Infinite Domains

PDEs · Axiom Academy

Understanding PDEs on unbounded regions and transform methods 1. Why Infinite Domains Need Transform Methods On a finite interval [0, L], boundary conditions lead to discrete eigenvalues λₙ = (nπ/L)². But when L → ∞, the spacing between eigenvalues approaches zero, and the discrete sum becomes a continuous integral. 2. No Boundary Eigenvalues — Continuous Spectrum Without finite boundaries, there are no boundary conditions to impose discrete eigenvalues. Instead of solving an eigenvalue problem with λₙ, we work with a continuous parameter k ranging over all real numbers. 3. Fourier Transform for -∞ < x < ∞ The Fourier transform decomposes a function into continuous frequency components. It's the infinite-domain analog of Fourier series. 4. Semi-Infinite Domains: x > 0 When the domain is semi-infinite (x ≥ 0), we have a boundary at x = 0 but infinity on the other end. The full Fourier transform isn't appropriate here — we need sine and cosine transforms. 5. Sine and Cosine Transforms for Semi-Infinite These transforms extend odd and even functions to the full real line, then apply the Fourier transform. 6. Decay Conditions at Infinity On infinite domains, we replace boundary conditions with decay conditions: the solution and its derivatives must vanish as |x| → ∞. This ensures the transform integrals converge and physical solutions are bounded.

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