A Geiger counter records a smoothed count rate produced by an unknown emission rate…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Conditions of Validity, Ch Integral equations, kernels and the Fredholm alternative for the last time

Problem

A Geiger counter records a smoothed count rate f(t)f(t) produced by an unknown emission rate ϕ(τ)\phi(\tau) through the relation ∫0t(t−τ)−1/3ϕ(τ) dτ=f(t),0≤t≤T,\int_0^t (t-\tau)^{-1/3}\phi(\tau)\,\mathrm{d}\tau = f(t), \qquad 0 \le t \le T, where the exponent −1/3-1/3 describes the detector's response and ϕ\phi is unknown. ((a)) Classify this equation: Fredholm or Volterra, first or second kind. ((b)) Predict, with a reason, whether recovering ϕ\phi from a noisy measured ff is a well-posed procedure.

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