#### Title

#### Date of Award

2022

#### First Advisor

Kaethe Minden

#### Second Advisor

Miha Habič

#### Abstract

The Heat Equation is a partial differential equation that describes the distribution of heat over a period of time. The heat equation is important in many fields of Mathematics. In statistics, the heat equation is connected with the study of Brownian motion via the Fokker-Planck equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes [4]. There are many methods that can be used to solve the Heat Equation. One of the most common methods used to solving the Heat Equation utilizes Fourier Series and Separation of Variables. According to [3], this method is originally discovered by Joseph Fourier in 1822. This study attempts to explore this method on 1D and 2D systems. In Section 2, we look at Fourier Series of periodic functions and functions defined on an interval in R. In Section 3, we look at the derivation of the 1-D Heat Equation, the significance of the steady state solution, different types of boundary conditions and initial conditions and attempt to solve the 1-D Heat Equation under some boundary conditions. In Section 4, we look at the 2D-Heat Equation, its steady state solution, and solving the 2D Heat Equation. In Section 5, we briefly talk about what follows the 2D Heat Equation and some other methods that can be used to solve the Heat Equation

#### Recommended Citation

Xu, Ethan, "Heat Equation" (2022). *Senior Theses*. 1599.

https://digitalcommons.bard.edu/sr-theses/1599

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