Let u(x, t) be the solution of utt − uxx = 0, 0 < x < 2, t > 0 u(0, t) = 0 = u(2, t), ∀ t > 0, u(x, 0) = sin (πx) + 2 sin(2πx), 0 ≤ x ≤ 2, ut(x, 0) = 0, 0 ≤ x ≤ 2. Which of the following is true?
u(1/2, 2) = 1.
Step 1 – solution:
\(u(x,t)=\sin(\pi x)\cos(\pi t)+2\sin(2\pi x)\cos(2\pi t)\)
These modes meet both boundary conditions and the initial displacement and zero initial velocity.
Step 2 – evaluate choices:
\(u(1,1)=0,\quad u(1/2,1)=-1,\quad u(1/2,2)=1,\quad u_t(1/2,1/2)=-\pi\)
Only the marked third statement is true.
Important Questions from Laplace Equation, Heat and Wave Equations
Which of the following are solutions of the Laplace equation uxx + uyy = 0 in the unit disk D = {(x, y) ∶ x2 + y2 < 1}?
The following partial differential equation
\(x^2 \frac{\partial^2 u}{\partial x^2}-2 x y \frac{\partial^2 u}{\partial x \partial y}\) - \(3 y^2 \frac{\partial^2 u}{\partial y^2}+\frac{\partial u}{\partial x}-\frac{\partial u}{\partial y}=0\)
is
