Question

A thin wire coinciding with the x-axis on the interval [-L, L] is bent into the shape of a circle so that the ends

X=-L and x = L are joined. Under certain conditions the temperature u(x, t) in the wire satisfies the boundary value problem
k∂²u/​∂x²=∂u/∂t​,−L0,
u(−L,t)=u(L,t),t>0
∂u/∂x​∣∣​x=−L​= ∂u/∂x​∣∣​x=L​,t>0
u(x,0)=f(x),−L

Answer 1

The problem involves a boundary value problem in physics, specifically concerning the temperature distribution in a bent wire. The temperature must satisfy a partial differential equation with given boundary conditions. The problem can be solved using the method of separation of variables.

Step-by-step explanation:

The given problem represents a boundary value problem in physics. The problem involves a thin wire bent into the shape of a circle and its temperature distribution over time. The temperature, represented by u(x, t), must satisfy the partial differential equation k∂²u/​∂x²=∂u/∂t​, along with specified boundary conditions.

To solve this problem, we can apply the method of separation of variables. This involves assuming a solution of the form u(x, t) = X(x)T(t) and substituting it into the partial differential equation. By separating the variables and solving the resulting ordinary differential equations, we can find the temperature distribution u(x, t) at any given time.

It's important to note that the given information also includes additional problems and examples related to magnetic fields, center of mass, and other physics concepts. However, the primary focus of the question is on the boundary value problem involving the temperature distribution in a bent wire.