The Robin boundary condition initial value problem for transient heat conduction with the time-fractional Caputo derivative in a semi-infinite domain with a convective heat transfer (Newton’s law) at the boundary has been solved and analyzed by two analytical approaches. The uniqueness and the stability of the solution on the half-axis have been analyzed. The problem solutions by application of the operational method (Laplace transform in the time domain) and the integral-balance method (double integration technique) have been developed analytically.

One of the promising approaches to the description of many physical processes is the use of the fractional derivative mathematical apparatus. Fractional dimensions very often arise when modeling various processes in fractal (multi-scale and self-similar) environments. In a fractal medium, in contrast to an ordinary continuous medium, a randomly wandering particle moves away from the reference point more slowly since not all directions of motion become available to it. The slowdown of the diffusion process in fractal media is so significant that physical quantities begin to change more slowly than in ordinary media.This effect can only be taken into account with the help of integral and differential equations containing a fractional derivative with respect to time. Here, the problem of heat and mass transfer in media with a fractal structure was posed and analytically solved when a heat flux was specified on one of the boundaries. The second initial boundary value problem for the heat equation with a fractional Caputo derivative with respect to time and the Riesz derivative with respect to the spatial variable was studied. A theorem on the semigroup property of the fractional Riesz derivative was proved. To find a solution, the problem was reduced to a boundary value problem with boundary conditions of the first kind. The solution to the problem was found by applying the Fourier transform in the spatial variable and the Laplace transform in time. A computational experiment was carried out to analyze the obtained solutions. Graphs of the temperature distribution dependent on the coordinate and time were constructed.

In this work, we have looked into the amplitude modulation and formation of enveloping soliton in a dense plasma with thermal degeneracy pressure expressions as a strong laser beam falls on it. We have focused on the situation where an intense laser interacts with plasma and we have made use of a new topological treatment of problems containing a set of coupled differential equations. At first, we have solved the equations to arrive at the dispersion relation. We have studied phase plots using particle-in-cell (PIC) simulation from where Kelvin–Helmholtz-type instabilities were observed. We have made use of the homotopy perturbation technique to find out more about the evolution of density, electric field, velocity streamlines, and the subsequent amplitude modulation with different types of nonlinearity. We identified the range of plasma and laser parameters in which such nonlinearity is important. The results obtained here will help interpret different phenomena that arise in laser–plasma interaction and plasma astrophysics where shocks solitons, and so on in dense hot plasma can be studied in laboratory scales.