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

Numerical Methods: Finite Difference and Finite Element

When Exact Solutions Are Hard

Sometimes, PDEs are too tricky to solve with pencil and paper. That's when computers and numerical methods come in!

Main Approaches

  • Finite Difference Method (FDM): Approximates derivatives by comparing values at nearby points.
  • Finite Element Method (FEM): Breaks the region up into small pieces and solves the PDE locally.

Why Use These Methods?

They let us handle complicated shapes, changing materials, or nonlinear equations.

Example Uses

  • Simulating airflow over a car.
  • Predicting how bridges flex in the wind.

Steps for FDM

  1. Replace derivatives with difference equations.
  2. Solve the resulting system of algebraic equations.

Steps for FEM

  1. Divide the domain into elements.
  2. Set up equations for each element.
  3. Assemble and solve the global system.

The Power of Computation

With numerical methods, we can explore PDEs too complex for exact solutions!

Examples

  • Modeling how heat spreads through a smartphone using FDM.

  • Designing airplane wings with FEM.

In a Nutshell

Numerical methods use computers to approximate solutions to tough PDEs.

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Modeling Heat Transfer