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Slide 01
Differential Equations
- The Language of Change and Motion
- Differential equations relate functions to their derivatives -- encoding how systems evolve through time, space, or any continuous parameter. From Newton's second law to the Black-Scholes equation, they are the mathematical backbone of physics, engineering, biology, and finance.
- This deck covers ordinary and partial differential equations: theory, solution methods, and applications across the sciences.
Slide 02
What Is a Differential Equation?
- An equation involving an unknown function and one or more of its derivatives. The goal: find the function(s) satisfying the equation.
- Ordinary (ODE)
- One independent variable. Example: dy/dx = ky models exponential growth. Newton's law: m*x'' = F(x,x',t).
- dy/dt = f(t, y)
- Partial (PDE)
- Multiple independent variables and partial derivatives. Example: heat equation, wave equation, Navier-Stokes.
- du/dt = k * d^2u/dx^2
Slide 03
Classification of ODEs
- PropertyDescriptionExample
- OrderHighest derivative presenty'' + y = 0 is 2nd order
- Lineary and derivatives appear linearlyy'' + p(t)y' + q(t)y = g(t)
- NonlinearProducts/powers of y or derivativesy' = y^2 (Riccati)
- Autonomoust does not appear explicitlyy' = y(1-y) (logistic)
- HomogeneousRight-hand side = 0y'' + y = 0
- Constant coefficientsCoefficients don't depend on ty'' + 3y' + 2y = 0
Slide 04
First-Order ODEs: Separable & Linear
- Separable Equations
- Form: dy/dx = f(x)g(y). Separate variables and integrate both sides.
- dy/g(y) = f(x) dx
- Example: dy/dx = xy gives y = Ce^(x^2/2). Works whenever the equation factors into a product of functions of x and y alone.
- First-Order Linear
- Form: y' + P(x)y = Q(x). Solved by integrating factor mu(x) = exp(integral P dx).
- y = (1/mu) * integral[mu * Q dx]
- Example: y' + 2y = e^(-x) gives y = e^(-x) + Ce^(-2x). Guaranteed existence and uniqueness when P, Q are continuous.
Slide 05
Existence and Uniqueness
- When does a solution exist? When is it unique? These foundational questions were settled in the 19th century.
- Picard-Lindelof Theorem
- If f(t,y) is continuous and Lipschitz in y, then y' = f(t,y) with y(t_0) = y_0 has a unique local solution. The Lipschitz condition prevents "branching."
- Peano's Theorem
- If f is merely continuous (no Lipschitz), existence is guaranteed but uniqueness may fail. Example: y' = y^(2/3), y(0)=0 has infinitely many solutions.
- Blow-up
- Solutions may not exist globally. y' = y^2, y(0)=1 gives y = 1/(1-t), which explodes at t=1. Finite-time singularities are physically meaningful.
Slide 06
Second-Order Linear ODEs
- The most important class: y'' + p(t)y' + q(t)y = g(t). Ubiquitous in mechanics, circuits, and wave phenomena.
- Homogeneous (g=0)
- Solution space is 2-dimensional (superposition principle)
- General solution: y = c1*y1 + c2*y2 (linearly independent)
- Wronskian W(y1,y2) != 0 certifies independence
- Constant coefficients: characteristic equation r^2 + pr + q = 0
- Characteristic Roots
- Distinct real roots r1, r2: y = c1*e^(r1*t) + c2*e^(r2*t)
- Repeated root r: y = (c1 + c2*t)*e^(r*t)
- Complex roots a +/- bi: y = e^(at)*(c1*cos(bt) + c2*sin(bt))
- Oscillatory behavior from complex roots (springs, circuits)
Slide 07
The Harmonic Oscillator
- The single most important differential equation in physics: mx'' + bx' + kx = F(t). Models springs, pendulums, circuits, molecular vibrations.
- Undamped (b=0)
- x'' + omega^2 * x = 0. Solution: x = A*cos(omega*t + phi). Pure oscillation at natural frequency omega = sqrt(k/m). Energy conserved forever.
- Underdamped (b^2 Oscillates with exponentially decaying amplitude. Envelope: e^(-bt/2m). Frequency slightly less than natural. Most physical oscillators.
- Overdamped (b^2 > 4mk)
- No oscillation -- exponential decay with two time constants. Door closers, shock absorbers. Returns to equilibrium without overshoot.
