Partial Differential Equations
A first course in partial differential equations that combines derivations and proofs with computation: what PDEs describe, order, linearity, superposition and well-posed problems, first-order equations solved by the method of characteristics with transport, traffic flow and shock formation in the Burgers equation, the classification of second-order equations as elliptic, parabolic or hyperbolic with characteristics and canonical forms, derivations of the heat, wave, Laplace and Poisson equations from conservation laws and Fourier's law, Fourier series with orthogonality, half-range expansions and Parseval's identity, Dirichlet's convergence theorem, coefficient decay and the Gibbs phenomenon, separation of variables for heat conduction with Dirichlet, Neumann and nonhomogeneous boundary conditions, the wave equation through d'Alembert's formula, standing waves, musical harmonics and energy conservation, harmonic functions with the mean value property and maximum principle, Poisson's integral, Bessel functions and the vibrating drum, finite difference schemes with von Neumann stability analysis, Crank–Nicolson and the Courant–Friedrichs–Lewy condition, and a capstone modelling seasonal heat flow in frozen ground with damping depths, geothermal gradients and Stefan's thaw model, with runnable Python code throughout.
MATH-336 Year 3 3 credits 12 lessons · 4 h 54 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
This is the Canadian version of this course. See the American version.
What you'll learn
- Classify partial differential equations by order and linearity, apply superposition, and state appropriate initial and boundary conditions for well-posed problems.
- Solve linear and quasilinear first-order equations by the method of characteristics, and predict shock formation in conservation laws such as traffic flow and the Burgers equation.
- Classify second-order linear equations as elliptic, parabolic or hyperbolic, find their characteristics and reduce them to canonical form.
- Derive the heat, wave, Laplace and Poisson equations from conservation laws and constitutive relations, and nondimensionalize them by scaling.
- Compute Fourier series, half-range sine and cosine series and complex Fourier series, and apply orthogonality and Parseval's identity.
- State and prove Dirichlet's convergence theorem, relate smoothness to coefficient decay and uniform convergence, and analyse the Gibbs phenomenon.
- Solve heat conduction problems with Dirichlet, Neumann, mixed and nonhomogeneous boundary conditions by separation of variables and steady-state subtraction.
- Solve the wave equation by d'Alembert's formula and by standing wave series, interpret domains of dependence, and prove energy conservation.
- Prove the mean value property and maximum principle for harmonic functions, use them for uniqueness and stability, and solve Laplace's equation on rectangles.
- Solve boundary value problems in polar coordinates using Poisson's integral, Bessel functions and Fourier–Bessel series, including the vibrating drum.
- Build explicit, implicit and Crank–Nicolson finite difference schemes, analyse their stability by the von Neumann method, and apply the CFL condition to waves.
- Model seasonal heat flow in frozen ground with exact periodic solutions, geothermal gradients, Stefan's thaw model and a verified numerical solver.
Course content
12 lessons · 4 h 54 min total · final exam
- 1 What partial differential equations describe 29 min · quiz
- 2 First-order PDEs and the method of characteristics 28 min · quiz
- 3 Classifying second-order PDEs: elliptic, parabolic and hyperbolic 25 min · quiz
- 4 Deriving the heat, wave and Laplace equations 27 min · quiz
- 5 Fourier series 26 min · quiz
- 6 Convergence of Fourier series and Gibbs phenomenon 22 min · quiz
- 7 Separation of variables for the heat equation 23 min · quiz
- 8 The wave equation: d'Alembert's solution and standing waves 25 min · quiz
- 9 Laplace's equation, harmonic functions and the maximum principle 23 min · quiz
- 10 Boundary value problems in polar coordinates and Bessel functions 23 min · quiz
- 11 Finite difference methods and stability 22 min · quiz
- 12 Capstone: modelling heat flow in frozen ground 21 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
A first course in partial differential equations that combines derivations and proofs with computation: what PDEs describe, order, linearity, superposition and well-posed problems, first-order equations solved by the method of characteristics with transport, traffic flow and shock formation in the Burgers equation, the classification of second-order equations as elliptic, parabolic or hyperbolic with characteristics and canonical forms, derivations of the heat, wave, Laplace and Poisson equations from conservation laws and Fourier's law, Fourier series with orthogonality, half-range expansions and Parseval's identity, Dirichlet's convergence theorem, coefficient decay and the Gibbs phenomenon, separation of variables for heat conduction with Dirichlet, Neumann and nonhomogeneous boundary conditions, the wave equation through d'Alembert's formula, standing waves, musical harmonics and energy conservation, harmonic functions with the mean value property and maximum principle, Poisson's integral, Bessel functions and the vibrating drum, finite difference schemes with von Neumann stability analysis, Crank–Nicolson and the Courant–Friedrichs–Lewy condition, and a capstone modelling seasonal heat flow in frozen ground with damping depths, geothermal gradients and Stefan's thaw model, with runnable Python code throughout.
Part of the Science stream, typically taken in year 3.