Part 1: Introduction to heat equations and one-dimensional finite difference method
The heat equation, also often called the diffusion equation, is a partial differential equation that describes how heat travels in a substance. Simply put, it helps us understand when a place suddenly gets hotter or cooler, how that heat or cooling affects the surrounding area. In mathematics and engineering, solving equations like this is very important because they help us predict and understand many natural phenomena, such as weather patterns, metallurgical processes, and various other engineering applications.
The finite difference method is a numerical method for solving heat equations. In simple terms, this method involves breaking down a substance or domain into discrete small parts or "points" and then calculating the heat transfer at each point. With this approach, we obtain a numerical solution rather than solving the actual partial differential equation. This is very useful in computer simulations because it allows us to easily simulate the spread of heat on the computer.
Now, let us show through C++ code how to solve the heat equation using the finite difference method.
#include <iostream>
#include <vector>
const double ALPHA