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We study time-dependent relativistic jets under the influence of radiation field of the accretion disk. The accretion disk consists of an inner compact corona and an outer sub-Keplerian disk. The thermodynamics of the fluid is governed by a relativistic equation of state (EoS) for multispecies fluid which enables to study the effect of composition on jet-dynamics. Jets originate from the vicinity of the central black hole where the effect of gravity is significant and traverses large distances where only special relativistic treatment is sufficient. So we have modified the flat metric to include the effect of gravity. In this modified relativistic framework we have developed a new total variation diminishing (TVD) routine along with multispecies EoS for the purpose. We show that the acceleration of jets crucially depends on flow composition. All the results presented are transonic in nature, starting from very low injection velocities, the jets can achieve high Lorentz factors. For sub-Eddington luminosities, lepton dominated jets can be accelerated to Lorentz factors > 50. The change in radiation field due to variation in the accretion disk dynamics will be propagated to the jet in a finite amount of time. Hence any change in radiation field due to a change in disk configuration will affect the lower part of the jet before it affects the outer part. This can drive shock transition in the jet flow. Depending upon the disk oscillation frequency, amplitude and jet parameters these shocks can collide with each other and may trigger shock cascades.