Applied Ordinary and Partial Differential Equations with Python

Second Edition (2026) · Stephen E. Moore · Department of Mathematics, University of Cape Coast, Ghana

🌐 Author's website: https://moorestephen.info/ 📘 The book (PDF) is available from the author; please get in touch through the website above.

This repository contains all of the Python programs of the book, organised by chapter. The book grew out of the course Ordinary and Applied Partial Differential Equations with Python taught at the African Institute for Mathematical Sciences (AIMS), Senegal, and at the University of Cape Coast. It is written for final-year (final-semester) undergraduates and beginning graduate students in mathematics, physics, engineering and data science.

About the book

The laws of nature are written as differential equations. This book teaches students to solve, understand and compute them: every method is derived by hand, analysed, implemented in Python, verified against exact solutions and applied to real problems (epidemics, pollution in a river, traffic flow, heat in a wall, vibrating strings and membranes, shock waves).

Every Python listing in the book has been run, and the output printed beneath it is the real output of the program in this repository.

Each chapter contains:

• Learning objectives and a chapter summary
• worked examples by hand and in Python, after every section
• highlighted Key Concept boxes (the ideas you must understand)
• Historical Notes on the people and problems behind the mathematics
• Caution boxes (common mistakes and failures of numerical methods)
• Python Corner boxes (libraries, idioms and good practice)
• Towards Graduate Studies boxes linking each topic to advanced mathematics
• exercises after every section, tagged Theory, Python and ★ Graduate, with answers and hints to selected exercises in Appendix B

The book has 13 chapters, 2 appendices, about 540 pages and 156 Python programs.

Contents and code

Part	Chapter	Code folder	Programs
I. Ordinary differential equations	1. General Overview of Differential Equations	code/ch01	verifying solutions, classifying ODEs, population of Ghana, cooling and carbon dating, free fall with drag, SIR epidemic, Lotka–Volterra, non-uniqueness and blow-up, direction fields, phase lines; Appendix A (appA_*): NumPy, Matplotlib, SymPy, SciPy quick reference
	2. Linear First-Order Equations and Initial Value Problems	code/ch02	separable equations, integrating factors, Picard iterates, exact equations and substitutions, reservoir pollution, Newton cooling, drug dosage, loans, logistic growth with harvesting, bifurcation
	3. Linear n-th Order Differential Equations	code/ch03	Wronskian and Abel, constant coefficients, undetermined coefficients, variation of parameters, Cauchy–Euler, vibrations, beats and resonance, systems and expm, phase portraits, Sturm–Liouville, Laplace transform
	4. Numerical Methods for ODEs	code/ch04	Euler and convergence, Heun/midpoint, Runge–Kutta tableaux, adaptive steps, solve_ivp (tolerances, events), pendulum, Lotka–Volterra, cholera SIR, Lorenz, stability regions, stiffness (Robertson), Adams methods, shooting, solve_bvp
II. First-order PDEs	5. Introduction to Partial Differential Equations	code/ch05	verifying PDE solutions, linearity test, classification, symbolic PDEs with SymPy (pdsolve), boundary conditions, Hadamard's example, backward heat equation, second-order classification and canonical forms
	6. Linear First-Order PDEs	code/ch06	transport snapshots, transformation to new coordinates, characteristic curves, transversality, integral surfaces, half-line problems, river pollutant, numerical characteristics, McKendrick age structure, Duhamel (Harmattan dust)
	7. Quasilinear Equations, Conservation Laws and Shock Waves	code/ch07	quasilinear characteristics, conservation form, Burgers breaking time, multivalued solutions, weak solutions, Riemann problems, shock paths, LWR traffic flow, Cole–Hopf
III. Second-order PDEs and finite differences	8. The Heat, Wave and Laplace Equations	code/ch08	Fourier series and Gibbs phenomenon, heat equation series solutions, heat kernel, energy, d'Alembert, plucked string, damped wave, Laplace on a rectangle, Poisson's formula, catalogue of exact solutions
	9. The Finite Difference Method	code/ch09	Taylor series and difference quotients, symbolic stencils (SymPy/Fornberg), round-off vs truncation, 1D Poisson, Neumann conditions, laterite wall, convection–diffusion, 2D Poisson (Kronecker products), L-shaped domain, discrete Laplacian eigenvalues
	10. Linear Parabolic Problems: the Heat Equation	code/ch10	method of lines, FTCS and blow-up, amplification factors, BTCS (Thomas, banded, sparse LU), Crank–Nicolson, convergence, Neumann conditions, heat with a source (AIMS project), 2D heat and ADI, Fisher–KPP
	11. Hyperbolic Problems: Transport and Wave Equations	code/ch11	FTCS vs upwind, CFL, Lax–Friedrichs, Lax–Wendroff, Beam–Warming, dispersion and diffusion, variable coefficients, damped transport (leapfrog), Burgers (conservative schemes, Godunov), traffic light, wave equation leapfrog, damped wave, 2D membrane, shallow-water dam break
	12. Approximation and Truncation Error Estimates	code/ch12	grid norms, truncation errors with SymPy, von Neumann analysis, BVP stability, matrix stability, global error, Lax equivalence, grid refinement and Richardson extrapolation, manufactured solutions, modified equations, balancing errors
	13. Project Works	code/ch13	worked sample project (convection–diffusion), starter codes: L-shaped domain, seasonal cholera SIR, traffic light

