Solving the 1D Heat Equation for a Fixed Rod

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 $T_0$.

The one-dimensional heat equation is a fundamental partial differential equation (PDE) that describes how heat diffuses through a given domain over time. It is critical to understanding various physical systems where thermal conduction is involved. When solving such equations, we often resort to analytical methods such as separation of variables, which is a powerful technique where we assume that the solution can be written as the product of functions, each depending only on one of the variables. This method enables us to break down the PDE into simpler ordinary differential equations (ODEs) that can be solved individually. The boundary conditions in this problem, where the rod ends are held at zero temperature, provide a straightforward scenario often referred to as Dirichlet boundary conditions. These are essential to consider as they help determine the form of the solution. Moreover, the initial condition of having a uniform temperature throughout the rod adds another layer, requiring the solution to meet conformity with both spatial and temporal constraints in the system. By using these methods, you not only solve the equation but also gain insight into the dynamic behavior of thermal distribution over time.