We consider a diffusion–wave equation with fractional derivative with respect to the time variable, defined on infinite interval, and with the starting point at minus infinity. For this equation, we solve an asympotic boundary value problem without initial conditions, construct a representation of its solution, find out sufficient conditions providing solvability and solution uniqueness, and give some applications in fractional electrodynamics.

We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using the Green function, we give a solution of the first boundary-value problem in a multidimensional cylindrical domain. The fractional differentiation is given by the Dzhrbashyan–Nersesyan fractional differentiation operator. In particular, this covers the cases of equations with the Riemann–Liouville and Caputo derivatives.

We consider the principle of least action in the context of fractional calculus. Namely, we derive the fractional Euler–Lagrange equation and the general equation of motion with the composition of the left and right fractional derivatives defined on infinite intervals. In addition, we construct an explicit representation of solutions to a model fractional oscillator equation containing the left and right Gerasimov–Caputo fractional derivatives with origins at plus and minus infinity. We derive a representation for the composition of the left and right derivatives with origins at plus and minus infinity in terms of the Riesz potential, and introduce special functions with which we give solutions to the model fractional oscillator equation with a complex coefficient. This approach can be useful for describing dissipative dynamical systems with the property of heredity.

We construct a representation of solutions for diffusion-wave equations as a sum of two solutions of the first order PDEs. Fractional differentiation is given by the Liouville fractional derivative. The representation is an analogue of the d’Alembert formula known for the wave equation. In the case of an infinite rectangular domain (half-strip), we give relations that connect the traces of the solutions involved in the representation on the boundary of the domain.