Integral transforms & integral equations
Laplace-type transforms and the Fredholm/Volterra theory of integral equations.
Founded: 1880 — Heaviside's operational calculus
The claim
William Burnside (Royal Society referee) & Peter Guthrie Tait, 1893: Burnside rejected a Heaviside paper for “irredeemable inadequacies in proof”; Tait attacked his methods in Nature.
Source: MacTutor History of Mathematics.
A rigor critique, not a usefulness one — Heaviside's methods were already solving real telegraphy problems while under attack; the rigorous justification (Bromwich) came only in the 1910s. A reversed pattern: applied first, proved later.
What happened
Feedback amplifier design (1927) — Bode/Nyquist/Black at Bell Labs codify Laplace-transform methods for circuit and control engineering.
Actuarial calculations (1904) — Fredholm himself applied his own integral equations at Skandia insurance.
0 years — Heaviside's methods were used by engineers before they were proved rigorous — applied essentially immediately
Verdict: Never dismissed, applied anyway.
Confidence: medium-high.