Courses / Mathematics / Partial differential equations and Fourier methods Mathematics · unit version 1 Partial differential equations and Fourier methods Thirty lessons connect characteristics, Fourier expansions, heat, wave and elliptic boundary problems with energy, maximum principles, transforms and numerical stability. Breadth is informed by MIT 18.152 Fall 2011 and Stanford Math 220 Autumn 2018; this is an original sequence, not a claim of equivalent assessment or complete graduate PDE theory. General existence, regularity and spectral theorem dependencies are labeled.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
For u(x,t)=e^(−4t)sin(2x), compute uₜ and uₓₓ. A proposed PDE solution must be checked against the equation and its data.
Compare reasoning and find a refresher Both are −4e^(−4t)sin(2x), so uₜ=uₓₓ. This checks the differential equation; initial and boundary conditions require separate substitution.
Review partial derivatives
Why does integration by parts produce a boundary term? Energy estimates and weak formulations depend on keeping or justifying removal of boundary contributions.
Compare reasoning and find a refresher It integrates the product rule: ∫f g′=[fg]−∫f′g. Boundary conditions may make [fg] zero, but that must be checked for the particular factors. Regularity must justify the calculation or a weak formulation must replace it.
Review integration by parts
When may you differentiate an infinite series term by term? Fourier solution formulas require more than formal manipulation.
Compare reasoning and find a refresher One sufficient route is convergence at one point and uniform convergence of the derivative series on the interval, with differentiable summands. Other settings use separate L² or distributional arguments. Pointwise convergence alone does not justify differentiation.
Review differentiation and convergence
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PDE: data, hypotheses and solution evidence
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Lesson details and objectives Separate PDE type, linearity and well-posedness.
Classify a second-order principal part. Specify initial and boundary data. Distinguish existence, uniqueness and continuous dependence. 6 practice tasks.
Lesson details and objectives Reduce a first-order PDE to an ODE on a path.
Derive constant-speed transport solutions. Include a source or reaction term. Check noncharacteristic data geometry. 6 practice tasks.
Lesson details and objectives Track nonlinear speeds and characteristic crossing.
Derive Burgers characteristics. Find the Jacobian of the label map. Explain gradient blow-up without amplitude blow-up. 6 practice tasks.
Lesson details and objectives Derive weak form and Rankine–Hugoniot.
Convert conservation into a test-function identity. Compute a shock speed. Explain weak nonuniqueness. 6 practice tasks.
Lesson details and objectives Solve Burgers Riemann problems and check entropy.
Distinguish compressive shocks and rarefactions. Verify a self-similar fan. State a scalar entropy inequality with its direction. 6 practice tasks.
Lesson details and objectives Derive coefficients and normalization before solving PDEs.
Compute trigonometric coefficients. Use orthogonality and parity. Distinguish projection from pointwise equality. 6 practice tasks.
Lesson details and objectives Separate L², pointwise and uniform conclusions.
State a Dirichlet convergence criterion. Explain Gibbs behavior. Use Parseval without unsupported differentiation. 8 practice tasks.
Lesson details and objectives Derive Green’s identity and the regular spectral framework.
Prove weighted orthogonality. Compute interval eigenpairs. State positivity and completeness hypotheses. 6 practice tasks.
Lesson details and objectives Solve homogeneous boundary heat problems by separation.
Derive exponential mode decay. Match initial data by Fourier coefficients. State convergence and compatibility requirements. 6 practice tasks.
Lesson details and objectives Use integration by parts and a parabolic maximum argument.
Derive heat energy decay. Prove uniqueness in a stated smooth class. Apply a parabolic maximum principle. 6 practice tasks.
Lesson details and objectives Verify the Gaussian fundamental solution and its initial limit.
Check normalization and the heat equation. Use convolution to solve a Cauchy problem. Distinguish smoothing from finite propagation. 6 practice tasks.
Lesson details and objectives Derive Duhamel and lift inhomogeneous boundaries.
Construct a boundary lift. Compute the transformed forcing. Apply a time-convolution solution formula. 6 practice tasks.
Lesson details and objectives Derive the whole-line wave formula and its data dependence.
Use characteristic coordinates for the wave equation. Apply d’Alembert’s formula. Identify the velocity integral. 6 practice tasks.
Lesson details and objectives Track flux through fixed and moving boundaries.
Derive local wave energy conservation. Prove uniqueness with homogeneous data. Explain a backward-cone estimate. 6 practice tasks.
Lesson details and objectives Solve fixed-end waves and explain reflections.
Derive sine-mode oscillations. Match both initial functions. Use odd reflection for a half-line boundary. 6 practice tasks.
Lesson details and objectives Interpret the three-dimensional Kirchhoff formula.
Apply spherical means to simple data. Reduce a radial three-dimensional wave. Distinguish domain of dependence from Huygens support. 6 practice tasks.
Lesson details and objectives Derive mean-value and rigidity mechanisms.
Check harmonic polynomials. Derive the mean-value identity. Explain Harnack and Liouville hypotheses. 6 practice tasks.
Lesson details and objectives Prove comparison before trusting a computed profile.
Apply the subharmonic maximum principle. Prove Dirichlet uniqueness. Distinguish homogeneous and forced bounds. 6 practice tasks.
Lesson details and objectives Solve rectangle and disk boundary problems.
Match separated boundary conditions. Explain decay of high angular frequencies. Identify convergence qualifications at corners. 6 practice tasks.
Lesson details and objectives Track Green-function signs and boundary terms.
Construct a one-dimensional Dirichlet Green function. Verify its derivative jump. Interpret fundamental solutions in higher dimensions. 6 practice tasks.
Lesson details and objectives Solve the structural obstacles in Neumann problems.
Derive the compatibility integral. Prove uniqueness modulo constants. Choose a normalization without inventing data. 6 practice tasks.
Lesson details and objectives Fix transform conventions before solving a PDE.
Compute derivative and shift rules. Connect Gaussian transforms to heat. Specify decay conditions for integration by parts. 6 practice tasks.
Lesson details and objectives Use inversion and norm identities with precise classes.
State inversion and Plancherel conventions. Solve heat modes in frequency space. Separate multiplier solutions from pointwise regularity. 6 practice tasks.
Lesson details and objectives Define weak derivatives through test functions.
Compute delta and derivative actions. Verify a jump derivative. Control when convolution is defined. 6 practice tasks.
Lesson details and objectives Solve the free Schrödinger equation by Fourier multipliers.
Verify plane-wave dispersion. Prove L² conservation for smooth decay. Explain oscillatory kernels without false positivity. 6 practice tasks.
Lesson details and objectives Separate weak existence from classical regularity.
Derive the weak Dirichlet equation. Compute a Galerkin coefficient. Apply a clearly stated existence theorem. 6 practice tasks.
Lesson details and objectives Connect stencil weights, Fourier amplification and convergence.
Compute an amplification factor. Derive the explicit stability threshold. Distinguish stability from accuracy. 7 practice tasks.
Lesson details and objectives Test upwinding, conservation and CFL limits.
Compute an upwind update. Compare numerical and physical dependence. Diagnose centered explicit instability. 6 practice tasks.
Lesson details and objectives Diagonalize a small system and preserve Maxwell constraints.
Find characteristic speeds. Derive symmetric-system energy. Identify vacuum Maxwell assumptions and constraints. 6 practice tasks.
Lesson details and objectives Close the course by connecting equations, boundaries and energy.
Compute an Euler–Lagrange PDE. Distinguish natural and imposed boundary conditions. Verify energy conservation and scope limits. 6 practice tasks.
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