This work presents an analytical approach to study the photothermal response of a functionally graded semiconducting thermoelastic half-space. The generalized thermoelastic heat conduction theory without energy dissipation is employed to extract the governing equations in which the properties of a half-space material are supposed to change in the longitudinal direction. In the context of photothermal transitional model, the interaction between heat-elastic plasma waves is described. The governing equations for the physical field variables are determined by Laplace transform procedure in the physical domain. The distribution of each field variable as well as the spread of thermo-elastic-plasma waves are plotted and discussed. Some illustrative examples are presented to determine the influence of characteristic parameters such as thermal relaxation and the power law (nonhomogeneity) index on the thermoelastic behavior of the considered medium. Finally, some interesting situations are derived from the current formulation.

Burr proposed twelve different forms of cumulative distribution functions for modeling data. Among those twelve distribution functions is the Burr X distribution. In statistical literature, a flexible family called the Burr X-G (BX-G) family is introduced. In this paper, we propose a bivariate extension of the BX-G family, in the so-called bivariate Burr X-G (BBX-G) family of distributions based on the Marshall–Olkin shock model. Important statistical properties of the BBX-G family are obtained, and a special sub-model of this bivariate family is presented. The maximum likelihood and Bayesian methods are used for estimating the bivariate family parameters based on complete and Type II censored data. A simulation study was carried out to assess the performance of the family parameters. Finally, two real data sets are analyzed to illustrate the importance and the flexibility of the proposed bivariate distribution, and it is found that the proposed model provides better fit than the competitive bivariate distributions.

The small size and clever design of nanoparticles can result in large surface areas. This gives nanoparticles enhanced properties such as greater sensitivity, strength, surface area, responsiveness, and stability. This research delves into the phenomenon of a nanobeam vibrating under the influence of a time-varying heat flow. The nanobeam is hypothesized to have material properties that vary throughout its thickness according to a unique exponential distribution law based on the volume fractions of metal and ceramic components. The top of the FG nanobeam is made entirely of ceramic, while the bottom is made of metal. To address this issue, we employ a nonlocal modified thermoelasticity theory based on a Moore–Gibson–Thompson (MGT) thermoelastic framework. By combining the Euler–Bernoulli beam idea with nonlocal Eringen’s theory, the fundamental equations that govern the proposed model have been constructed based on the extended variation principle. The fractional integral form, utilizing Atangana–Baleanu fractional operators, is also used to formulate the heat transfer equation in the suggested model. The strength of a thermoelastic nanobeam is improved by performing detailed parametric studies to determine the effect of many physical factors, such as the fractional order, the small-scale parameter, the volume fraction indicator, and the periodic frequency of the heat flow.