This chapter examines a one-dimensional solid issue to assess the quasi-static thermoelastic response. The one-dimensional heat conduction equation contains the space fractional ordered and using memory-dependent derivative on time. The heat conduction solution is obtained using the integral transform technique with the help of memory-dependent derivatives and finite Reisz fractional derivatives. Visual demonstrations of numerical investigations for non-dimensional temperature and stress flow for time and spatial direction are shown for different time delay and Riesz fractional derivative parameters. Successful discussions of a few pertinent cases are presented. By combining diverse parameters and functions with memory-dependent responses, this study’s thermal fluctuations may correctly describe material behavior. In engineering, these insights may be used to improve structures or build a variety of structural designs in a realistic manner.