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 SM ISO690:2012BIHUN, Yaroslav, PETRYSHYN, Roman, SKUTAR, Ihor. The Multifrequency Systems with Linearly Transformed Arguments and Multipoint and Local-Integral Conditions. In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 25-27 august 2022, Chişinău. Chișinău, Republica Moldova: Casa Editorial-Poligrafică „Bons Offices”, 2022, Ediţia a 29, pp. 45-47. ISBN 978-9975-81-074-6. |
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| Conference on Applied and Industrial Mathematics Ediţia a 29, 2022 |
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Conferința "Conference on Applied and Industrial Mathematics" 29, Chişinău, Moldova, 25-27 august 2022 | ||||||
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A problem of existence and uniqueness of the solution and its approximate construction for the system of differential equations with a vector of slow variables a ∈ Δ ⊂ Rn and fast variables φ ∈ Tm of the form da dτ = X(τ, aΛ,φΘ), dφ dτ = ω(τ ) ε + Y (τ, aΛ,φΘ), (1) are investigated in the paper. Here τ ∈ [0,L], small parameter ε ∈ (0, ε0], 0 < λ1 < · · · < λp ≤ 1, 0 < θ1 < · · · < θq ≤ 1, aΛ = (aλ1, . . . ,aλp ), aλi (τ) = a(λiτ ), φΘ = (φθ1, . . . ,φθq ), φθj (τ) = φ(θjτ ). Multifrequency ODE systems are researched in detail in [1], systems with delayed argument were studied in [2, 3], etc. Conditions are set for the system (1) r ν=1 ανa(τν) = s ν=1 ην ξν fν(τ, aΛ,φΘ)dτ, r ν=1 βνφ(τν) = s ν=1 ην ξν gν(τ, aΛ,φΘ)dτ, (2) where [ξν, ην] ⊂ [0,L], ∩ν [ξν, ην] = ∅. For the problem (1), (2) a much simpler problem is constructed by averaging over fast variables φθν on the cube of periods [0, 2π]mq. A sufficient condition for the system of equations (1) to exit a small circumference of the resonance of frequencies ω(τ ) was found, the condition of which in the point τ ∈ [0,L] is q ν=1 θν kν,ω(θντ ) = 0,kν ∈ Zm, ∥k1∥ + · · · + ∥kq∥ ̸= 0. It is proved that for the smooth enough right parts of the system (1) and sub-integral functions under conditions (2), the condition to exit the circumference of resonances and for small enough ε0 > 0 there exists a unique solution for the problem (1), (2) and an estimation is found ∥a(τ ; y,ψ, ε)−a(τ ; y)∥+∥φ(τ ; y,ψ, ε)−φ(τ ; y,ψ, ε)∥ ≤ c1εα,α = (mq)−1, where c1 > 0 and does not depend on ε, a(0; y,ψ, ε),φ(0; y,ψ, ε) = (y,ψ), a(τ ; y), φ(τ ; y,ψ, ε) – the solution of the averaged problem with initial conditions (y, ψ), while ∥y − y∥ + ∥ψ − ψ∥ ≤ c2εα. |
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<?xml version='1.0' encoding='utf-8'?> <oai_dc:dc xmlns:dc='http://purl.org/dc/elements/1.1/' xmlns:oai_dc='http://www.openarchives.org/OAI/2.0/oai_dc/' xmlns:xsi='http://www.w3.org/2001/XMLSchema-instance' xsi:schemaLocation='http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd'> <dc:creator>Bihun, Y.</dc:creator> <dc:creator>Petryshyn, R.</dc:creator> <dc:creator>Skutar, I.</dc:creator> <dc:date>2022</dc:date> <dc:description xml:lang='en'><p>A problem of existence and uniqueness of the solution and its approximate construction for the system of differential equations with a vector of slow variables a ∈ Δ ⊂ Rn and fast variables φ ∈ Tm of the form da dτ = X(τ, aΛ,φΘ), dφ dτ = ω(τ ) ε + Y (τ, aΛ,φΘ), (1) are investigated in the paper. Here τ ∈ [0,L], small parameter ε ∈ (0, ε0], 0 < λ1 < · · · < λp ≤ 1, 0 < θ1 < · · · < θq ≤ 1, aΛ = (aλ1, . . . ,aλp ), aλi (τ) = a(λiτ ), φΘ = (φθ1, . . . ,φθq ), φθj (τ) = φ(θjτ ). Multifrequency ODE systems are researched in detail in [1], systems with delayed argument were studied in [2, 3], etc. Conditions are set for the system (1) r ν=1 ανa(τν) = s ν=1 ην ξν fν(τ, aΛ,φΘ)dτ, r ν=1 βνφ(τν) = s ν=1 ην ξν gν(τ, aΛ,φΘ)dτ, (2) where [ξν, ην] ⊂ [0,L], ∩ν [ξν, ην] = ∅. For the problem (1), (2) a much simpler problem is constructed by averaging over fast variables φθν on the cube of periods [0, 2π]mq. A sufficient condition for the system of equations (1) to exit a small circumference of the resonance of frequencies ω(τ ) was found, the condition of which in the point τ ∈ [0,L] is q ν=1 θν kν,ω(θντ ) = 0,kν ∈ Zm, ∥k1∥ + · · · + ∥kq∥ ̸= 0. It is proved that for the smooth enough right parts of the system (1) and sub-integral functions under conditions (2), the condition to exit the circumference of resonances and for small enough ε0 > 0 there exists a unique solution for the problem (1), (2) and an estimation is found ∥a(τ ; y,ψ, ε)−a(τ ; y)∥+∥φ(τ ; y,ψ, ε)−φ(τ ; y,ψ, ε)∥ ≤ c1εα,α = (mq)−1, where c1 > 0 and does not depend on ε, a(0; y,ψ, ε),φ(0; y,ψ, ε) = (y,ψ), a(τ ; y), φ(τ ; y,ψ, ε) – the solution of the averaged problem with initial conditions (y, ψ), while ∥y − y∥ + ∥ψ − ψ∥ ≤ c2εα.</p></dc:description> <dc:source>Conference on Applied and Industrial Mathematics (Ediţia a 29) 45-47</dc:source> <dc:title>The Multifrequency Systems with Linearly Transformed Arguments and Multipoint and Local-Integral Conditions</dc:title> <dc:type>info:eu-repo/semantics/article</dc:type> </oai_dc:dc>
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