Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields

Journal of Mathematical Analysis and Applications

Shelkovich

2011

We solve the Cauchy problems for p-adic linear and semi-linear evolutionary pseudodifferential equations (the time-variable t ∈ R and the space-variable x ∈ Q n p ). Among the equations under consideration there are the heat type equation and the Schrödinger type equations (linear and nonlinear). To solve these problems, we develop the "variable separation method" (an analog of the classical Fourier method) which reduces solving evolutionary pseudo-differential equations to solving ordinary differential equations with respect to real variable t. The problem of stabilization for solutions of the Cauchy problems as t → ∞ is also studied. These results give significant advance in the theory of p-adic pseudo-differential equations and can be used in applications.

Section: P-adic Pseudo-differential Operators On the Lizorkin Spacesmentioning

confidence: 99%

The Cauchy problems for evolutionary pseudo-differential equations over p-adic field and the wavelet theory

Albeverio

Journal of Mathematical Analysis and Applications

Shelkovich

2011

“…Since p-adic wavelets are eigenvectors of integral operators, wavelets were used for the investigation of integral equations [179,178,150,9,12,13,14,335,336,55,223,57,337,221].…”

Section: P-adic Waveletsmentioning

confidence: 99%

p-Adic mathematical physics: the first 30 years

Драгович

P-Adic Num Ultrametr Anal Appl

Козырев

et al. 2017

135

p-Adic mathematical physics is a branch of modern mathematical physics based on the application of p-adic mathematical methods in modeling physical and related phenomena. It emerged in 1987 as a result of efforts to find a non-Archimedean approach to the spacetime and string dynamics at the Planck scale, but then was extended to many other areas including biology. This paper contains a brief review of main achievements in some selected topics of p-adic mathematical physics and its applications, especially in the last decade. Attention is mainly paid to developments with promising future prospects.

“…We will consider the solution with f 0 = 0, i.e. the solution which satisfies the initial condition Discretization F = Π 0 f of f given by (13) belongs to the space of linear functionals over V 0 . We call this discretization the time series over Q p /Z p .…”

Section: Time Series and P-adic Brownian Motionmentioning

confidence: 99%

APPLICATION OF p-ADIC ANALYSIS TO TIME SERIES

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

Козырев

Oleschko

et al. 2013

Time series defined by a p-adic pseudo-differential equation is investigated using the expansion of the time series over p-adic wavelets. Quadratic correlation function is computed. This correlation function shows a degree-like behavior and is locally constant for some time periods. It is natural to apply this kind of models for the investigation of avalanche processes and punctuated equilibrium as well as fractal-like analysis of time series generated by measurement of pressure in oil wells.