Efficient numerical methods for the solution
of Initial and Boundary Value Problems is of continued
importance in astrodynamics are of continued import-
ance in astrodynamics. This paper proposes a novel
Galerkin method for the solution of IVPs and BVPs. The
method works by projecting the solution onto a Cheby-
shev basis, and by finding the projection coefficients that
zero the ODE residual through Newton-Rhapson iter-
ations. The method is tested against classical Runge-
Kutta methods and Modified Chebyshev-Picard Itera-
tions (MCPI), which is a similar pseudo-spectral method
for the solution of IVPs and BVPs. The Galerkin method
is up to 300 times faster than state-of-the-art Runge-Kutta
solvers for IVPs, and up to 80 times faster than MCPI in
the solution of BVPs.