AdvancedTopic 72 of 104Engineering Mathematics
Partial differential equations
Equations with several independent variables, such as heat, wave and Laplace equations.
In depth
A PDE contains partial derivatives with respect to two or more variables. Classification: elliptic (Laplace), parabolic (heat) and hyperbolic (wave). They model diffusion, vibration, electrostatics, stress and fluid flow, usually requiring boundary and initial conditions.
Key points
- Linear PDEs allow superposition
- Analytical solutions are limited to simple geometry
- Numerical methods handle the rest
ExampleLaplace's equation ∇²φ=0 gives steady-state temperature.
Many physical processes depend on both space and time, and PDEs are the mathematical language for heat, waves, fluids and electromagnetic fields.
What is Partial differential equations?
Equations with several independent variables, such as heat, wave and Laplace equations.
Why is it important to understand partial differential equations?
Many physical processes depend on both space and time, and PDEs are the mathematical language for heat, waves, fluids and electromagnetic fields.
What is the difference between an ODE and a PDE?
An ODE has one independent variable. A PDE has several, such as space and time, and involves partial derivatives.
Why are PDEs hard to solve?
Exact solutions exist only for simple geometries, so most practical problems need numerical methods such as finite differences or finite elements.
How is Partial differential equations used in practice?
Typical applications include predicting the temperature inside a heated plate over time, modelling the vibration of a bridge deck and simulating airflow around a car body.
What should I learn before Partial differential equations?
Study "Fourier transform" first (topic 71 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Partial differential equations?
Linear PDEs allow superposition. Analytical solutions are limited to simple geometry. Numerical methods handle the rest.
Can you give an example of Partial differential equations?
Laplace's equation ∇²φ=0 gives steady-state temperature.
What comes after Partial differential equations?
The next topic is "Heat equation", which builds directly on this one.
How difficult is Partial differential equations?
It is classed as Advanced in Engineering Mathematics (topic 72 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.