How does a Muslim captive perform his prayers while imprisoned, shackled, and tortured, and not given time to perform prayer?

Prayer is obligatory upon the captive at its appointed time, to the best of their ability. What they are unable to do is waived for them, due to the general Islamic proofs regarding the waiving of obligation due to inability. So, they should pray in their condition. They should perform tayammum (dry ablution) if they are unable to perform wudu (ablution with water). If they are unable to perform tayammum, they should pray without purification. If they are unable to face the qiblah, they should pray facing another direction. They should bow at the rukoo' (bowing) and sujood (prostration) positions, and cover their 'awrah (private parts) to the best of their ability. The majority of scholars hold that it is obligatory for one who lacks both means of purification (water and soil for tayammum) to pray, out of respect for the prayer's time. The strongest opinion is that they should pray in their condition and not repeat the prayer.