Sobolev Type Equations Research Articles

Asymptotic behaviors of solutions to Sobolev-type stochastic differential equations

This paper is devoted to studying the Sobolev-type stochastic differential equations with Lévy noise and mixed fractional Brownian motion. Applying a method (principle) of comparability of functions by character of Shcherbakov recurrence, it characters at least one (or exactly one) solution with the same properties as the coefficients of the equation. We establish the existence of Poisson stable solutions for the Sobolev-type equation, which includes periodic solutions, quasi-periodic solutions, almost periodic solutions, almost automorphic solutions, etc. We also obtain the global asymptotical stability of bounded Poisson stable solutions and present an example to illustrate our theoretical results.

Analysis on nonlinear differential equation with a deviating argument via Faedo–Galerkin method

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

Study of Uniqueness and Ulam-Type Stability of Abstract Hadamard Fractional Differential Equations of Sobolev Type via Resolvent Operators

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives.

Painleve analysis of travelling wave solutions and analysis of energy estimates for a nonlinear Sobolev-type equation

Painleve analysis of travelling wave solutions and analysis of energy estimates for a nonlinear Sobolev-type equation

Approximate controllability of neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results.

Instantaneous blow-up solutions for nonlinear Sobolev-type equations on the Heisenberg groups

Instantaneous blow-up solutions for nonlinear Sobolev-type equations on the Heisenberg groups

Novel waves structures for the nonclassical Sobolev-type equation in unipolar semiconductor with its stability analysis

In this study, the Sobolev-type equation is considered analytically to investigate the solitary wave solutions. The Sobolev-type equations are found in a broad range of fields, such as ecology, fluid dynamics, soil mechanics, and thermodynamics. There are two novel techniques used to explore the solitary wave structures namely as; generalized Riccati equation mapping and modified auxiliary equation (MAE) methods. The different types of abundant families of solutions in the form of dark soliton, bright soliton, solitary wave solutions, mixed singular soliton, mixed dark-bright soliton, periodic wave, and mixed periodic solutions. The linearized stability of the model has been investigated. Solitons behave differently in different circumstances, and their behaviour can be better understood by building unique physical problems with particular boundary conditions (BCs) and starting conditions (ICs) based on accurate soliton solutions. So, the choice of unique physical problems from various solutions is also carried out. The 3D, line graphs and corresponding contours are drawn with the help of the Mathematica software that explains the physical behavior of the state variable. This information can help the researchers in their understanding of the physical conditions.

A Time-Fractional Differential Inequality of Sobolev Type on an Annulus

Several phenomena from natural sciences can be described by partial differential equations of Sobolev-type. On the other hand, it was shown that in many cases, the use of fractional derivatives provides a more realistic model than the use of standard derivatives. The goal of this paper is to study the nonexistence of weak solutions to a time-fractional differential inequality of Sobolev-type. Namely, we give sufficient conditions for the nonexistence or equivalently necessary conditions for the existence. Our method makes use of the nonlinear capacity method, which consists in making an appropriate choice of test functions in the weak formulation of the problem. This technique has been employed in previous papers for some classes of time-fractional differential inequalities of Sobolev-type posed on the whole space RN. The originality of this work is that the considered problem is posed on an annulus domain, which leads to some difficulties concerning the choice of adequate test functions.

Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems

Three Cauchy problems for Sobolev-type equations with a common linear part from the theory of ion acoustic and drift waves in a plasma are considered. The problems are reduced to equivalent integral equations. We prove the existence of unextendable solutions for two problems and the existence of a local-in-time solution for the third problem. For one of the problems, by applying a modified method of Kh.A. Levin, sufficient conditions for finite time blow-up of solutions are obtained and an upper bound for the solution blow-up time is found. For another problem, S.I. Pohozaev’s nonlinear capacity method is used to obtain a finite time blow-up result and two results concerning the nonexistence of even local solutions, and an upper bound for the solution blow-up time is obtained as well.

Inverse problem for an abstract neutral differential equation of Sobolev-type

In this manuscript, we consider an inverse problem for the neutral differential equation of Sobolev type with a deviated argument and identify the parameter using an overdetermined condition on a mild solution. A direct approach using Volterra integral equations for sufficiently regular data and an optimal control approach for less regular data are the main techniques to find the result. Under certain hypotheses, the characterization of the limit of the sequence of approximate solutions demonstrates that it is a solution to the original inverse problem under specific hypotheses. At last, an example is provided in the support of our results.

An analysis of approximate controllability for Hilfer fractional delay differential equations of Sobolev type without uniqueness

This study focused on the approximate controllability results for the Hilfer fractional delay evolution equations of the Sobolev type without uniqueness. Initially, the Lipschitz condition is derived from the hypothesis, which is represented by a measure of noncompactness, in particular, nonlinearity. We also examined the continuity of the solution map of the Sobolev type of Hilfer fractional delay evolution equation and the topological structure of the solution set. Furthermore, we prove the approximate controllability of the fractional evolution equation of the Sobolev type with delay. Finally, we provided an example to illustrate the theoretical results.

Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.

