Numerical solution of the heat equation using the Finite Difference Method

This paper focuses on solving the one-dimensional heat equation numerically through the Finite Difference Method (FDM), highlighting the application of the Crank-Nicolson approach. The heat equation, a key partial differential equation in physics and engineering, describes how heat diffuses over time within a material. Although exact analytical solutions exist for simple scenarios, they are often inadequate for more complex problems, making numerical techniques essential. In this study, the continuous domain is discretized in both time and space, converting the heat equation into a set of algebraic expressions. The Crank-Nicolson method, well-regarded for its numerical stability and second-order precision, is applied to examine temperature variations under different types of boundary conditions, such as Dirichlet and Neumann. Implementation is carried out using Wolfram Mathematica, which also enables dynamic visualizations through animated plots and 3D surfaces. The accuracy of the numerical results is checked by comparing them to known exact solutions, using measures like the L2-Norm and maximum absolute error. The analysis demonstrates that the Crank-Nicolson method is an effective and accurate tool for simulating heat transfer, offering a reliable solution strategy for practical thermal conduction problems.