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...
Study Strategies

Analyze and Check Boundary Conditions Carefully

Don’t Forget the Edges!

Many mistakes in solving PDEs come from incorrect or missing boundary conditions. Always make sure you understand what happens at the edges of your system and how it affects your solution.

  • Write down all boundary and initial conditions before starting to solve.
  • Double check if the solution satisfies these conditions.
  • If in doubt, sketch the domain and mark the boundaries.

Careful attention to boundaries leads to accurate, meaningful solutions.

Examples

  • Ensuring the solution to a vibrating string is zero at the endpoints.

  • Checking that the temperature at the edge of a metal plate matches insulation requirements.