Solving Nonlinear p -Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point Theor
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Solving Nonlinear p-Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point Theorem Ehsan Pourhadi1 · Andrei Yu. Khrennikov1 · Klaudia Oleschko2 · María de Jesús Correa Lopez3 Received: 25 December 2019 / Published online: 14 August 2020 © The Author(s) 2020
Abstract 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. Keywords Pseudo-differential equations · p-adic field · p-adic wavelet basis · Schauder fixed point theorem · Arzelà–Ascoli theorem
Communicated by Hans G. Feichtinger.
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Andrei Yu. Khrennikov [email protected] Ehsan Pourhadi [email protected] Klaudia Oleschko [email protected] María de Jesús Correa Lopez [email protected]
1
International Center for Mathematical Modelling in Physics and Cognitive Sciences MSI, Linnaeus University, 351 95 Växjö, Sweden
2
Centro de Geociencias, Campus UNAM Juriquilla, Universidad Nacional Autonoma de Mexico (UNAM), Blvd. Juriquilla 3001, 76230 Querétaro, Mexico
3
Edificio Piramide, Boulevard Adolfo Ruiz Cortines 1202, Oropeza, 86030 Villahermosa, Tabasco, Mexico
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Journal of Fourier Analysis and Applications (2020) 26:70
Mathematics Subject Classification 35S10 · 42B35 · 47H10
1 Introduction During recent 30 years, p-adic analysis has received a lot of attention through its applications to mathematical physics, string theory, quantum mechanics, dynamical systems, turbulence, cognitive sciences, and recently geophysics, see e.g. [5–7,11,13, 15,16,19–21,26,28,31,39,40] and references therein. It is well-known that the theory of p-adic distributions (generalized functions) and the corresponding Fourier and wavelet analysis play an important role in solving mathematical problems and applications in aforementioned fields. In p-adic analysis, which is associated with maps Q p → C, the operation of differentiation is well not defined. For such reason, p-adic modeling widely utilizes the calculus of pseudo-differential operators. In this calculus, the crucial role is played by the fractional differentiation operator D α (the Vladimirov operator). The pseudodifferential equations over p-adic fields have been studied in numerous publications [3,8,12,17–19,22,24,25,27,32,34–37]. But up to now, in almost all models, only linear and semilinear pseudo-differential equations have been considered (
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