Solved: In 710, a string of length L coincides with the interval [0, L] on the x-axis

In Problems 1-6, a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t). The left end is held at temperature zero, and the right end is insulated. The initial temperature is f(x) throughout.

In Problems 1-6, a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t). The left end is held at temperature \(u_{0}\), and the right end is held at temperature \(u_{1}\). The initial temperature is zero throughout.

In Problems 1-6, a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t). The left end is held at temperature \(100^{\circ}\), and there is heat transfer from the right end into the surrounding medium at temperature zero. The initial temperature is f(x) throughout.

In Problems 1-6, a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t). There is heat transfer from the left end into a surrounding medium at temperature \(20^{\circ}\), and the right end is insulated. The initial temperature is f(x) throughout.

In Problems 1-6, a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t). The left end is at temperature \(\sin (\pi t / L)\), the right end is held at zero, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature zero. The initial temperature is f(x) throughout.

In Problems 1-6, a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t). The ends are insulated, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature \(50^{\circ}\). The initial temperature is \(100^{\circ}\) throughout.

In Problems 7-10, a string of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the displacement u(x, t). The ends are secured to the x-axis. The string is released from rest from the initial displacement x(L-x).

In Problems 7-10, a string of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the displacement u(x, t). The ends are secured to the x-axis. Initially the string is undisplaced but has the initial velocity \(\sin (\pi x / L)\).

In Problems 7-10, a string of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the displacement u(x, t). The left end is secured to the x-axis, but the right end moves in a transverse manner according to \(\sin \pi t\). The string is released from rest from the initial displacement f(x). For t>0 the transverse vibrations are damped with a force proportional to the instantaneous velocity.

In Problems 7-10, a string of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the displacement u(x, t). The ends are secured to the x-axis, and the string is initially at rest on that axis. An external vertical force proportional to the horizontal distance from the left end acts on the string for t>0.

In Problems 11 and 12, set up the boundary-value problem for the steady-state temperature u(x, y). A thin rectangular plate coincides with the region in the xy-plane defined by \(0 \leq x \leq 4, 0 \leq y \leq 2\). The left end and the bottom of the plate are insulated. The top of the plate is held at temperature zero, and the right end of the plate is held at temperature f(y).

In Problems 11 and 12, set up the boundary-value problem for the steady-state temperature u(x, y). A semi-infinite plate coincides with the region defined by \(0 \leq x \leq \pi, y \geq 0\). The left end is held at temperature \(e^{-y}\), and the right end is held at temperature \(100^{\circ}\) for \(0<y \leq 1\) and temperature zero for y>1. The bottom of the plate is held at temperature f(x).