Nonseparated Boundary Conditions Research Articles

Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions

Fractional Sequential Coupled Systems of Hilfer and Caputo Integro-Differential Equations with Non-Separated Boundary Conditions

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained.

Solvability of the inverse spectral problem for Sturm-Liouville operator with spectral parameter in the boundary condition

This paper deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs, and some number.

SOLVABILITY OF THE INVERSE PROBLEM FOR DIFFERENTIAL OPERATOR WITH SPECTRAL PARAMETER IN THE BOUNDARY CONDITION

Abstract. The paper considers the inverse spectral problem of reconstructing the diffusion operator with nonseparated boundary conditions, one of which depends quadratically on the spectral parameter. Sufficient conditions for the solvability of the inverse problem are obtained. The spectra of two boundary value problems and a certain sequence of signs are used as spectral data.

Inverse Sturm–Liouville problems with polynomials in nonseparated boundary conditions

An nonself-adjoint Sturm–Liouville problem with two polynomials in nonseparated boundary conditions are considered. It is shown that this problem have an infinite countable spectrum. The corresponding inverse problems is solved. Criterions for unique reconstruction of the nonself-adjoint Sturm-Liouville problem by eigenvalues of this problem and the spectral data of an additional problem with separated boundary conditions are proved. Schemes for unique reconstruction of the Sturm-Liouville problems with polynomials in nonseparated boundary conditions and corresponding examples are given

Eigenvalue problems for a class of Sturm-Liouville operators on two different time scales

In this study, we consider a boundary value problem generated by the Sturm-Liouville equation with a frozen argument and with non-separated boundary conditions on a time scale. Firstly, we present some solutions and the characteristic function of the problem on an arbitrary bounded time scale. Secondly, we prove some properties of eigenvalues and obtain a formulation for the eigenvalues-number on a finite time scale. Finally, we give an asymptotic formula for eigenvalues of the problem on another special time scale: $\mathbb{T}=[\alpha,\delta_{1}]\bigcup[\delta_{2},\beta].$

Solving nonlinear boundary value problems by a boundary shape function method and a splitting and linearizing method

Abstract In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method.

Some Existence Results for a System of Nonlinear Sequential Fractional Differential Equations with Coupled Nonseparated Boundary Conditions

This article concerns with the existence and uniqueness theory of solutions for sequential fractional differential system involving Caputo fractional derivatives of order 1<alpha, beta<2 with coupled nonseparated boundary conditions. The standard tools of the fixed point theory were used to establish the main results. Application is introduced to show the validity of our results.

Analysis of a Coupled System of Nonlinear Fractional Langevin Equations with Certain Nonlocal and Nonseparated Boundary Conditions

In this article, we use some fixed point theorems to discuss the existence and uniqueness of solutions to a coupled system of a nonlinear Langevin differential equation which involves Caputo fractional derivatives of different orders and is governed by new type of nonlocal and nonseparated boundary conditions consisting of fractional integrals and derivatives. The considered boundary conditions are totally dissimilar than the ones already handled in the literature. Additionally, we modify the Adams-type predictor-corrector method by implicitly implementing the Gauss–Seidel method in order to solve some specific particular cases of the system.

New Existence of Solutions for Fractional Integro-Differential Equations with Nonseparated Boundary Conditions

Results reported in this article prove the existence and uniqueness of solutions for a class of nonlinear fractional integro-differential equations supplemented by nonseparated boundary value conditions. We consider a new norm to establish the existence of solution via Krasnoselskii fixed point theorem; however, the uniqueness results are obtained by applying the contraction mapping principle. Some examples are provided to illustrate the results.

Stability analysis of boundary value problems for Caputo proportional fractional derivative of a function with respect to another function via impulsive Langevin equation

In this paper, we discuss existence and stability results for a new class of impulsive fractional boundary value problems with non-separated boundary conditions containing the Caputo proportional fractional derivative of a function with respect to another function. The uniqueness result is discussed via Banach's contraction mapping principle, and the existence of solutions is proved by using Schaefer's fixed point theorem. Furthermore, we utilize the theory of stability for presenting different kinds of Ulam's stability results of the proposed problem. Finally, an example is also constructed to demonstrate the application of the main results.

