Bosonic boundary states at finite temperature are constructed as solutions of boundary conditions at $T\ne 0$ for bosonic open strings with a constant gauge field $F_{\mathrm{ab}}$ coupled to the boundary. The construction is done in the framework of thermo field dynamics where a thermal Bogoliubov transformation maps states and operators to finite temperature. Boundary states are given in terms of states from the direct product space between the Fock space of the closed string and another identical copy of it. By analogy with zero temperature, the boundary states have the interpretation of $\mathrm{D}p$-branes at finite temperature. The boundary conditions admit two different solutions. The entropy of the closed string in a $\mathrm{D}p$-brane state is computed and analyzed. It is interpreted as the entropy of the $\mathrm{D}p$-brane at finite temperature.

All possible Bogoliubov operators that generate the thermal transformations in thermo field dynamics form an $\mathrm{SU}(1,1)\mathrm{}$ group. We discuss this construction in the bosonic string theory. In particular, the transformation of the Fock space and string operators generated by the most general $\mathrm{SU}(1,1)\mathrm{}$ unitary Bogoliubov transformation and the entropy of the corresponding thermal string are computed. Also, we construct the thermal D-brane generated by the $\mathrm{SU}(1,1)\mathrm{}$ transformation in a constant Kalb-Ramond field and compute its entropy.

Fractional calculus provides powerful tools for modeling nonlocality, dissipative systems, and, when defined in the time representation, provides an interesting memory effect in mathematical physics. In this paper, we review four standard fractional approaches: the Riemann–Liouville, Gerasimov–Caputo, Grünwald–Letnikov, and Riesz formulations. We present their definitions, basic properties, Weyl–Marchaud, and physical interpretations. We also give a brief review of related operators that have been used recently in applications but have received less attention in the physical literature: the fractional Laplacian, conformable derivatives, and the Fractional Action-Like Variational Approach (FALVA) for variational principles with fractional action weights. Our emphasis is on how these operators are, and can be, applied in physical problems rather than on exhaustive coverage of the field. This review is intended as an accessible introduction for physicists working in diverse areas interested in fractional calculus and fractional methods. For deeper technical or domain-specific treatments, readers are encouraged to consult the works in the corresponding fields, for which the bibliography suggests a starting point.