Abstract: A method called complete Orthogonal Function Series Expansion (OFSE) in Hilbert space is proposed to solve the non-dissipative torsional Alfvén wave in coronal loops. Every base function corresponds to an intrinsic angular frequency w_n of every magnetic field line in coronal loops. Torsional Alfvén wave resonance of a magnetic field line in coronal loops comes out when the driven angular frequency equals to its intrinsic angular frequency. With the method, we present a new form of Torsional Alfvén wave evolution solution with two-footpoint driven boundary condition. There exists a resonant term in the solution, from which it could be found that: near the resonant place with an angular frequency w,a discontinuity profile appears at times t equal to the multiples of n/w and a 1/x discontinuity profile appears at times t equal to the odd multiples of n/(2w). It is also found that the wave amplitude at resonant place increases linearly with time and the slope is proportional to Alfvén wave speed, inverse proportional to loop length and independent of driven frequency.