Solvability issues of a pseudo‐parabolic fractional order equation with a nonlinear boundary condition
Aitzhanov S.E. Berdyshev A.S. Bekenayeva K.S.
October 2021MDPI
Fractal and Fractional
2021#5Issue 4
This paper is devoted to the fundamental problem of investigating the solvability of ini-tial‐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 prob-lems, 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 re-quirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc.
A priori estimates , Galerkin approximations , Global solvability , Mittag–Leffler function , Pseudo‐parabolic equation , The Caputo fractional derivative , Uniqueness of solution , Weak solution
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Department of Mathematics, Al‐Farabi Kazakh National University, Almaty, A15E3B6, Kazakhstan
Department of Mathematics and Mathematical Modeling, International University of Information Technologies, Almaty, A15M0F0, Kazakhstan
Department of Mathematics and Mathematical Modeling, Abai Kazakh National Pedagogical University, Almaty, 050010, Kazakhstan
Department of Mathematics
Department of Mathematics and Mathematical Modeling
Department of Mathematics and Mathematical Modeling
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