We are interested in a linearized inverse scattering problem considered by Symes and Nolan [10] for the marine data acquisition geometry. This arises as follows: a vessel carries an acoustic source (airgun) and recording instruments (hydrophones). The airgun sends acoustic waves through the ocean to the subsurface and the reflections occur when the sound waves encounter singularities in the mat...

We derive certain estimates and asymptotics for oscillatory integral operators with degenerate phase functions, concentrating our attention on the situation when one of the projections from the associated canonical relation is a Whitney fold. We discuss related topics: convexity, higher order degeneracy of canonical relations, and almost orthogonality. In the second part of the paper, we apply ...

An elliptic Fourier integral operator of order 0, associated to a homogeneous canonical diffeomorphism, on a compact manifold is Fredholm on L2. The index may be expressed as the sum of a term, which we call the contact degree, associated to the canonical diffeomorphism and a term, computable by the Atiyah-Singer theorem, associated to the symbol. The contact degree is shown to be defined for a...

We extend the direct approach to semiclassical Bergman kernel asymptotics, developed recently in Deleporte et al. (Ann Fac Sci Toulouse Math, 2020) for real analytic exponential weights, smooth case. Similar (2020), our avoids use of Kuranishi trick and it allows us construct amplitude asymptotic projection by means an inversion explicit Fourier integral operator.

The present paper proposes a fast numerical method for the linear Volterra integral equations withregular and weakly singular kernels having smooth solutions. This method is based on the approx-imation of the kernel, to simplify the integral operator and then discretization of the simpliedoperator using a forward dierence formula. To analyze and verify the accuracy of the method, weexamine samp...

In this work, a non-integer order Airy equation involving Liouville differential operator is considered. Proposing an undetermined integral solution to the left fractional Airy differential equation, we utilize some basic fractional calculus tools to clarify the closed form. A similar suggestion to the right FADE, converts it into an equation in the Laplace domain. An illustration t...