Solving the Heat Equation Using Separation of Variables

Ahmad M. D. Al-Eybani

Abstract: The heat equation is one of the fundamental partial differential equations in mathematical physics, describing how heat diffuses through a medium over time. It appears in various fields, such as thermodynamics, fluid dynamics, and materials science, and is a cornerstone of understanding diffusive processes. Mathematically, the heat equation in one spatial dimension is expressed as:

∂u∂t=α∂2u∂x ,

where (u(x, t) represents the temperature at position (x) and time (t) , and (α) is the thermal diffusivity, a positive constant that depends on the material properties. Solving this equation analytically can be challenging due to its dependence on both space and time, but one of the most elegant and widely applicable methods is separation of variables. This technique reduces the PDE into a set of ordinary differential equations (ODEs), which are easier to solve. In this article, we will explore the separation of variables method in detail, including its assumptions, step-by-step derivation, and a practical example applied to a finite rod with fixed boundary conditions.