Question

1. Solve the differential equation analytically $\frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2}$ on the domain $0 \le x \le 3$, $t \ge 0$ with homogeneous Dirichlet boundary conditions $u(0, t) = u(3, t) = 0$ and initial conditions (a) $u(x, 0) = 3 \sin(2\pi x/3)$ (b) $u(x, 0) = 3 \sin(2\pi x/3) - 5 \sin(\pi x)$ For both (a) and (b) plot: solution of the equation at $t = 0$, solution of the equation at $t = 0.1$, solution of the equation at $t = 0.2$ (on the same plot, but different plots for (a) and (b)). Describe the temporal evolution of the solution. Explain why the temporal evolution looks conceptually different for parts (a) and (b).

          1. Solve the differential equation analytically
$\frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2}$
on the domain $0 \le x \le 3$, $t \ge 0$ with homogeneous Dirichlet boundary conditions $u(0, t) = u(3, t) = 0$
and initial conditions
(a) $u(x, 0) = 3 \sin(2\pi x/3)$
(b) $u(x, 0) = 3 \sin(2\pi x/3) - 5 \sin(\pi x)$
For both (a) and (b) plot: solution of the equation at $t = 0$, solution of the equation at $t = 0.1$, solution
of the equation at $t = 0.2$ (on the same plot, but different plots for (a) and (b)). Describe the temporal
evolution of the solution. Explain why the temporal evolution looks conceptually different for parts
(a) and (b).

1. Solve the differential equation analytically
(∂ u)/(∂ t) = 2 (∂^2 u)/(∂ x^2)
on the domain 0 ≤ x ≤ 3, t ≥ 0 with homogeneous Dirichlet boundary conditions u(0, t) = u(3, t) = 0
and initial conditions
(a) u(x, 0) = 3 sin(2π x/3)
(b) u(x, 0) = 3 sin(2π x/3) - 5 sin(π x)
For both (a) and (b) plot: solution of the equation at t = 0, solution of the equation at t = 0.1, solution
of the equation at t = 0.2 (on the same plot, but different plots for (a) and (b)). Describe the temporal
evolution of the solution. Explain why the temporal evolution looks conceptually different for parts
(a) and (b).

Added by Derrick A.

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