Partial Differential Equations
Equations with partial derivatives.
Key idea
Equations with partial derivatives.
In depth
PDEs involve partial derivatives of functions of several variables. Main types: elliptic (Laplace, steady), parabolic (heat, diffusion), hyperbolic (wave). The type determines the right boundary conditions and solution methods.
Worked example
u_t = k u_xx is parabolic; u_tt = c²u_xx is hyperbolic; u_xx + u_yy = 0 is elliptic.
Real-world uses
- Describing weather, sound, heat and fluid flow in engineering and science
Common mistake
Treating all PDEs alike.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Partial Differential Equations?
Equations with partial derivatives. It is a topic at the expert level in Differential Equations, Abstract Algebra & Applied Math.
Where is Partial Differential Equations used in real life?
Real-world examples: Describing weather, sound, heat and fluid flow in engineering and science.
What should I learn before Partial Differential Equations?
Make sure you are comfortable with Boundary Value Problems, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Partial Differential Equations?
Treating all PDEs alike.
How can I practise Partial Differential Equations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Partial Differential Equations?
Move on to Heat Equation next.
Related expert topics in Differential Equations, Abstract Algebra & Applied Math
- Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
- Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
- Runge–Kutta Methods – Higher-accuracy numerical solvers.
- Second-Order Homogeneous Equations – Use the characteristic equation.
- Undetermined Coefficients – Guess the particular solution form.
- Variation of Parameters – General method for particular solutions.