- Critically Damped (b^2 = 4mk)
- Fastest return to equilibrium without oscillation. (c1 + c2*t)*e^(-bt/2m). Optimal for instruments needing quick settling.
Slide 08
Systems of ODEs
- Multiple coupled equations arise naturally. Any nth-order ODE can be rewritten as a first-order system.
- dx/dt = Ax + b(t), where x is a vector and A is a matrix
- For constant A, the solution involves the matrix exponential: x(t) = e^(At) * x(0). Eigenvalues of A determine behavior:
- All eigenvalues with negative real part: stable node/spiral (decays to 0)
- All positive real part: unstable (solutions diverge)
- Pure imaginary: center (neutrally stable oscillation)
- Mixed signs: saddle point (unstable)
- Phase portraits visualize trajectories in state space
Slide 09
Phase Plane Analysis
- For 2D autonomous systems x' = f(x,y), y' = g(x,y), the phase plane reveals qualitative behavior without solving explicitly.
- Equilibria
- Points where f = g = 0. Classified by eigenvalues of Jacobian: nodes, spirals, saddles, centers. Stability determined by sign of real parts.
- Nullclines
- Curves where f=0 (horizontal flow) or g=0 (vertical flow). Intersections are equilibria. Flow direction between nullclines reveals global dynamics.
- Limit Cycles
- Isolated periodic orbits. Poincare-Bendixson theorem: bounded planar trajectories must approach equilibrium or limit cycle. Van der Pol oscillator is classic example.
Slide 10
Laplace Transform
- Converts differential equations into algebraic equations. Especially powerful for initial-value problems with discontinuous forcing.
- L{f(t)} = F(s) = integral_0^inf e^(-st) f(t) dt
- Key Properties
- L{f'} = sF(s) - f(0): derivatives become polynomials in s
- L{e^(at)f} = F(s-a): frequency shifting
- L{f*g} = F(s)*G(s): convolution becomes multiplication
- L{delta(t-a)} = e^(-as): impulse response
- Solution Method
- Transform the ODE (algebraic in s)
- Solve for Y(s) using algebra
- Partial fractions decomposition
- Inverse transform to get y(t)
Slide 11
Series Solutions and Special Functions
- When coefficients are not constant, power series methods generate solutions term by term, often yielding named special functions.
- Bessel's equation: x^2*y'' + x*y' + (x^2 - n^2)*y = 0. Solutions: J_n(x), Y_n(x). Arise in cylindrical geometries (drum vibrations, electromagnetic waveguides).
- Legendre's equation: (1-x^2)*y'' - 2x*y' + l(l+1)*y = 0. Legendre polynomials P_l(x). Spherical harmonics in quantum mechanics.
- Hermite equation: y'' - 2x*y' + 2n*y = 0. Hermite polynomials H_n. Quantum harmonic oscillator wavefunctions.
- Airy equation: y'' - x*y = 0. Airy functions Ai, Bi. Quantum tunneling, optics near caustics.
- Hypergeometric equation: unifies many special functions into a single framework (Gauss, 1812).
Slide 12
Sturm-Liouville Theory
- The spectral theory of second-order linear operators. Provides the mathematical foundation for separation of variables in PDEs.
- d/dx[p(x)*y'] + [q(x) + lambda*w(x)]*y = 0 with boundary conditions
- Eigenvalues lambda_n form an infinite increasing sequence approaching infinity
- Eigenfunctions y_n form a complete orthogonal set (generalized Fourier series)
- Any "nice" function can be expanded: f(x) = sum c_n * y_n(x)
- Fourier series is the special case p=1, q=0, w=1 on [0,L]
- Foundation of quantum mechanics: Schrodinger equation is a Sturm-Liouville problem
Slide 13
Partial Differential Equations
- PDEs involve multiple independent variables. The "big three" of classical mathematical physics:
- Heat Equation (Parabolic)
- u_t = k * u_xx. Diffusion, smoothing. Solution: initial temperature distribution smooths out exponentially. Solved by Fourier (1807).
- Wave Equation (Hyperbolic)
- u_tt = c^2 * u_xx. Propagation at finite speed c. D'Alembert solution: u = f(x-ct) + g(x+ct). Vibrating strings, sound, electromagnetics.
