Question

Consider the following initial boundary value problem

∂u/∂t ​ = ∂²u/∂x², ​ ,0≤x≤L,t>0 (1a)
∂u/∂x (0,t)=T₁​ ,t>0 (1b)
u(L,t)=T₂ ,t>0 (1c)
u(x,0)=f(x), 0≤x≤L, (1d)​
where T 1 ​ and T 2 ​ are some constants, and f(x) a given function of space.
(a) Classify the partial differential equation (1a).

Answer 1

The equation ∂u/∂t = ∂²u/∂x² is a second-order linear parabolic PDE, specifically the heat equation, and the initial boundary value problem involves finding a solution that satisfies the given boundary and initial conditions.

Step-by-step explanation:

The partial differential equation (PDE) described is ∂u/∂t = ∂²u/∂x², where the function u(x,t) denotes the temperature at a position x and time t within a rod of length L. This PDE, which models the heat distribution over time within a rod with specific initial and boundary conditions, is a classical example of the heat equation, which is a second-order linear PDE. Due to its form and the occurrence of the second derivative with respect to both time and space, the heat equation is classified as a parabolic PDE.

Considering the boundary conditions, (1b) ∂u/∂x (0,t)=T₁, and (1c) u(L,t)=T₂, together with the initial condition (1d) u(x,0)=f(x), we have an initial boundary value problem where the solution u(x,t) would be expected to satisfy both the equation and the specified conditions.