Sobolev Type Equations Research Articles

Existence of mild solutions for Sobolev-type Hilfer fractional evolution equations with boundary conditions

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of the properties of Hilfer fractional calculus, the theory of propagation families as well as the theory of the measure of noncompactness and fixed point methods, we obtain the existence results of mild solutions for Sobolev-type fractional evolution differential equations involving the Hilfer fractional derivative. Finally, an example is presented to illustrate the main result.

Краевая задача для уравнения третьего порядка, не разрешенного относительно старшей производной

Рассматривается краевая задача для уравнения третьего порядка, не разрешенного относительно старшей производной. Уравнения такого типа, часто называемые уравнениями соболевского типа, встречаются во многих прикладных задачах. С помощью нестационарного метода Галёркина и метода регуляризации доказана теорема существования и единственности регулярного решения краевой задачи. Также получена оценка погрешности метода Галёркина через параметр регуляризации и собственные значения спектральной задачи для оператора Лапласа. We consider a boundary value problem for the third-order equation not solvable with respect to the highest-order derivative. Equations of this type, often called Sobolev type equations, occur in many applied problems. The nonstationary Galerkin method and regularization method are applied to prove the existence and uniqueness theorem for a regular solution of the boundary value problem. Also we obtain an error estimate via regularization parameter and in terms of eigenvalues of the spectral problem for the Laplace operator.

Boundary Value Problems for Degenerating and Nondegenerating Sobolev-Type Equations with a Nonlocal Source in Differential and Difference Forms

Boundary value problems are considered for degenerating and nondegenerating differential equations of the Sobolev type with a nonlocal source as well as finite-difference methods for solving these problems. A priori estimates are derived for solving the problems posed in differential and difference interpretations. These a priori estimates entail the uniqueness and stability of the solution with respect to the initial data and the right-hand side on a layer as well as the convergence of the solution of each difference problem to that of the counterpart differential problem.

Numerical Study of a Mathematical Model of an Autocatalytic Reaction with Diffusion in a Tubular Reactor

The article analyzes analytically and numerically the model of the autocatalytic reaction with diffusion in the degenerate case on a finite connected directed graph G with the Showalter -- Sidorov condition. The mathematical model of the autocatalytic reaction with diffusion is based on the system of distributed Brusselator equations. The system of degenerate equations of a distributed Brusselator whose functions satisfy the conditions of continuity and flow balance belongs to a wide class of semilinear Sobolev type equations. To investigate the existence of a solution of this system of equations, the phase space method which was developed by G.A.Sviridyuk and his students to study the solvability of Sobolev type equations will be used. We will show the simplicity of the phase space and the existence of a unique local solution of the given Showalter -- Sidorov problem. The theoretical results of this article served as the basis for developing an algorithm for numerical study of the model in a Maple environment. The algorithm of numerical investigation is based on the Galerkin method, which allows us to take into account the phenomenon of degeneracy of the equation. The article gives several examples illustrating the results of the computational experiment obtained on the three-ribbed and five-ribbed graphs.

Controllability of a Stochastic Neutral Functional Differential Equation Driven by a fBm

In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.

Faedo-Galerkin approximate solutions for nonlocal fractional differential equation of Sobolev type

Faedo-Galerkin approximate solutions for nonlocal fractional differential equation of Sobolev type

Numerical Solution for Non-Stationary Linearized Hoff Equation Defined on Geometrical Graph

The non-stationary linearized Hoff equation is considered in the article. For this equation, a solution is obtained both on the domain and on the geometric graph. For the five-edged graph, the Sturm -- Liouville problem is solved to obtain a numerical solution of the non-stationary linearized Hoff equation on the graph. A numerical method for solving this equation on a graph is described. The graphics for obtained numerical solution are constructed at different instants of time for given values of the equation parameters and functions. The article besides the introduction and the bibliography contains four parts. The first part contains information on abstract non-stationary Sobolev type equations, and solutions for the non-stationary linearized Hoff equation on the domain are constructed. In the second one we consider the Sturm -- Liouville problem on a graph and construct necessary spaces and operators on graphs. In the third one we study the solvability of the non-stationary linearized Hoff equation on the five-edged graph, and finally, in the last part we describe the numerical solution of the equation on the graph and the graphics of these solutions at different instants of time.

ОБ ОБОБЩЕННОЙ КРАЕВОЙ ЗАДАЧЕ ДЛЯ ЛИНЕЙНЫХ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА НА ГРАФЕ

On the geometric graph, where in addition to the continuity conditions and balance flow, condition of immobility is first introduced into the vertices of the graph, which is converted to a Dirichlet condition when the graph has one edge with two vertices. To solve this problem we first consider the corresponding Sturm–Liouville problem, and the results are then used to solve the Cauchy problem for two linear models, defined on the graph: Hoff equation and Barenblatt–Zheltov–Kochina equation. A feature of the work is the fact that on each edge of the graph given by the equation with different coefficients, which coupled with the introduction of vertices, is fixed for the first time in this problem. Both models relate to Sobolev type equations, the study of which is experiencing an era of its heyday. Reduction of these equations to an abstract Sobolev type equation makes it possible to apply the method of degenerate semigroups of operators. The phase space of solutions is determined by the phase space method, which consists in reducing the singular equation to a regular equation defined on some subspace of the original space. The obtained results of theorems can be used in consideration of inverse problems, optimal control problems, the initial-end and multipoint problems, and also in consideration of stochastic equations for the models set in a geometric graph.