Existence of solutions for a coupled system of Caputo type fractional-order differential inclusions with non-separated boundary conditions on multivalued maps

Existence of solutions for a coupled system of Caputo type fractional-order differential inclusions with non-separated boundary conditions on multivalued maps

Analysis of a coupled system of fractional differential equations with non-separated boundary conditions

Solutions to fractional differential equations is an emerging part of current research, since such equations appear in different applied fields. A study of existence, uniqueness, and stability of solutions to a coupled system of fractional differential equations with non-separated boundary conditions is the main target of this paper. The existence and uniqueness results are obtained by employing the Leray–Schauder fixed point theorem and the Banach contraction principle. Additionally, we examine different types of stabilities in the sense of Ulam–Hyers such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. To prove the effectiveness of our main results, we study a few interesting examples.

Boundary shape function method for nonlinear BVP, automatically satisfying prescribed multipoint boundary conditions

It is difficult to exactly and automatically satisfy nonseparable multipoint boundary conditions by numerical methods. With this in mind, we develop a novel algorithm to find solution for a second-order nonlinear boundary value problem (BVP), which automatically satisfies the multipoint boundary conditions prescribed. A novel concept of boundary shape function (BSF) is introduced, whose existence is proven, and it can satisfy the multipoint boundary conditions a priori. In the BSF, there exists a free function, from which we can develop an iterative algorithm by letting the BSF be the solution of the BVP and the free function be another variable. Hence, the multipoint nonlinear BVP is properly transformed to an initial value problem for the new variable, whose initial conditions are given arbitrarily. The BSF method (BSFM) can find very accurate solution through a few iterations.

Inverse spectral problems for differential operators with non-separated boundary conditions

Abstract We give a short review of results on inverse spectral problems for second-order differential operators on an interval with non-separated boundary conditions. We pay the main attention to the most important nonlinear inverse problems of recovering coefficients of differential operators from given spectral characteristics. In the first part of the review, we provide the main results and methods related to inverse problems for Sturm–Liouville operators with non-separated boundary conditions: periodic, quasi-periodic and Robin-type boundary conditions. At the end, we present the main results on inverse problems for differential pencils with non-separated boundary conditions.

On the Generalization of a Solution for a Class of Integro-Differential Equations with Nonseparated Integral Boundary Conditions

In this paper, the existence and uniqueness results of the generalization nonlinear fractional integro-differential equations with nonseparated type integral boundary conditions are investigated. A natural formula of solutions is derived and some new existence and uniqueness results are obtained under some conditions for this class of problems by using standard fixed point theorems and Leray–Schauder degree theory, which extend and supplement some known results. Some examples are discussed for the illustration of the main work.

An Inverse Problem for Discontinuous Sturm–Liouville Equations with Non-separated Boundary Conditions

In this article, a discontinuous Sturm–Liouville boundary value problem with non-separated boundary conditions is considered. It is proven that the potential function is uniquely established by either the Weyl function or by two spectra.

Uniqueness of Reconstruction of an nth-Order Differential Operator with Nonseparated Boundary Conditions by Several Spectra

A uniqueness theorem for reconstruction of an nth-order differential operator with nonseparated boundary conditions from several spectra is proved. This theorem is based on the results of E.A. Baranova.

Existence of solutions for equations and inclusions of multiterm fractional q-integro-differential with nonseparated and initial boundary conditions

The goal of this paper is to investigate existence of solutions for the multiterm nonlinear fractional q-integro-differential {}^{c}D_{q}^{alpha } u(t) in two modes equations and inclusions of order alphain(n -1, n], with non-separated boundary and initial boundary conditions where the natural number n is more than or equal to five. We consider a Carathéodory multivalued map and use Leray–Schauder and Covitz–Nadler famous fixed point theorems for finding solutions of the inclusion problems. Besides, we present results whenever the multifunctions are convex and nonconvex. Lastly, we give some examples illustrating the primary effects.

Identification of Nonseparated Boundary Conditions

We consider the problem of identification of nonseparated boundary conditions by five eigenvalues. Based on the Plucker conditions that appear in the problem of matrix recovery by its maximal-size minors, we construct the well-posedness set for this problem. We solve the problem of identification of the matrix of nonseparated boundary conditions in terms of the characteristic determinant of the corresponding spectral problem. The corresponding examples are presented.