A novel model of the generalized magneto-thermoelasticity with two-temperature theory is investigated. The memory-depended derivative (MDD) during the excitation processes by pulsed laser is established for a time-dependent material. The overlapping between an isotropic homogeneous thermoelastic half-space medium and the non-Gaussian laser pulse in one-dimensional space with time delay is obtained. The semi-infinite bounding surface of the elastic medium is taken as traction free and subject to a thermal shock problem in a time-dependent case. Laplace transformation technique is used to obtain the initial solutions of the main physical field which defined in the governing equations. The temporal complete solutions in Laplace time domain obtained by using the inversion method of the Laplace transform, to obtain the general solution (exact solution) for some physical quantities. Numerical results are performed with discussed for suitable elastic medium and illustrated graphically, taken into consideration the time-delay parameters. © 2020 Informa UK Limited, trading as Taylor & Francis Group.