Green’s functions, positive solutions, and a Lyapunov inequality for a caputo fractional-derivative boundary value problem

We consider a nonlinear boundary problem whose highest-order derivative is a Caputo derivative of order α with 1 < α < 2. Properties of its associated Green’s function are derived. These properties enable us to deduce sufficient conditions for the existence of a positive solution to the boundary value problem and to prove a Lyapunov inequality for the problem. Our results sharpen and extend earlier results of other authors.

Current-Phase Relation in Josephson Junctions with Complex Ferromagnetic/Normal Metal Interlayers

In the frame of Usadel equation the two-dimensional nonlinear boundary problem for heterostructures with complex ferromagnetic (F)/normal metal (N) interlayer has been solved numerically for arbitrary values of electrodes spacing L, thickness dN and dF of N and F films. We calculate the current-phase relations for ramp-type S-NF-S and overlap-type SFN-FN-NFS junctions. We have found the range of parameters providing an opportunity for realization of Josephson φ-junction based on the considered structures.