Partial differential equations form tools for modelling, predicting and understanding our world. Scientists and engineers use them in the analysis of advanced problems. In this eBook, award-winning educator Dr Chris Tisdell demystifies these advanced equations. Highlights of this eBook include an integration of the lessons with YouTube videos; and the design of active learning spaces. By engaging with this eBook, its examples and Chris’s YouTube videos, you’ll be well-placed to better understand partial differential equations and their solutions techniques. Download now!

## Content

- How to use this book
- What makes this book different?
- Acknowledgement

- The Transport Equation
- Introduction – Where are we going?
- Solution Method to Transport Equation via Directional Derivatives
- Derivation of the Most Basic Transport Equation
- Transport Equation Derivation

- Solve PDE via a Change of Variables
- Change of Co-ordinates and PDE
- Non-constant Co-efficients

- PDE and the method of characteristics
- Introduction
- The semi-linear Cauchy problem
- Quasi-linear Case

- Solving the Wave Equation
- Introduction
- A Factoring Approach
- Initial Value Problem: Wave Equation
- Inhomogenous Case
- Duhamel’s Principle
- Derivation of Wave Equation
- Second-Order PDEs: Classication and Solution Method
- Independent Learning – Reflection Method: Initial/Boundary Value Problem
- Independent Learning – Reflection method: Nonhomogenous PDE
- PDE with Purely Second-Order Derivatives and Classication
- Classifying Second-Order PDE with First-Order Derivatives

- The Heat Equation
- Introduction
- Diffusion on the Whole Line
- The Modified Problem
- Independent Learning – Heat Equation: Inhomogenous Case
- Independent Learning – Duhamel’s Principle
- Solving Heat Equation on Half Line
- Derivation of Heat Equation in 1-Dimension
- Similarity Solutions to PDE

- Laplace Transforms
- Introduction
- Inverse Laplace Transforms
- Using Tables to Calculate Transforms
- First Shifting Theorem
- Introduction to Heaviside Functions
- Second Shifting Theorem: Laplace Transforms
- Transform of Derivatives
- Solving IVPs with Laplace Transforms
- Applications of Laplace Transforms to PDE

- Applications of Fourier Transforms to PDE
- Introduction to Fourier Transforms
- First Shifting Theorem of Fourier Transforms
- Second Shifting Theorem of Fourier Transforms
- Convolution Theorem of Fourier Transforms

- Introduction to Green’s Functions
- Introduction
- Green’s First Identity
- Uniqueness for the Dirichlet Problem
- Existence for the Dirichlet Problem

- Harmonic Functions and Maximum Principles
- Introduction
- First Mean Value Theorem for Harmonic Functions
- Maximum Principle for Subharmonic Functions
- Independent Learning – More Properties of Harmonic Functions

- Bibliography

## About the author

“With more than a million YouTube hits, Dr Chris Tisdell is the equivalent of a best-selling author or chart-topping musician. And the unlikely subject of this mass popularity? University mathematics.” [Sydney Morning Herald, 14/6/2012].

Chris Tisdell has been inspiring, motivating and engaging large mathematics classes at UNSW, Sydney for over a decade. His lectures are performance-like, with emphasis on contextualisation, clarity in presentation and a strong connection between student and teacher.

He blends the live experience with out-of-class learning, underpinned by flexibility, sharing and openness. Enabling this has been his creation, freely sharing and management of future-oriented online learning resources, known as Open Educational Resources (OER). They are designed to empower learners by granting them unlimited access to knowledge at a time, location and pace that suits their needs. This includes: hundreds of YouTube educational videos of his lectures and tutorials; an e-textbook with each section strategically linked with his online videos; and live interactive classes streamed over the internet.

His approach has changed the way students learn mathematics, moving from a traditional closed classroom environment to an open, flexible and forward-looking learning model.

Indicators of esteem include a prestigious educational partnership with Google; an e-textbook with over 500,000 unique downloads; mathematics videos enjoying millions of hits from over 200 countries; a UNSW Vice-Chancellor’s Award for Teaching Excellence; and 100% student satisfaction rating in teaching surveys across 15 different courses at UNSW over eight years.

Chris has been an educational consultant to The Australian Broadcasting Corporation and has advised the Chief Scientist of Australia on educational policy.