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Partial Differential Equations

∂u

Partial Differential Equations

Heat, wave, and Laplace equations and beyond

147Modules

8Units

1

Introduction to PDEs

Classification and basic concepts

20 modules

What are PDEs?INTRO

ODEs vs PDEsINTRO

PDEs in NatureINTRO

PDE NotationLESSON

PDE ClassifierACTIVITY

Order and DegreeLESSON

Linear vs Nonlinear PDEsLESSON

Homogeneous PDEsLESSON

Classification of Second-Order PDEsLESSON

Classification ExamplesEXAMPLE

Classification PracticePRACTICE

Elliptic, Parabolic, HyperbolicLESSON

Boundary ConditionsLESSON

Boundary and Initial Condition ExamplesEXAMPLE

Boundary Conditions PracticePRACTICE

Initial ConditionsLESSON

Well-PosednessLESSON

Introduction SummarySUMMARY

Introduction ReferenceFORMULA SHEET

Introduction to PDEs QuizQUIZ

2

First-Order PDEs

Method of characteristics

19 modules

First-Order PDEsINTRO

Transport PhenomenaINTRO

Linear First-Order PDEsLESSON

First-Order PDEs PracticePRACTICE

Method of CharacteristicsLESSON

Method of Characteristics ExamplesEXAMPLE

Characteristics PlotterACTIVITY

Method of Characteristics PracticePRACTICE

Characteristic CurvesLESSON

Quasilinear PDEsLESSON

Transport EquationLESSON

Transport Equation ExamplesEXAMPLE

Burgers' EquationLESSON

Shock WavesLESSON

Rarefaction WavesLESSON

Traffic FlowREAL WORLD

First-Order PDEs SummarySUMMARY

First-Order PDEs ReferenceFORMULA SHEET

First-Order PDEs QuizQUIZ

3

The Heat Equation

Parabolic PDEs and diffusion

22 modules

The Heat EquationINTRO

Diffusion ProcessesINTRO

Derivation of Heat EquationLESSON

Heat in a Rod ExamplesEXAMPLE

Heat in a Plate ExamplesEXAMPLE

Heat Equation SimulatorACTIVITY

1D Heat Equation PracticePRACTICE

2D Heat Equation PracticePRACTICE

Heat Equation in 1DLESSON

Separation of VariablesLESSON

Separation of Variables SolverACTIVITY

Fourier Series SolutionsLESSON

Dirichlet Boundary ConditionsLESSON

Neumann Boundary ConditionsLESSON

Robin Boundary ConditionsLESSON

Heat KernelLESSON

Maximum PrincipleLESSON

Heat Equation in 2DLESSON

Thermal EngineeringREAL WORLD

Heat Equation SummarySUMMARY

Heat Equation ReferenceFORMULA SHEET

Heat Equation QuizQUIZ

4

The Wave Equation

Hyperbolic PDEs and waves

22 modules

The Wave EquationINTRO

Waves in NatureINTRO

Derivation of Wave EquationLESSON

Wave Equation SimulatorACTIVITY

1D Wave Equation PracticePRACTICE

Wave Equation in 1DLESSON

D'Alembert's SolutionLESSON

D'Alembert Solution PracticePRACTICE

Traveling WavesLESSON

Standing WavesLESSON

Separation of VariablesLESSON

Boundary Conditions for WavesLESSON

Wave ReflectionLESSON

Wave Equation in 2DLESSON

Wave Equation in 3DLESSON

Huygens' PrincipleLESSON

Musical InstrumentsREAL WORLD

Vibrating String ExamplesEXAMPLE

Vibrating Membrane ExamplesEXAMPLE

Wave Equation SummarySUMMARY

Wave Equation ReferenceFORMULA SHEET

Wave Equation QuizQUIZ

5

Laplace and Poisson Equations

Elliptic PDEs and potential theory

22 modules

Laplace's EquationINTRO

Steady-State ProblemsINTRO

Laplace Equation DerivationLESSON

Laplace Equation ExamplesEXAMPLE

Laplace Equation SolverACTIVITY

Laplace Equation PracticePRACTICE

Harmonic FunctionsLESSON

Maximum PrincipleLESSON

Uniqueness of SolutionsLESSON

Laplace on a RectangleLESSON

Laplace on a CircleLESSON

Poisson's EquationLESSON

Poisson Equation ExamplesEXAMPLE

Poisson Equation PracticePRACTICE

Green's FunctionsLESSON

Potential TheoryLESSON

Laplace Equation in 3DLESSON

ElectrostaticsREAL WORLD

Fluid FlowREAL WORLD

Laplace and Poisson SummarySUMMARY

Laplace/Poisson ReferenceFORMULA SHEET

Laplace and Poisson QuizQUIZ

6

Transform Methods

Fourier and Laplace transforms for PDEs

18 modules

Transform MethodsINTRO

Integral TransformsINTRO

Fourier Transform for PDEsLESSON

Fourier Transform PDE ExamplesEXAMPLE

Transform Method SolverACTIVITY

Fourier Transform PracticePRACTICE

Heat Equation via Fourier TransformLESSON

Wave Equation via Fourier TransformLESSON

Laplace Transform for PDEsLESSON

Laplace Transform PDE ExamplesEXAMPLE

Laplace Transform PracticePRACTICE

Heat Equation via Laplace TransformLESSON

Wave Equation via Laplace TransformLESSON

Duhamel's PrincipleLESSON

Infinite DomainsLESSON

Transform Methods SummarySUMMARY

Transform Methods ReferenceFORMULA SHEET

Transform Methods QuizQUIZ

7

Numerical Methods for PDEs

Computational approaches

19 modules

Numerical Methods for PDEsINTRO

Why Numerical Methods?INTRO

Finite Differences for PDEsLESSON

Finite Difference for Heat EquationEXAMPLE

Finite Difference SolverACTIVITY

Explicit MethodsLESSON

Explicit/Implicit PracticePRACTICE

Implicit MethodsLESSON

Crank-Nicolson MethodLESSON

Stability AnalysisLESSON

Stability PracticePRACTICE

CFL ConditionLESSON

Finite Element Method IntroductionLESSON

Spectral MethodsLESSON

Computational Fluid DynamicsREAL WORLD

Numerical Wave EquationEXAMPLE

Numerical Methods SummarySUMMARY

Numerical Methods ReferenceFORMULA SHEET

Numerical Methods QuizQUIZ

8

Course Resources

Comprehensive references and practice materials

5 modules

Problem Solving GuideSUMMARY

PDEs Master ReferenceFORMULA SHEET

Complete Formula SheetFORMULA SHEET

Solution Techniques SummaryFORMULA SHEET

Comprehensive PDEs TestQUIZ