Solvability of Pseudoparabolic Equations with Non-Linear Boundary Condition

Aitzhanov, S. E.; Zhumagul, G. О.

doi:10.1134/s1995080220090061

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…For integer values α = 1 and α = 2, the theorems on the existence and uniqueness of a weak solution of the analogue of the problem K (1)-(3) are proved in [30][31][32].…”

Section: Discussionmentioning

confidence: 99%

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Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

Aitzhanov

Bekenayeva

2021

Fractal Fract

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This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc.

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“…For integer values α = 1 and α = 2, the theorems on the existence and uniqueness of a weak solution of the analogue of the problem K (1)-(3) are proved in [30][31][32].…”

Section: Discussionmentioning

confidence: 99%

“…where Γ = ∂Ω × (0, T) and σ is a positive constant. We note that the solvability issues of the problem K in the case of α = 1 have been studied in [30]. For the integer parameter α, problems with Dirichlet-Neumann boundary conditions have been investigated in [31,32].…”

Section: Statement Of the Problemmentioning

confidence: 99%

Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

Aitzhanov

Bekenayeva

2021

Fractal Fract

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“…The study of problems for the pseudo‐parabolic type began in the late 1970s. A large number of works are devoted to the study of nonlinear equations of pseudoparabolic type 2–37 . The modeling of physical processes leading to equations of the Sobolev type and, in particular, of the pseudoparabolic type is devoted to the works 2,3,5,6,14–19,23 …”

Section: Introductionmentioning

confidence: 99%

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

Antont︠s︡ev

Aitzhanov

Zhanuzakova

2022

Math Methods in App Sciences

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An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann-Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann-Dirichlet boundary condition are obtained. KEYWORDS asymptotic behavior of the solution, blow-up of the solution, Galerkin method, nonlinear boundary conditions, pseudoparabolic equations, the existence of a solution, uniqueness of the solution MSC CLASSIFICATION

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“…PPPDEs appear in several fields of mechanics and physics. ey are used to investigate homogeneous fluid flow in fractured rocks, quasi-stationary paths in semiconductors, thermodynamic and transportation phenomena, and different physical systems (see, e.g., [4][5][6]).…”

Section: Introductionmentioning

confidence: 99%

Double Laplace Decomposition Method and Finite Difference Method of Time-fractional Schrödinger Pseudoparabolic Partial Differential Equation with Caputo Derivative

Modanlı

Bajjah

2021

Journal of Mathematics

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In this paper, an initial-boundary value problem for a one-dimensional linear time-dependent fractional Schrödinger pseudoparabolic partial differential equation with Caputo derivative of order α ∈ 0,1 is being considered. Two strong numerical methods are employed to acquire the solutions to the problem. The first method used is the double Laplace decomposition method where closed-form solutions are obtained for any α ∈ 0,1 . As the second method, the implicit finite difference scheme is applied to obtain the approximate solutions. To clarify the performance of these two methods, numerical results are presented. The stability of the problem is also investigated.

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Solvability of Pseudoparabolic Equations with Non-Linear Boundary Condition

Cited by 4 publications

References 11 publications

Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

Double Laplace Decomposition Method and Finite Difference Method of Time-fractional Schrödinger Pseudoparabolic Partial Differential Equation with Caputo Derivative

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