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Brilliant

Differential Equations II

via Brilliant

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Overview

Drum vibrations, heat flow, the quantum nature of matter, and the dynamics of competing species are just a few real-world examples involving advanced differential equations.

These models and many others from across the sciences, engineering, and finance have nonlinear terms or several independent variables. Their equations hold many surprises, and their solutions draw on other areas of math like linear algebra and vector calculus.
This course provides an immersive and interactive experience covering the essentials of nonlinear and partial differential equations.

Syllabus

  • Introduction: What are nonlinear and partial differential equations?
    • Nonlinear Equations in a Nutshell: What makes an equation nonlinear, and why does it matter?
    • PDEs in a Nutshell: Tour the equations governing heat, waves, diffusion, and the quantum realm.
  • Nonlinear Equations: The essential toolbox for systems of nonlinear equations.
    • Lotka-Volterra I: Learn how to visualize nonlinear pairs with a direction field.
    • Lotka-Volterra II: Apply multivariable calculus ideas to an important pair of nonlinear equations.
    • Linearization: Use linear algebra to figure out the nature of equilibria.
    • The Hartman-Grobman Theorem: Find out why linearization works so well by borrowing ideas from topology.
    • Application: Get Pumped for Lasers!: Sharpen your nonlinear skills on a space age application.
    • Challenge: Liapunov Functions: Learn how to classify nonlinear equilibria with energy-like functions.
    • Cycles and The Pursuit of Happiness: Find out what limit cycles are all about with a classic pursuit problem.
    • The Poincaré-Bendixson Theorem: Deduce the existence of a limit cycle in a real-world chemical oscillator.
    • Chaos and the Lorenz Equations: Discover how nonlinear equations changed our views on science and predictability forever.
  • Partial Differential Equations: The basics of differential equations with many variables.
    • 1D Waves & d'Alembert's Formula: Begin your journey into partial differential equations (pdes) with waves on a length of rope.
    • Sources & Boundary Conditions: What happens when a 1D wave reaches the end of its rope?
    • Challenge: 2D & 3D Waves: Step into higher dimensions by modeling surface ripples and sound waves.
    • Separation of Variables & Waves: Learn how to split a difficult linear pde into a set of easier problems.
    • Fundamental Solutions: Investigate and master the equation which governs diffusion and heat conduction.
    • Challenge: Fun with Functionals: Discover how to squeeze useful info out of a pde without actually solving it.
    • Laplace's Equation: Reconstruct a pde common in science and engineering from a classic geometry problem.
    • Approximating Laplace: How can computers help solve a pde?
  • Transform Methods: Learn how to turn hard diff eq problems into easy ones.
    • The Fourier Transform: Find out what signals analysis and linear pdes have in common.
    • Practice: Fourier & The Heat Equation: Learn to wield the power of the Fourier transform by practicing on the heat equation.
    • Practice: Fourier & Laplace's Equation: Continue to develop your Fourier transform skills with Laplace's equation.
    • Challenge: Fourier & 3D Waves: Rise to the challenge of modeling 3D compression waves with the Fourier transform.
    • Schrödinger's Equation: Find deeper meaning in the Fourier transform by going on a trip to the quantum realm.
    • Conformal Maps: Sometimes, it's not the equation that needs to change, it's space itself!
    • The Laplace Transform: Come full circle and see how transform methods work on ordinary differential equations.
    • Laplace Transform Applications: Apply the Laplace transform to interesting electrical engineering problems.
  • Power Series: Discover the power of unending sums.
    • Series Solutions I: Learn how infinite sums clear the path to solving difficult differential equations.
    • Series Solutions II: Explore some quirks and special features of infinite series.
    • The Airy Equation: Use power series to solve an important problem from quantum mechanics.
    • Interlude: Return of the Wronskian: What's a reliable way for telling power series apart?
    • Cauchy-Euler Equation: Practice power series method by working on a fluid dynamics problem.
    • Bessel's Equation: Apply the experience you gained solving the Cauchy-Euler equation to model a circular vibrating drumhead.
    • Hermite's Equation: Return to the subatomic realm and use power series to solve the quantum oscillator.
    • Capstone: Hydrogen Atom I: Pull everything together to reproduce one of the greatest scientific achievements of the 20th century.
    • Capstone: Hydrogen Atom II: Complete the hydrogen atom model and sketch an electron orbital, a staple of chem classes the world over.

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