Keep watch on Duke's health by using a Magic Lens until Dukes HP gets to about 10,000-20,000 (If you're around level 60-70) and set everyone on Defend, then use your Mystic Arte by doing the following:
for level 3 Overlimit (requires Limit Trio from Synthesis).
2) While in Overlimit, use an arcane, altered, or burst arte. While you are performing the Arte, hold down the attack button. (This will be whatever button you have mapped for normal attacks)
3) Once the Arte finishes and you are holding down the attack button, your character should perform the Mystic Arte. (Requires Special Skill, which is gained through characters various weapons).
The easiest character to do this on is Yuri because his Mystic Arte is pure damage and you should have learned this from his Dein Nomos
or Second Star
weapons. Estelle, for example, has a healing part to her Mystic Arte so you may
not do enough damage to kill Duke if you misjudge the damage.
Just use the Mystic Arte on whatever character you're most comfortable with judging their damage numbers on.
You will know if you got the mission done because the boss will die immediately after using your Mystic Arte.