Warp equations were used to adjust the warp fields and subspace bubble patterns that enveloped the starship, distorting the local spacetime continuum and moving the starship at velocities that exceeded the speed of light. Bessel functions were a component that calculated energy required for a warp jump.
In 2364, Starfleet propulsion expert Kosinski claimed to have discovered a revolutionary new method of warp equations, based in the simply test different ways of entering warp speed and different intermix formulas, associated with the nature of elementals like space and time.
Using warp equation simulations and a three-dimensional geometric display which built and grew, Kosinski obtained a wonderfully complex and incredibly beautiful pattern of a warp field and a theoretical representation of a large increase in the warp power of the starship.
However, the three percent efficiency observed in two other Starfleet starships, the USS Ajax and USS Fearless, could still have been caused by other things such as simply working more carefully and better station coordination. It later transpired that it was his mysterious assistant, The Traveler who was effecting the upgrades and not him. (TNG: "Where No One Has Gone Before")
Some of Kosinski's theories were still valid, and Ensign Wesley Crusher tested his warp equations aboard the Enterprise while it was docked at Starbase 133 in 2367. When Crusher inadvertently created a warp bubble which trapped his mother, Doctor Beverly Crusher, in an alternate universe, The Traveler appeared and he urged Wesley to concentrate his own thoughts in order to facilitate the rescue. (TNG: "Remember Me")