Integral Transforms and Spectral Methods

Why Transition to Frequency Space?

Differential equations in the "x-t" space contain derivatives—which is not trivial. In the "k-ω" space (space of frequencies and wavenumbers), the derivative ∂/∂t is replaced with multiplication by −iω, and ∂/∂x by ik. A differential equation turns into an algebraic one! This is the magic of integral transforms.

Additional bonus: many physical phenomena are more transparent in frequency space. Oscillation period, bandwidth, dispersion—all these are properties of the frequency spectrum.

Fourier Transform

Direct: F̂(k) = ∫_{−∞}^{+∞} f(x) e^{−ikx} dx

Inverse: f(x) = (1/2π) ∫_{−∞}^{+∞} F̂(k) e^{ikx} dk

Key properties: derivative → multiplication (f'(x) ↔ ikF̂(k)); convolution f*g → product F̂·Ĝ; Parseval’s theorem: ∫|f(x)|²dx = (1/2π)∫|F̂(k)|²dk (norm conservation).

Numerical example: wave equation. ∂²u/∂t² = c² ∂²u/∂x². Fourier in x: ∂²û(k,t)/∂t² = −c²k²û. This is an ordinary ODE in t! Solution: û(k,t) = A(k)e^{ickt} + B(k)e^{−ickt}. Inverse Fourier gives the d'Alembert formula: u(x,t) = f(x+ct) + g(x−ct)—waves propagating in both directions.

Heat conduction equation. ∂u/∂t = κ ∂²u/∂x². Fourier in x: ∂û/∂t = −κk²û. Solution: û(k,t) = û(k,0)e^{−κk²t}. For t > 0: higher frequencies decay faster → profile diffuses. Inverse Fourier: u(x,t) = (1/√(4πκt)) ∫ u(x',0) e^{−(x−x')²/(4κt)} dx'—the Green's function for heat conduction in action.

Laplace Transform

Definition: F(s) = L{f(t)} = ∫₀^{+∞} f(t) e^{−st} dt, s = σ + iω ∈ ℂ

Table of key pairs: L{1} = 1/s, L{e^{at}} = 1/(s−a), L{cos(ωt)} = s/(s²+ω²), L{f'(t)} = sF(s) − f(0).

Numerical example: Cauchy problem for ODE. y'' + 5y' + 6y = 0, y(0) = 1, y'(0) = 0.

Step 1. Apply L: (s²Y − s − 0) + 5(sY − 1) + 6Y = 0

Step 2. Y(s²+ 5s + 6) = s + 5 → Y = (s+5)/((s+2)(s+3))

Step 3. Decomposition: Y = A/(s+2) + B/(s+3). A = (2+5)/(2−3+5)|{s=−2} = 3/(−1+3) → A = 3, B = −2. (Calculating: A = lim{s→−2}(s+2)Y = (−2+5)/(−2+3) = 3, B = lim_{s→−3}(s+3)Y = (−3+5)/(−3+2) = −2.)

Step 4. y(t) = 3e^{−2t} − 2e^{−3t}. Check: y(0) = 3−2 = 1 ✓, y'(0) = −6+6 = 0 ✓.

Spectral Methods for Numerical Solution of DEs

Idea: expand the unknown function u(x,t) in a basis:

u(x,t) ≈ Σₙ aₙ(t) φₙ(x)

By substituting into the DE we obtain a system of ODEs for the coefficients aₙ(t). Bases: {e^{ikx/L}}—Fourier (for periodic problems); Chebyshev polynomials {Tₙ(x)}—for non-periodic on an interval; Legendre polynomials {Pₙ(x)}—for physical problems with spherical symmetry.

Advantage of spectral methods: For smooth functions, convergence is exponential: error ~ e^{−αN} for N terms in the series. Finite-difference schemes provide only algebraic convergence ~ N^{−p}. Therefore, spectral methods are used in atmospheric modeling, computational hydrodynamics (codes ORCL, DEDALE).

Fast Fourier Transform (FFT): the Cooley–Tukey algorithm computes an N-point DFT in O(N log N) operations instead of O(N²). For N = 10⁶: speed up by a factor of 50. FFT is the cornerstone of digital signal processing (MP3, JPEG, 4G/5G communications).

Fourier Series Expansion: Physical Example

Standing wave in a closed resonator: u(x,t) on [0,L], u(0) = u(L) = 0. Expansion: u(x,t) = Σₙ aₙ(t) sin(nπx/L). Each mode sin(nπx/L) is a "normal mode." Its frequency: ωₙ = nπc/L. For n=1: fundamental mode. For n=2: first overtone with λ/2 = L.

Application: optical resonators in lasers; acoustics of musical instruments (flute, organ); microwave resonators in accelerators.

Real Applications

Wi-Fi (OFDM): the signal is divided into ~50 subcarrier frequencies (modes). Each subcarrier independently encodes data—via IFFT/FFT, transmitter/receiver switch between time and frequency domains. Spectral efficiency of OFDM is a direct result of the orthogonality of the trigonometric basis.

Seismology: Fourier analysis of earthquake signals allows separation of P- and S-waves, determination of the frequency content, and estimation of source characteristics.

Integral Transforms in Digital Signal Processing

The Fourier transform and its generalizations are fundamental to digital technologies. The Fast Fourier Transform (FFT, Cooley–Tukey algorithm, 1965) reduced the computational complexity of the DFT from O(N²) to O(N log N) and revolutionized digital signal processing. FFT is used in: audio codecs (MP3, AAC—computing modified DCT for frequency decomposition), OFDM modulation (LTE, Wi-Fi, 5G—each subcarrier channel is a frequency component from FFT), radar (target range—via inverse FFT of the convolution of returned and reference signals), medical tomography (CT image reconstruction—via 2D FFT and the central slice theorem). The Laplace transform is a tool for analysis of linear dynamic systems in control: transfer functions, Nyquist stability criterion, and PID controller synthesis are formulated in the s-domain. BIBO-stability of a filter is equivalent to all poles of H(s) lying in the left half-plane—a direct consequence of complex analysis theory. In computational chemistry, fast algorithms for interaction potentials (PME—Particle Mesh Ewald) use 3D FFT to accelerate molecular dynamics.

Spectral methods are used in computational hydrodynamics (pseudo-spectral schemes for the Navier–Stokes equations), where accuracy increases exponentially with the number of modes for smooth solutions. Fast sine/cosine transform algorithms (DCT/DST) accelerate Fourier decomposition on bounded domains and are used in LAPACK and FFTW packages, which underpin the entire scientific computing ecosystem—from climate models to turbulence computations in aerospace engineering.

Assignment: (a) Solve ∂u/∂t = κ ∂²u/∂x² on [0,L], u(0)=u(L)=0, u(x,0)=sin(πx/L). Find u(x,t) via Fourier expansion. How does the "cooling time" depend on L and κ? (b) Using the Laplace transform: y'' − y = e^{−t}, y(0)=y'(0)=0. (c) Estimate for which N = number of modes the Fourier series for the function f(x) = x(L−x) approximates it with accuracy to 1% on [0,L].