On convergence of difference schemes of high accuracy for one pseudo-parabolic Sobolev type equation

Difference schemes of the finite difference method and the finite element method of high-order accuracy in time and space are proposed and investigated for a pseudo-parabolic Sobolev type equation. The order of accuracy in space is improved in two ways using the finite difference method and the finite element method. The order of accuracy of the scheme in time is improved by a special discretization of the time variable. The corresponding a priori estimates are determined and, on their basis, the accuracy estimates of the proposed difference schemes are obtained with sufficient smoothness of the solution to the original differential problem. Algorithms for the implementation of the constructed difference schemes are proposed.

A NUMERICAL SCHEME ON S-MESH FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE SOBOLEV PROBLEMS WITH LARGE TIME DELAY

The purpose of this article is to provide a numerical method for time delay singularly perturbed Sobolev type equations. First, asymptotic estimates for the Sobolev problem solution with singular perturbation and delay parameters were obtained. This estimate showed that the solution depends on the initial data. It is constructed and examined to solve this problem using a finite difference technique on a specific piecewise uniform mesh (Shishkin mesh)whose solution converges pointwise independent of the singular perturbation parameter. A discrete norm was used to investigate the stability of difference schemes. It is showed that the completely discrete scheme converges with order $O\left( \tau ^{2}+N_{l}^{-2}\ln ^{2}N_{l}\right) $ in both space and time, independent of the perturbation parameter. Finally, with a test problem and numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.

Extraction of soliton for the confirmable time-fractional nonlinear Sobolev-type equations in semiconductor by [formula omitted]-modal expansion method

The current study deals with the exact solutions of nonlinear confirmable time fractional Sobolev type equations. Such equations have applications in thermodynamics, the flow of fluid through fractured rock. The underlying models are 2D equation of a semi-conductor with heating and Sobolev equation in 2D unbounded domain. These equation are used to describe the different aspects in semi-conductor. The analytical solutions of underlying models is not addressed yet or it is difficult to find. We gain the exact solutions of such models with help of analytical technique namely ϕ6-model expansion method. The abundant families of solutions are obtained by the Jacobi elliptic function and it will give us soliton and solitary wave solutions. So, we extract the different types of solutions such as, dark, bright, singular, combine, periodic and mixed periodic. The unique physical problems are selected from a variety of the solutions that will help the reader for the verification and data experiment. The graphical behavior of the underlying models is represented in the form of 3D, line graphs and their corresponding contours for the various values of the parameters.

Two-grid finite difference method for 1D fourth-order Sobolev-type equation with Burgers’ type nonlinearity

This paper proposes a time two-grid finite difference (TTGFD) technique for computing numerical solution of the one-dimensional (1D) fourth-order Sobolev-type equation with Burgers-type nonlinearity. The proposed strategy mainly contains three computational stages. First, the fully nonlinear problem is approximated over a coarse grid with grid size τC. Second, an approximate solution is obtained on a fine grid with grid size τF based on the solution of the coarse grid using the Lagrangian interpolation formula. Finally, the linear problem is solved on the fine grid. Compared with the standard finite difference (SFD) scheme, an advantage of the TTGFD method is that it can maintain optimal accuracy while reducing the computational cost. Meanwhile, based on the reduced order and the discrete energy method, the conservative invariant and uniqueness of proposed method are demonstrated. In addition, the stability and convergence with the order O(τC2+τF2+h2) are evaluated in the L∞-norm, where h is the spatial step size. Finally, numerical results show the validity of the proposed strategy and support the theoretical results.

Extraction of soliton solutions for the time–space fractional order nonclassical Sobolev-type equation with unique physical problems

In this paper, we investigate the soliton solutions of the time–space fractional order nonclassical Sobolev-type (TSFNST) equation. Sobolev equations are used in thermodynamics, the flow of fluid through fractured rock, and other fields. There are certain significant partial differential equations having a third-order mixed derivative with regard to time and space that are now being used in real-world applications. The study aims to analyze the TSFNST equation analytically by employing the ϕ6-model expansion and modified G′/G2 expansion techniques respectively. The solution is obtained successfully in the Jacobi elliptic function solutions that will prove to us the hyperbolic, trigonometric, and rational solutions. Additionally, we select some unique physical problems of the solution. Lastly, the 3-dimensional and their corresponding contour plots are drawn by choosing the different values of constants

Existence, uniqueness and approximation of nonlocal fractional differential equation of sobolev type with impulses

<abstract><p>This paper is concerned with the study of nonlocal fractional differential equation of sobolev type with impulsive conditions. An associated integral equation is obtained and then considered a sequence of approximate integral equations. By utilizing the techniques of Banach fixed point approach and analytic semigroup, we obtain the existence and uniqueness of mild solutions to every approximate solution. Then, Faedo-Galerkin approximation is used to establish certain convergence outcome for approximate solutions. In order to illustrate the abstract results, we present an application as a conclusion.</p></abstract>

Optimal Control of Solutions to the Cauchy Problem for an Incomplete Semilinear Sobolev Type Equation of the Second Order

Optimal Control of Solutions to the Cauchy Problem for an Incomplete Semilinear Sobolev Type Equation of the Second Order

Stability of a Stationary Solution to One Class of Non-Autonomous Sobolev Type Equations

Stability of a Stationary Solution to One Class of Non-Autonomous Sobolev Type Equations