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This work gives some results and develops the derivation of memory dependence heat conduction model with dual phase-lags using the modified definition of the fractional derivative of Caputo and higher-order memory-dependent derivative (MDD). The introduced model was derived by improving the heat conduction equation replacing the conventional Fourier's law and generalizing some of the previous models. As an application of the proposed model, we will consider an infinite thermoelastic long annular cylinder whose inner surface is thermally insulated and restricted, while the exterior surface is due to thermal shock and traction free. The numerical results of the physical fields are shown graphically against the position, and the effects of some basic parameters on them are estimated. Theoretical analysis and mathematical results confirm that the selection of an appropriate kernel, time delays and memory response can affect the turbulence in the thermal fields. [ABSTRACT FROM AUTHOR]