The Cauchy Problem for the Sobolev Type Equation of Higher Order

The Cauchy Problem for the Sobolev Type Equation of Higher Order

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

ABSTRACTIn this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.

Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair (A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.

Vragov’s boundary value problem for an implicit equation of mixed type

We study a Vragov boundary value problem for a third-order implicit equation of mixed type with an arbitrary manifold of type switch. These Sobolev-type equations arise in many important applied problems. Given certain constraints on the coefficients and the right-hand side of the equation, we demonstrate, using nonstationary Galerkin method and regularization method, the unique regular solvability of the boundary value problem. We also obtain an error estimate for approximate solutions of the boundary value problem in terms of the regularization parameter and the eigenvalues of the Dirichlet spectral problem for the Laplace operator.

Asymptotic Stability of Neutral Impulsive Stochastic Partial Differential Equation of Sobolev Type with Poisson Jumps

This paper is concerned with the p-th asymptotical stability of the mild solution of an impulsive neutral stochastic integro-differential equation of Sobolev type involving Poisson jumps and non-instantaneous impulses. By utilizing Leray–Schauder fixed point theorem and compact semigroup, a set of sufficient conditions establishing the asymptotical stability of the mild solution to stochastic system in the p-moment is derived. An example is also considered for illustrating abstract results.

An expanded mixed covolume element method for integro-differential equation of Sobolev type on triangular grids

The expanded mixed covolume Element (EMCVE) method is studied for the two-dimensional integro-differential equation of Sobolev type. We use a piecewise constant function space and the lowest order Raviart-Thomas (mathit{RT}_{0}) space as the trial function spaces of the scalar unknown u and its gradient σ and flux λ, respectively. The semi-discrete and backward Euler fully-discrete EMCVE schemes are constructed, and the optimal a priori error estimates are derived. Moreover, numerical results are given to verify the theoretical analysis.

On exact solutions of a Sobolev equation

A nonlinear Sobolev-type equation that can be used to describe nonstationary processes in the semiconductor medium is studied. A number of families of exact solutions of this equation that can be expressed in terms of elementary functions and quadratures is obtained; some of these families contain arbitrary sufficiently smooth functions of one argument. The qualitative behavior of the resulting solutions is analyzed.

Spectral Problems for One Mathematical Model of Hydrodynamics

Spectral problems of the form $ (T + P) u = \lambda u $ have a huge range of applications: hydrodynamic stability problems, elastic vibrations of a membrane, a set of possible states of systems in quantum mechanics, and so forth. The self-adjoint operators perturbed by bounded operators are most thoroughly studied. In applications, the perturbed operator is usually represented by the Sturm~--~Liouville or Schrodinger operator. At present moment, the researchers are very interested in the equations not solved with respect to the highest derivative $ L \dot {u} = T u + f $, which are known as Sobolev type equations. The study of Sobolev type equations leads to spectral problems of the form $ T u = \lambda L u $. In many cases, the~operator~$ T $ can be perturbed by an operator $ P $, and then the spectral problem takes the~form \mbox{$ (T + P) u = \lambda L u $.} The study of such problems allows to construct a solution of the equation, as well as to investigate various parameters of mathematical models. Previously, such spectral problems with the perturbed operator were not studied. In this paper, we propose the method for investigating and solving the direct spectral problem for a hydrodynamic model.

ВЫРОЖДЕННЫЕ ПОТОКИ РАЗРЕШАЮЩИХ ОПЕРАТОРОВ ДЛЯ НЕСТАЦИОНАРНЫХ УРАВНЕНИЙ СОБОЛЕВСКОГО ТИПА

М.А. Сагадеева, Южно-Уральский государственный университет, г. Челябинск, Российская Федерация E-mail: sagadeevama@susu.ru. M.A. Sagadeeva South Ural State University, Chelyabinsk, Russian Federation E-mail: sagadeevama@susu.ru

Sobolev Type Equation in (n,p)-Sectorial Case

Sobolev Type Equation in (n,p)-Sectorial Case

On a certain nonlinear nonlocal Sobolev-type wave equation

The initial boundary-value problem for the nonlinear nonlocal Sobolev equation is studied. Sufficient conditions for local and for global (with respect to time) solvability are obtained. For the case of local (not global) solvability, upper and lower bounds for the lifespan of the solution are obtained in the form of explicit and implicit formulas.

The Bounded Solutions on a Semiaxis for the Linearized Hoff Equation in Quasi-Sobolev Spaces

In this paper we investigate the properties of the linearized Hoff equation in quasi-Sobolev spaces. Hoff equation, specified on the interval, describes the buckling of the H-beam. Due to the fact that for certain values of the parameters in the equation may be missing the derivative with respect to time, this equation refers in a frame of class of nonclassical equations of mathematical physics. The article by relatively spectral theorem describes the morphology of the phase space and the existence of invariant spaces of solutions. Using these results, we prove the existence of bounded on the semiaxis solutions for homogeneous evolution equations of Sobolev type in quasi-Sobolev spaces. Apart of the introduction and bibliography the article contains three parts. The first one shows the results on the solvability of the investigated class of equations. The second part shows the existence of bounded on the semiaxis solutions for the homogeneous equations of the research class. Finally, the third part presents the results of the existence of solutions bounded on the semiaxis for analog linearized Hoff equation in quasi-Sobolev spaces.