Solving Nonlinear p-Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point Theorem

• Published in: The Journal of fourier analysis and applications Vol. 26; no. 4
• Main Authors: Pourhadi, Ehsan, Khrennikov, Andrei Yu, Oleschko, Klaudia, de Jesús Correa Lopez, María
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-08-2020; Springer; Springer Nature B.V
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Capillary flow; Cauchy problems; Computational fluid dynamics; Derivatives; Differential equations; Exact solutions; Fixed points (mathematics); Fluid flow; Fourier Analysis; Geophysics; Mathematical analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Nonlinear equations; Partial Differential Equations; Porous media; Real time; Schauder fixpoint theorem; Signal,Image and Speech Processing; 42B35; 47H10; Schauder fixed point theorem; Arzelà–Ascoli theorem; adic wavelet basis; 35S10; adic field; Pseudo-differential equations
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-020-09779-x

Recently theory of
p
-adic wavelets started to be actively used to study of the Cauchy problem for nonlinear pseudo-differential equations for functions depending on the real time and
p
-adic spatial variable. These mathematical studies were motivated by applications to problems of geophysics (fluids flows through capillary networks in porous disordered media) and the turbulence theory. In this article, using this wavelet technique in combination with the Schauder fixed point theorem, we study the solvability of nonlinear equations with mixed derivatives,
p
-adic (fractional) spatial and real time derivatives. Furthermore, in the linear case we find the exact solution for the Cauchy problem. Some examples are provided to illustrate the main results.