- Laplace Equation (Elliptic)
- u_xx + u_yy = 0. Steady-state. Harmonic functions satisfy maximum principle. Electrostatics, fluid flow, gravitational potential.
Slide 14
Separation of Variables
- The most powerful elementary method for linear PDEs. Assume u(x,t) = X(x)*T(t) and separate into ODEs.
- Substitute product form into PDE
- Separate: each side depends on one variable only = constant (lambda)
- Solve resulting ODEs (often Sturm-Liouville problems)
- Apply boundary conditions to determine eigenvalues
- Superpose: u = sum c_n * X_n(x) * T_n(t)
- Use initial conditions to determine coefficients c_n (Fourier coefficients)
- Works for heat, wave, and Laplace equations on regular geometries (rectangles, circles, spheres).
Slide 15
Fourier Series and Transforms
- Joseph Fourier's 1807 insight: any periodic function can be decomposed into sines and cosines. This revolutionized both mathematics and physics.
- Fourier Series
- f(x) = a_0/2 + sum[a_n*cos(nx) + b_n*sin(nx)]
- Coefficients: projections onto basis functions
- Convergence: pointwise for piecewise smooth f
- Gibbs phenomenon at discontinuities (~9% overshoot)
- Fourier Transform
- F(omega) = integral f(t)*e^(-i*omega*t) dt
- Extends to non-periodic functions on R
- Convolution theorem: F(f*g) = F(f)*F(g)
- Parseval's: energy in time = energy in frequency
- Uncertainty principle: narrow in time = wide in frequency
Slide 16
The Navier-Stokes Equations
- The governing equations of fluid mechanics. One of the Clay Millennium Prize Problems ($1M) -- existence and smoothness of solutions in 3D remains unproven.
- rho*(du/dt + u*grad(u)) = -grad(p) + mu*laplacian(u) + f
- Nonlinear PDE system coupling velocity u, pressure p, and density rho
- Conservation of momentum + incompressibility constraint (div u = 0)
- Reynolds number Re = rho*U*L/mu determines laminar vs. turbulent flow
- Turbulence: still no complete mathematical theory. Kolmogorov's 1941 scaling law is empirical.
- Computational fluid dynamics (CFD) solves approximately on grids. Weather prediction, aircraft design, blood flow.
Slide 17
Nonlinear Dynamics and Chaos
- Small changes in initial conditions can lead to wildly different outcomes. Deterministic systems can behave unpredictably.
- Lorenz System (1963)
- x'=sigma(y-x), y'=x(rho-z)-y, z'=xy-beta*z. Discovered sensitive dependence while modeling atmospheric convection. The "butterfly effect."
- Strange Attractors
- Fractal structures in phase space that attract trajectories. Lorenz attractor has Hausdorff dimension ~2.06. Bounded but never repeating.
- Lyapunov Exponents
- Quantify rate of divergence of nearby trajectories. Positive exponent = chaos. Sum of all exponents negative = dissipative system (attractor exists).
Slide 18
Bifurcation Theory
- How do solutions change qualitatively as a parameter varies? Bifurcation points mark transitions between different dynamical regimes.
- Saddle-node bifurcation: two equilibria collide and annihilate. x' = r + x^2. Equilibria exist only for r Transcritical bifurcation: equilibria exchange stability. x' = rx - x^2. Origin stable for r 0.
- Pitchfork bifurcation: symmetry-breaking. x' = rx - x^3. One equilibrium splits into three at r = 0.
- Hopf bifurcation: equilibrium loses stability, periodic orbit born. Models onset of oscillation in circuits, chemical reactions, ecology.
- Period-doubling cascade: route to chaos. Successive doublings of period as parameter changes. Universal Feigenbaum constants (delta = 4.669...).
Slide 19
Numerical Methods for ODEs
- Most differential equations have no closed-form solution. Numerical methods approximate solutions on discrete time steps.
- MethodOrderProperties
- Euler (Forward)1Simplest. Error O(h). Often unstable for stiff problems.
- Midpoint (RK2)2One midpoint evaluation. Error O(h^2). Better accuracy.
- Classical RK44Four evaluations per step. Error O(h^4). Workhorse method.
- Dormand-Prince (RK45)4-5Adaptive step size. Embedded pair. MATLAB ode45 default.
