Introduction
Heat equations are partial differential equations in mathematics and physics. Solutions of the heat equation are sometimes known as caloric functions. In 1822, Joseph Fourier developed the heat equation to model how a quantity, such as heat, diffuses through a region.
Heat equations are among the most widely studied topics in pure mathematics, and their analysis is considered fundamental to partial differential equations in general. In addition to Riemannian manifolds, the heat equation has many geometric applications. According to Subbaramiah Minakshisundaram and Ke Pleijel, the heat equation is closely related to spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry by James Eells and Joseph Sampson in 1964, which inspired Richard Hamilton to introduce the Ricci flow in 1982 and Grigori Perelman to prove the Poincaré conjecture in 2003. By applying the Atiyah–Singer index theorem to certain solutions of the heat equation, heat kernels provide subtle information about the region in which they are defined.
Wave propagation in elastic media and heat propagation in bodies are important topics in physics. Using basic physical laws, we can model wave motion and heat diffusion as partial differential equations, PDEs. Although there is no general method for solving PDEs, the wave and heat equations have unique solutions. Using traveling waves and superposition of standing waves, we will solve the wave equation. The Fourier series theory can be obtained by using the latter method. Next, we will discuss the heat equation. After discussing the maximum principle, we will solve the heat equation by finding a particular solution and then constructing the general solution. Separation of variables and eigenfunction expansion will be used to solve the heat equation for a bounded interval. The purpose of this paper is to present some basic results from the theory of PDEs. According to the theorems and method of proof, We rely on Partial Differential Equations by Walter A. Strauss, Fourier Analysis by Elias M. Stein and Rami Shakarchi, and Partial Differential Equations by Fritz John. To highlight the connection between physics and Fourier Analysis, We present the material from a different perspective.