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Well, it depends on what you want you are looking at in terms of "absorption" and "heating." If you mean at the macroscopic level (like in the "real world"), then the wave theory is what will explain the transmission of energy most conveniently. For example, Maxwell's equations, the 4 classic equations of electrodynamics that best characterize wave theory, are still used in figuring out how to transmit radio waves like WiFi or cellphone signals.
Quantum theory (new name for corpuscular theory with a little bit of wave theory mixed in) is best used in terms of individual atoms, molecules, photons, or anything else at a ridiculously small scale. Absorption and raising energy levels (heating, kind of) in quantum theory involves only being able to absorb photons (corpuscle) of light at discrete energies. Wave theory on its own cannot explain this experimentally proven fact, so we need quantum (corpuscular) theory to figure out what's going on.
For example, a hydrogen atom in the ground state can only accept a photon that has an energy of roughly 10.2 eV (to bring it up to the next energy level) or other energies that bring it up the exact amount to reach a higher energy level. Hydrogen in its ground state cannot absorb a photon with less energy than 10.2 eV. This example is kind of like how you can't exchange parts of a penny. You can get $0.01, $0.02, or any other amount of pennies, but you cannot exchange half of a cent normally, just like how a Hydrogen atom can't do anything with a 5 eV photon.
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