High-Frequency Geometrical-Optics Approximations for Helmholtz and Fractional Helmholtz Equations with Fox H-Functions

This work presents novel geometrical-optics approximations for both Helmholtz equation and fractional Helmholtz equation using Fox H-functions. The asymptotic approximations with Fox H-functions extend the results in [Gao & Luo, J. Sci. Comput. 103, 70 (2025)] by providing not only accurate solutions in non-caustic regions that are away from the source point, but also uniform accuracy near the source point. Moreover, when the fractional order approaches two, the asymptotic approximations with Fox H-functions include the Babich’s form with Hankel functions of first kind for the standard Helmholtz equation as a particular case. In the asymptotic approximations, the phase and amplitude of the wavefunction are shown to be determined by an eikonal equation and a transport equation, respectively, even for the nonlocal fractional problem. Numerical experiments verify the effectiveness of the proposed asymptotic approximations, where the phase and amplitude are computed by solving the eikonal and transport equations numerically with fast sweeping Lax-Friedrichs schemes and high-order weighted essentially non-oscillatory spatial finite-difference approximations.