Solutions of the Cahn-Hilliard Equations: A practical guide to the mathematics, numerical methods, and computer implementation of the Cahn–Hilliard equation.
About this ebook
Second Revised and Expanded Edition
A practical guide to the mathematics, numerical methods, and computer implementation of the Cahn–Hilliard equation.
The Cahn–Hilliard equation is a fundamental model for phase separation, microstructure evolution, and diffusion-driven phenomena in materials science. This book takes the subject from mathematical formulation to practical numerical simulation, making it useful both for students learning phase-field modelling and for researchers implementing Cahn–Hilliard models.
The book develops the equation systematically and explains how to solve it using a range of computational approaches, including:
Finite-difference methods
Semi-implicit and energy-stable time integration
Sparse-matrix and iterative solvers
Conjugate-gradient methods
Fourier spectral methods
Mixed finite-element methods
One- and two-dimensional simulations
Verification through mass conservation and energy dissipation
A major feature of the book is its emphasis on actual computation rather than theory alone. Examples and programs are provided using MATLAB, Python (NumPy/SciPy), and Wolfram Language, allowing readers to implement and reproduce Cahn–Hilliard simulations.
Applications discussed include binary-alloy phase separation, spinodal decomposition, coarsening, Fe–45Cr alloys, thin-film dewetting, grain growth, solidification, and lithium-ion electrodes. The book also introduces emerging computational approaches including physics-informed neural networks (PINNs) and high-performance computing.
What you will learn
The book connects the complete computational workflow:
Cahn–Hilliard equation → mathematical formulation → numerical discretization → solution algorithm → computer program → verification → physical application
This makes it particularly suitable for readers who want to move from understanding the equation to actually solving and simulating it.
Who should read this book?
Postgraduate students in mathematics, physics, materials science, and engineering
Researchers working with phase-field models and nonlinear PDEs
Engineers involved in computational materials science
Researchers studying phase separation and microstructure evolution
Readers seeking practical MATLAB and Python implementations of Cahn–Hilliard models
Second Revised and Expanded Edition — suitable as a reference, computational guide, and study aid for advanced students and researchers working with the Cahn–Hilliard equation and phase-field modelling.
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About the author
Rahul Basu is Emeritus Professor of Mechanical Engineering in Bangalore, India. His research focuses on the Cahn–Hilliard equation, phase-field modelling, and numerical methods for nonlinear diffusion problems in materials science. He emphasises matrix formulations, stability analysis, and reproducible computer implementations in MATLAB, Python, and the Wolfram Language. Solutions of the Cahn–Hilliard Equations collects his approach to the theory and computation of phase separation and microstructure evolution for graduate students, researchers, and engineers.
