Implicit Solution of Parabolic Partial Differential Equations

This MATLAB implementation presents a method for solving partial differential equations using an implicit approach specifically designed for parabolic PDEs. The method employs various numerical techniques such as finite difference methods and finite element methods to discretize and solve the equations. Key implementation aspects include time step control algorithms for numerical stability, boundary condition handling through specialized functions, and precision enhancement techniques using iterative solvers. The core algorithm typically involves constructing a system of linear equations at each time step, where matrices represent spatial discretization and time derivatives are handled implicitly using methods like backward Euler or Crank-Nicolson schemes. MATLAB functions such as sparse matrix operations and linear system solvers (e.g., backslash operator or specialized iterative methods) are crucial for efficient computation. This powerful mathematical tool finds applications across various engineering and scientific domains including fluid dynamics simulations using Navier-Stokes equations, heat transfer analysis through Fourier's law implementations, and electromagnetic field calculations based on Maxwell's equations. The implementation allows for customizable parameters and adaptive mesh refinement to handle complex physical phenomena with high accuracy.