- Adams-BashforthkMultistep. Reuses past evaluations. Efficient per step.
- BDF (Backward Diff)kImplicit. Stable for stiff systems. MATLAB ode15s.
Slide 20
Stiff Systems
- A system is "stiff" when solution components evolve on vastly different timescales. Explicit methods require impractically small steps.
- The Problem
- Fast transients decay quickly but force tiny step sizes
- Explicit Euler on y'=-1000y with h>0.002 is unstable
- Chemical kinetics: reactions span 10^-12 to 10^3 seconds
- Circuit simulation: fast switching + slow RC decay
- The Solution
- Implicit methods: solve nonlinear systems each step
- BDF methods (up to order 6): A-stable or stiffly stable
- Implicit Runge-Kutta: RADAU, SDIRK methods
- Exponential integrators: exact for linear part
- Cost per step higher, but enormously fewer steps needed
Slide 21
Numerical Methods for PDEs
- Finite Differences (FDM)
- Replace derivatives with discrete approximations on a grid. u_xx ~ (u_{i+1} - 2u_i + u_{i-1})/h^2. Simple, flexible, widely used for regular geometries.
- Finite Elements (FEM)
- Decompose domain into triangles/tetrahedra. Approximate solution as sum of basis functions. Handles complex geometry. Dominant in structural mechanics.
- Spectral Methods
- Expand solution in global basis (Fourier, Chebyshev). Exponential convergence for smooth solutions. Used in weather/climate models and turbulence DNS.
- Finite Volumes (FVM)
- Conserve fluxes across cell boundaries. Natural for conservation laws. Dominant in CFD (fluid dynamics). Handles shocks via Riemann solvers.
Slide 22
Maxwell's Equations
- James Clerk Maxwell's 1865 PDEs unify electricity, magnetism, and optics. They predicted electromagnetic waves traveling at the speed of light.
- curl E = -dB/dt, curl B = mu_0*J + mu_0*epsilon_0*dE/dt, div E = rho/epsilon_0, div B = 0
- Four coupled PDEs relating electric field E and magnetic field B
- In vacuum: wave equation for E and B with speed c = 1/sqrt(mu_0*epsilon_0)
- Predicted radio waves (verified by Hertz, 1887)
- Special relativity emerges from their Lorentz invariance
- Numerical solution: FDTD (Yee's algorithm, 1966) used in antenna and chip design
Slide 23
The Schrodinger Equation
- The fundamental equation of quantum mechanics (1926). A PDE governing the wave function psi(x,t) of a quantum system.
- i*hbar * dpsi/dt = -(hbar^2/2m) * d^2psi/dx^2 + V(x)*psi
- Linear PDE: superposition principle holds (quantum superposition)
- Time-independent form: eigenvalue problem for energy levels E_n
- Hydrogen atom: exact solution yields spectral lines (Bohr's model explained)
- Harmonic oscillator: equally spaced energy levels (phonons, photons)
- Tunneling: classically forbidden regions have exponentially decaying psi
- Many-body version: N particles in 3D requires solving in 3N dimensions
Slide 24
Einstein's Field Equations
- General relativity (1915): gravity is the curvature of spacetime, described by a system of 10 coupled nonlinear PDEs.
- G_mu_nu + Lambda*g_mu_nu = (8*pi*G/c^4) * T_mu_nu
- G_mu_nu (Einstein tensor) encodes spacetime curvature
- T_mu_nu (stress-energy tensor) encodes matter/energy distribution
- Exact solutions: Schwarzschild (black holes), Friedmann (expanding universe), Kerr (rotating black holes)
- Numerical relativity: simulate binary black hole mergers (gravitational waves detected 2015)
- The cosmological constant Lambda: dark energy driving accelerated expansion
Slide 25
Biological Applications
- Population Dynamics
- Lotka-Volterra: predator-prey cycles. Logistic growth: dN/dt = rN(1-N/K). SIR model: epidemic spread as coupled ODEs. Competitive exclusion principle.
- Reaction-Diffusion
- Turing patterns (1952): morphogenesis from instability in coupled diffusing chemicals. Explains animal coat patterns, coral structures, chemical oscillations.
- Neuroscience
- Hodgkin-Huxley model (1952, Nobel Prize): 4 coupled ODEs for action potential. FitzHugh-Nagumo simplification. Neural network dynamics.
