Partial Differential Equations

Partial Differential Equations explores the mathematical techniques for solving equations involving multivariable functions and their partial derivatives.

What Are Partial Differential Equations?First-Order PDEs and Methods of SolutionCommon Second-Order PDEs: Heat, Wave, and Laplace Equations
Boundary and Initial Value ProblemsFourier Series and Transform MethodsNumerical Methods: Finite Difference and Finite Element
Modeling Heat TransferSimulating Waves and VibrationsModeling Fluid Flow and Weather
Practice with Simple Problems FirstVisualize and Interpret SolutionsAnalyze and Check Boundary Conditions Carefully
What Are Partial Dif...First-Order PDEs and...Common Second-Order ...Boundary and Initial...Fourier Series and T...Numerical Methods: F...Modeling Heat Transf...Simulating Waves and...Modeling Fluid Flow ...Practice with Simple...Visualize and Interp...Analyze and Check Bo...
Advanced Topics

Fourier Series and Transform Methods

Breaking Down Complex Problems

Many PDEs get easier when we use clever tricks like Fourier series and transforms.

  • Fourier Series: Expresses a function as a sum of sines and cosines.
  • Fourier Transform: Converts functions between time (or space) and frequency domains.

Why Use These Methods?

They turn differential equations into algebraic equations, which are much easier to solve.

Applications

  • Solving the heat equation on a finite or infinite domain.
  • Analyzing vibrations in complex systems.

Key Steps

  • Expand the initial condition in a Fourier series.
  • Solve for each term.
  • Add up all the solutions to get the final answer.

Beyond Simple Shapes

These techniques can be extended to more complicated geometries and higher dimensions.

Examples

  • Finding the temperature in a metal plate using Fourier series.

  • Analyzing sound waves with the Fourier transform.

In a Nutshell

Fourier techniques simplify PDEs by turning them into algebra problems.

Key Terms

Fourier Series
Representation of a function as a sum of sines and cosines.
Eigenfunction
A function that remains essentially unchanged by a given operation, except for a constant factor.
Next
Numerical Methods: Finite Difference and Finite Element