Assignment $1$: Heating a Rod

Problem Statement:

A metal rod, insulated along its sides, is $5$ meters long. It is initially at a uniform temperature of $20C.$

At time $t=0,$ one end of the rod is placed in a furnace at $100C.$ The other end remains insulated. Find

an expression for the temperature at a point $P$ located $x$ meters from the heated end at any time $t$

seconds after $t=0.$

Instructions:

Mathematical Formulation:

Formulate the one$-$dimensional heat equation based on the given scenario.

Define the terms and parameters involved.

Boundary and Initial Conditions:

Specify the boundary conditions for the rod.

Discuss the physical significance of the conditions.

Analytical Solution:

Solve the one$-$dimensional heat equation analytically to find the expression for the

temperature distribution at any point $P$ and time $t.$

Visualization:

Provide visualizations of the temperature distribution along the rod for different time

instances.

Discussion:

Discuss the behavior of the rod as it heats up over time.

Analyze the impact of changing parameters $($e$.$g$.,$ initial temperature, rod length$)$ on

the temperature distribution.