- Cardiac Modeling
- Bidomain equations: coupled PDEs for electrical propagation in heart tissue. Used to understand arrhythmias and design defibrillators.
Slide 26
Financial Mathematics
- Stochastic differential equations (SDEs) model financial markets, incorporating randomness through Brownian motion.
- Black-Scholes Equation (1973)
- dV/dt + (1/2)*sigma^2*S^2*d^2V/dS^2 + r*S*dV/dS - r*V = 0
- PDE for option pricing. Nobel Prize 1997. Assumes log-normal stock prices, constant volatility, no transaction costs.
- Stochastic Calculus
- Geometric Brownian motion: dS = mu*S*dt + sigma*S*dW
- Ito's lemma: chain rule for stochastic processes
- Risk-neutral pricing: change measure, discount at risk-free rate
- Monte Carlo simulation: sample paths of SDEs
Slide 27
Variational Methods
- Many physical laws arise from minimizing a functional (action, energy). The Euler-Lagrange equation bridges variational principles and differential equations.
- Euler-Lagrange: d/dt(dL/dq') - dL/dq = 0 for Lagrangian L = T - V
- Hamilton's principle: true trajectory minimizes the action S = integral L dt
- Brachistochrone problem (1696): curve of fastest descent is a cycloid
- Geodesics: shortest paths on curved surfaces satisfy E-L equations
- Noether's theorem (1918): every continuous symmetry gives a conservation law (time->energy, space->momentum, rotation->angular momentum)
- Modern physics: Standard Model Lagrangian encodes all known particle interactions
Slide 28
Historical Timeline
- 1687
- Newton's Principia: F = ma is a second-order ODE. Calculus invented (simultaneously by Leibniz) to solve it.
- 1807
- Fourier presents heat equation solution via trigonometric series. Rejected initially, later transforms mathematics.
- 1822
- Cauchy proves existence theorems. Rigor enters the field.
- 1890
- Poincare's qualitative theory: phase portraits, index theory, chaos foreshadowed.
- 1926
- Schrodinger equation. Quantum mechanics formulated as a PDE eigenvalue problem.
- 1963
- Lorenz discovers deterministic chaos in a 3-ODE weather model.
- 2000
- Navier-Stokes existence and smoothness named Millennium Prize Problem.
Slide 29
Modern Frontiers
- Machine Learning + DEs
- Neural ODEs (2018): continuous-depth networks. Physics-informed neural networks (PINNs) solve PDEs without grids. Learned simulators for turbulence.
- Geometric Integration
- Structure-preserving numerical methods. Symplectic integrators for Hamiltonian systems. Conserve energy over astronomical timescales.
- Infinite-Dimensional Dynamics
- PDEs as dynamical systems in function spaces. Attractor theory, inertial manifolds. Understanding turbulence through infinite-dimensional chaos.
- Quantum Computing for DEs
- HHL algorithm promises exponential speedup for linear systems. Variational quantum eigensolvers for molecular Schrodinger equation. Still early stage.
Slide 30
Software Ecosystem
- ToolStrengths
- MATLAB / Simulinkode45/ode15s, block diagrams, industry standard for control systems
- Python (SciPy)solve_ivp, integrate.odeint, open source, ML integration
- Julia (DifferentialEquations.jl)Fastest general-purpose solver suite, 300+ algorithms, stiff/non-stiff
- Mathematica / MapleSymbolic solutions, DSolve/dsolve, visualization
- COMSOL / ANSYSMultiphysics FEM, industrial PDE simulation
- FEniCS / deal.IIOpen-source FEM frameworks for research
- OpenFOAMCFD (Navier-Stokes), open source, industrial adoption
Slide 31
Key Takeaways
- Language of Nature
- Differential equations express the fundamental laws of physics, biology, and economics. They translate "rates of change" into precise predictions.
- Theory Meets Computation
- Existence theorems guarantee solutions exist; numerical methods let us compute them. The interplay drives modern applied mathematics.
- Chaos Is Deterministic
- Simple differential equations can produce unpredictable behavior. Chaos theory revealed that complexity needs not arise from complicated rules.
- Open Frontiers
- Millennium Prize Problems (Navier-Stokes), turbulence, neural DEs, and quantum simulation ensure the field remains vibrant and central to science.
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