  In this work we develop an analytic approach to study pulsar spindown. We use
the monopolar spindown model by Alvarez and Carrami\~{n}ana (2004), which
assumes an inverse linear law of magnetic field decay of the pulsar, to extract
an all-order formula for the spindown parameters which are expressed in terms
of modified Bessel functions. We further extend the analytic model to
incorporate the quadrupole term that accounts for the emission of gravitational
radiation, and obtain expressions for the period $P$ and frequency $f$ in terms
of transcendental equations. We derive the period of the pulsar evolution as an
approximate first order solution in the small parameter present in the full
solution. We find that the first three spindown parameters of the Crab, PSR
B1509-58, PSR B0540-69 and Vela pulsars are within their known bounds providing
a consistency check on our approach. After the four detections of gravitational
waves from binary black hole coalescence and a binary neutron merger 170814,
which was a novel joint gravitational and electromagnetic detection, a
detection of gravitational waves from pulsars will be the next landmark in the
field of multi-messenger gravitational wave astronomy.