Appendix A is a Python quick reference for differential equations; Appendix B contains answers and hints to selected exercises.

Each program code/chNN/name.py is self-contained. The file code/chNN/name.out next to it contains the output printed in the book, so you can check your results against it.

Getting started

You need Python 3 with NumPy, SciPy, SymPy and Matplotlib.

• Easiest: install Anaconda, which includes everything.
• No installation: open Google Colab, upload a .py file (or paste its contents into a cell) and run it.
• With pip:

git clone https://github.com/moorekwesi/applied-ode-pde-with-python.git
cd applied-ode-pde-with-python
pip install -r requirements.txt

Run a single program, for example the explicit scheme for the heat equation from Chapter 10:

python code/ch10/heat_ftcs.py

Programs that draw figures save them as PDF files in the folder you run them from. To run every program in the book and regenerate all outputs and figures (in figures/):

python run_all.py          # all chapters (about 4 minutes)
python run_all.py ch10     # only Chapter 10

How to learn from this repository

1. Read a section of the book with the corresponding program open.
2. Run the program and compare your output with the .out file.
3. Change something: the initial data, the step sizes, the scheme, the boundary conditions. Predict what will happen first (Will it be stable? What order of convergence?), then check.
4. Attempt the Python exercises at the end of each section; most of them start from one of these programs.
5. Try a project from Chapter 13: model, analyse, discretise, implement, verify, report.

Editions

Edition	Year
First edition	2024
Second edition	2026

Citing the book

S. E. Moore, Applied Ordinary and Partial Differential Equations with Python, 2nd ed., Department of Mathematics, University of Cape Coast, Ghana, 2026. Available at https://github.com/moorekwesi/applied-ode-pde-with-python

A machine-readable citation is in CITATION.cff.

Licence and permission

The author, Stephen E. Moore, grants permission to anyone to use, copy, share, print, translate, adapt and redistribute the book and the programs, for teaching, learning, research or any other purpose, under the following licences. No further written permission is required; please give credit to the author.

• The book (Applied_ODE_and_PDE_with_Python.pdf) is licensed under the Creative Commons Attribution 4.0 International Licence (CC BY 4.0).
• The Python programs (code/ and run_all.py) are licensed under the MIT Licence.

Author

Stephen E. Moore, Department of Mathematics, University of Cape Coast, Ghana Website: https://moorestephen.info/

Corrections and suggestions are welcome: please open an issue.