factor: $\cos / \sin(\frac{m\pi a}{L}t)$
Laplace Equation
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$
Steady-state; no time dependence
Dirichlet Eigenfunctions
$$P_m(x) = \sin\!\left(\frac{m\pi x}{L}\right)$$
Fourier Sine Series for $C_m$
Neumann Eigenfunctions
$$P_0 = 1,\quad P_m = \cos\!\left(\frac{m\pi x}{L}\right)$$
Fourier Cosine Series (incl. DC)
Non-Homog. Strategy
$$u(x,t) = V(x) + w(x,t)$$
Reduce to homogeneous problem for $w$
Three-Case Rule
Dirichlet: only $\lambda > 0$ works.
Neumann: $\lambda = 0$ (constant) AND $\lambda > 0$ work.
Sine Coefficients
$$C_m = \frac{2}{L}\int_0^L f(x)\sin\!\left(\frac{m\pi x}{L}\right)dx$$
Cosine Coefficients
$$C_m = \frac{2}{L}\int_0^L f(x)\cos\!\left(\frac{m\pi x}{L}\right)dx$$