Question

Translate the following into a PDE with boundary conditions The left end of a rod of length L is held at temperature 83 , and there is heat transfer from the right end into the surrounding medium, which has constant temperature zero. The initial temperature x units from the left end of the rod is 83ᵉˣˡ⁻ˣ². How to enter the answers. The PDE is The boundary values / initial conditions are: Left end is held at temperature 83 : Heat transfer from the right end into the surrounding medium at temperature zero: Initial temperature is 8 :

Answer 1

The problem is about deriving the PDE and boundary conditions for the temperature distribution in a rod under certain constraints. The heat equation is used as the PDE with specified boundary and initial conditions reflecting constant temperature at one end, convective heat transfer at the other, and initial temperature variation along the rod.

Step-by-step explanation:

We are dealing with the heat conduction in a rod, which can be described by the heat equation, a partial differential equation (PDE). The PDE for the temperature T(x,t) as a function of position x and time t in the rod is given by:

∂T/∂t = α∂²T/∂x²

where α is the thermal diffusivity of the material. The boundary conditions and initial condition for this problem are:

1. Left end is held at temperature 83: T(0,t) = 83 for all t > 0.
2. Heat transfer from the right end into the surrounding medium at temperature zero: This describes a convective boundary condition, typically modeled as ∂T/∂x(L,t) = -h(T(L,t) - T_{∞}), where h is the heat transfer coefficient and T_{∞} is the surrounding medium's temperature, which in this case is 0.
3. Initial temperature is 83ᵉʰ₁⁻ᵢ²: T(x,0) = 83e^{x(1-x^2)} for 0 ≤ x ≤ L.

The first boundary condition ensures that the temperature at the left end remains constant at 83 degrees, while the second models heat loss at the right end to the surrounding environment. The third sets the initial temperature distribution along the length of the rod.