Partial Differential Equations and Boundary Value Problems for Materials Science
Partial differential equations (PDEs) are essential for the mathematical description of diffusion, heat conduction, wave propagation, and phase transformations in materials science. This series covers theory from the heat equation, wave equation, and Laplace equation to separation of variables, Fourier series expansion, Green's function method, and numerical methods for boundary value problems, learning both theory and implementation (Python/NumPy/SciPy) in pairs.
Knowledge of basic calculus and vector analysis (partial derivatives, multiple integrals) and linear algebra (eigenvalue problems) is sufficient for learning. Understanding of basic Python usage is desirable.
Learn from fundamental theory of 1D and multidimensional heat conduction to solution methods for initial value and boundary value problems, and temperature distribution calculations in materials. Implement analytical and numerical solutions of the diffusion equation based on Fourier's law, with applications to heat treatment processes.
Learn wave propagation and d'Alembert's solution, standing wave formation, and vibration analysis of materials. Starting from string vibration, understand wave energy conservation laws and effects of boundary conditions, and implement applications to ultrasonic testing.
Learn electrostatic potential, properties of harmonic functions, and the maximum principle. Implement solution methods for Dirichlet and Neumann problems, construction and application of Green's functions, and analysis of steady-state heat conduction problems.
Learn separation of variables techniques, solution construction by Fourier series expansion, and Sturm-Liouville theory and eigenvalue problems. Implement practical solution methods including orthogonality of eigenfunctions, applications to boundary value problems, and convergence arguments.
Learn fundamentals of finite difference methods, time evolution using Crank-Nicolson method, and stability and convergence analysis. Implement applications to materials process simulations (heat treatment, diffusion, phase transformation) and master practical solution methods.
Upon completing this series, you will achieve:
For more advanced study in this field:
Expand your knowledge with related topics:
Apply your skills to hands-on projects: