Continuous Dependence of the Singular Nonlinear Van der Pol Equation Solutions with Respect to the Boundary Conditions: Elements of p-regularity Theory

Medak, Beata; Tretyakov, А. А.

doi:10.1007/s10884-020-09849-0

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…Our research was inspired by works [9,10]. We obtain analytical formulas for solving the nonlinear Burgers equation of the form (1) based on the p-regularity theory.…”

Section: Discussionmentioning

confidence: 99%

“…Above theorem is proved in [10]. This is an analogue of the Lyusternik theorem on the tangent cone, which concerns the existence of continuous solutions of the equation F(v, y) = 0, where…”

Section: Introductionmentioning

confidence: 98%

“…The basis for our practical applications will be the following analogue of Lyusternik theorem on tangent cone (see [10]).…”

Section: Introductionmentioning

confidence: 99%

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The nonlinear singular Burgers equation with small parameter and p-regularity theory

Medak

Tretyakov

2022

Lett Math Phys

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In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation $$\begin{aligned} F(u,\varepsilon )=u_{t}'-u_{xx}''+uu_{x}'+\varepsilon u^{2}=f(x,t), \end{aligned}$$ F ( u , ε ) = u t ′ - u xx ′ ′ + u u x ′ + ε u 2 = f ( x , t ) , where $$F: \Omega \rightarrow \mathcal {C}([0,\pi ]\times [0,\infty ))$$ F : Ω → C ( [ 0 , π ] × [ 0 , ∞ ) ) , $$\Omega = \mathcal {C}^{2}([0,\pi ]\times [0,\infty ))\times \mathbb {R}$$ Ω = C 2 ( [ 0 , π ] × [ 0 , ∞ ) ) × R , $$u(0,t)=u(\pi ,t) =0$$ u ( 0 , t ) = u ( π , t ) = 0 , $$u(x,0)=g(x)$$ u ( x , 0 ) = g ( x ) , and F, f(x, t), g(x) will be describe in the text. The first derivative of operator F at the solution point is degenerate. By virtue of p-regularity theory and Michael selection theorem, we prove the existence of continuous solution for this nonlinear problem.

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“…Our research was inspired by works [9,10]. We obtain analytical formulas for solving the nonlinear Burgers equation of the form (1) based on the p-regularity theory.…”

Section: Discussionmentioning

confidence: 99%

“…Above theorem is proved in [10]. This is an analogue of the Lyusternik theorem on the tangent cone, which concerns the existence of continuous solutions of the equation F(v, y) = 0, where…”

Section: Introductionmentioning

confidence: 98%

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The nonlinear singular Burgers equation with small parameter and p-regularity theory

Medak

Tretyakov

2022

Lett Math Phys

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On Solution Existence for a Singular Nonlinear Burgers Equation with Small Parameter and p-Regularity Theory

Medak,

Tret’yakov

2023

Dokl. Math.

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ON THE APPLICATION OF THE SOLUTION OF THE DEGENERATE NONLINEAR BURGERS EQUATION WITH A SMALL PARAMETER AND THE THEORY OF <i>p</i>-REGULARITY

Medak,

Tret’yakov

2023

Doklady Rossijskoj akademii nauk. Matematika, informatika, proc

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The article discusses various modifications of the nonlinear Burgers equation with small parameter and degenerate in solution of the form$F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} - f(x,t) = 0,$ (1)where $F:\Omega \to C([0,\pi ] \times [0,T])$, $T 0$, $\Omega = {{C}^{2}}([0,\pi ] \times [0,T]\,)\,\mathbb{R}$ and $u(0,t) = u(\pi ,t) = 0$, $u(x,0) = \varphi (x)$, $f(x,t) \in C([0,\pi ] \times [0,T])$, $\varphi (x) \in C[0,\pi ]$. We will be interested in the most important in applications case of a small parameter ε with oscillating initial conditions of the form $\varphi (x) = k\sin x$, where k –some, generally speaking, constant depending on ε, and study the question of the existence of a solution in neighborhood of the trivial $(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)$, which corresponds to $k = k{\kern 1pt} * = 0$ and at what initial Under certain conditions on the values of k, it is possible to construct an analytical approximation of this solution for small ε.We will look for a solution in the traditional way of separation of variables on a subspace of functions of the form $u(x,t) = v(t)u(x)$, where $v(t) = c{{e}^{{ - t}}}$, $u(x) \in {{\mathcal{C}}^{2}}([0,\pi ])$. In this case, the problem under consideration is degenerate at the point $(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)$, since ${\text{Im}}F_{u}^{'}(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) \ne Z = \mathcal{C}([0,\pi ] \times [0,T])$. This follows from the Sturm-Liouville theory. To achieve our goals, we apply the apparatus of p-regularity theory [6, 7, 15, 16] and show that the mapping $F(u,\varepsilon )$ is 3-regular at the point $(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)$, т.е. p = 3.

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Continuous Dependence of the Singular Nonlinear Van der Pol Equation Solutions with Respect to the Boundary Conditions: Elements of p-regularity Theory

Cited by 3 publications

References 7 publications

The nonlinear singular Burgers equation with small parameter and p-regularity theory

The nonlinear singular Burgers equation with small parameter and p-regularity theory

On Solution Existence for a Singular Nonlinear Burgers Equation with Small Parameter and p-Regularity Theory

ON THE APPLICATION OF THE SOLUTION OF THE DEGENERATE NONLINEAR BURGERS EQUATION WITH A SMALL PARAMETER AND THE THEORY OF <i>p</i>-REGULARITY

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