Abstract
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.
Highlights
It is known that one of the most effective ways to study environmental processes using mathematical methods is their modeling in the form of differential equations
The solutions of many practically important problems arising in the study of liquid filtration processes in fractured porous media; the movement of underground water with a free surface in multilayer media; the transfer of moisture, heat, and salts in porous media; and so on are connected with the need to study boundary value problems for pseudo-parabolic equations of the third order [1,2,3,4,5,6,7]
This paper provides the first study of the problems of unique solvability of initial boundary value problems for a quasi-linear pseudo-parabolic fractional equation with nonlinear boundary conditions with Caputo fractional differentiation operators
Summary
It is known that one of the most effective ways to study environmental processes using mathematical methods is their modeling in the form of differential equations. Applying the iterative process obtained for a linear problem, the existence, uniqueness, and continuous dependence of a weak generalized solution of a nonlinear problem are proved. For loaded pseudo-parabolic equations of fractional order, we can note the work [17], where, with the help of the obtained priori estimates, the uniqueness of the solution and its stability with respect to the initial data and the right part are proved, as well as the convergence of the associated difference problem’s solutions to the solution of the differential problem with speed O h2 + τ , where h and τ are steps in spatial and temporal variables. The novelty of this work is the study of the issues of unique solvability of initial boundary value problems for a quasi-linear pseudo-parabolic fractional equation with nonlinear boundary conditions with Caputo fractional differentiation operators
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