Hypercomplex operator calculus for the fractional Helmholtz equation

Abstract

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equations where fractional derivatives in the sense of Caputo and Riemann-Liouville are applied. Our method extends the recently proposed Fractional Reduced Differential Transform Method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady-state solutions or spatial blow-ups of the fractional Helmholtz equation in a better way.

More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady-solutions and when spatial blow-ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting.

Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann-Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel-Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems.