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Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 6, Pages 163–194 (Mi izv4281)  

This article is cited in 3 scientific papers (total in 3 papers)

$p$-adic evolution pseudo-differential equations and $p$-adic wavelets

V. M. Shelkovich

St. Petersburg State University of Architecture and Civil Engineering

Abstract: In the theory of $p$-adic evolution pseudo-differential equations (with time variable $t\in\mathbb{R}$ and space variable $x\in \mathbb{Q}_p^n$), we suggest a method of separation of variables (analogous to the classical Fourier method) which enables us to solve the Cauchy problems for a wide class of such equations. It reduces the solution of evolution pseudo-differential equations to that of ordinary differential equations with respect to the real variable $t$. Using this method, we solve the Cauchy problems for linear evolution pseudo-differential equations and systems of the first order in $t$, linear evolution pseudo-differential equations of the second and higher orders in $t$, and semilinear evolution pseudo-differential equations. We derive a stabilization condition for solutions of linear equations of the first and second orders as $t\to \infty$. Among the equations considered are analogues of the heat equation and linear or non-linear Schrödinger equations. The results obtained develop the theory of $p$-adic pseudo-differential equations and can be used in applications.

Keywords: $p$-adic pseudo-differential operator, $p$-adic fractional operator, $p$-adic wavelet bases, $p$-adic pseudo-differential equations.

DOI: https://doi.org/10.4213/im4281

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English version:
Izvestiya: Mathematics, 2011, 75:6, 1249–1278

Bibliographic databases:

UDC: 517.983.37+517.984.57+512.625.5
MSC: Primary 47G30, 42C40, 11F85; Secondary 26A33
Received: 31.12.2009
Revised: 12.07.2010

Citation: V. M. Shelkovich, “$p$-adic evolution pseudo-differential equations and $p$-adic wavelets”, Izv. RAN. Ser. Mat., 75:6 (2011), 163–194; Izv. Math., 75:6 (2011), 1249–1278

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Kosyak A.V. Khrennikov A.Yu. Shelkovich V.M., “Pseudodifferential operators on adele rings and wavelet bases”, Dokl. Math., 85:3 (2012), 358–362  crossref  mathscinet  zmath  isi  elib  elib  scopus
    2. Lebedeva E., Skopina M., “Walsh and Wavelet Methods For Differential Equations on the Cantor Group”, J. Math. Anal. Appl., 430:2 (2015), 593–613  crossref  mathscinet  zmath  isi  elib  scopus
    3. Evdokimov S., “On non-compactly supported p-adic wavelets”, J. Math. Anal. Appl., 443:2 (2016), 1260–1266  crossref  mathscinet  zmath  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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