Differential Equations: Partial Differential Equations and Fourier Series

Using the heat equation, determine how the temperature distribution along a one-dimensional rod changes over time given certain boundary conditions and an initial temperature function.

Using the heat equation, determine how the temperature distribution along a one-dimensional rod changes over time given certain boundary conditions and an initial temperature function.

Use the Fourier Transform to solve the 1D heat equation $U_T = \alpha^2 U_{XX}$, assuming an infinite 1D piece of metal with an initial temperature distribution $U(X, 0)$ that dies out to zero at $\pm\infty$.

Given a material bar with ends kept at temperature zero, analyze the heat flow and temperature distribution over time using the heat equation.

Suppose we have a metal bar of length $L$. The ends of the bar are attached to some heat baths, so their temperatures are fixed. The center of the bar is heated and becomes very hot. Describe how the temperature distribution changes and eventually stabilizes over time as the bar cools off using Fourier series.

Dirichlet problem for the Laplace equation with the region defined as the upper half plane.

Newman's problem for the Laplace equation in the upper half plane.

Robin's problem example: $u_{xx} + u_{yy} = 0$, in polar coordinates $\Delta^2 u = 0$ on a disk with radius 1. Boundary condition: $\frac{\partial u}{\partial r}$ at $r = 1$ plus $h$ into Function $f(x)$.

Given a circular region, with the top half at a temperature of 1 and the bottom half at a temperature of -1, find the temperature distribution inside the circle that reaches equilibrium.

Solve the Laplace equation for a heated plate with the following boundary conditions: The temperature is zero on the left and right edges, the bottom edge is insulated (derivative is zero), and the top edge has a given function describing the temperature distribution.

Solve the Laplace equation for a region where non-zero functions exist on all four boundaries, using the concept of superposition to combine the solutions from two separate problems, each addressing different boundary conditions.

Find the solution for the Laplace equation $\Delta^2 u = 0$ in two dimensions with the given boundary conditions.

What is the general solution to this differential equation: $\frac{\partial U}{\partial t} = \alpha \frac{\partial^2 U}{\partial x^2}$ with boundary conditions $U(0, t) = 0$ and $U(L, t) = 0$?

Given a circular rod where the temperature at one end is the same as the temperature at the other end ($U(-L, T) = U(L, T)$) and the heat flow at both ends is also equal, solve the heat equation with the initial conditions provided.

Solve the heat equation for a one-dimensional iron rod with prescribed temperature of 0 Celsius at both ends and a known starting temperature across the rod using the separation of variables technique.

Solve the 1-D heat equation for a rod with length $L$, where the ends of the rod are held at a fixed temperature of zero, starting with an initial temperature $T_0$.

Solve the homogeneous wave equation using separation of variables under the given initial and boundary conditions.

Using separation of variables, solve the one-dimensional wave equation for an electric field, represented as a partial differential equation.