Hybrid Boundary Value Problems Research Articles

The Existence and Uniqueness Results of Solutions for a Fractional Hybrid Integro-differential System

This paper discuss two important results for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the researches and the advance in this field and also the importance of this subject in the modeling of nonlinear real phenomena corresponding to a great variety of events gives the motivation to study this boundary value problem. The results are as follow, the first result consider the existence and uniqueness results of solutions for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential system this result based on Krasnoslskii fixed point theorem for a sum of two operators, the second result is the uniqueness of solution for fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the main result is based on Banach fixed point theorem, both results comes after transforming the system into Volterra integral system then transform again into operator system, then using fixed point theory to prove the results, this articule was ended buy an example to well illustrat the results and ideas of proof.

Study of Hybrid Problems under Exponential Type Fractional-Order Derivatives

In this investigation, we develop a theory for the hybrid boundary value problem for fractional differential equations subject to three-point boundary conditions, including the antiperiodic hybrid boundary condition. On suggested problems, the third-order Caputo–Fabrizio derivative is the fractional operator applied. In this regard, the corresponding hybrid fractional integral equation is obtained by the Caputo–Fabrizio operator’s properties with the Green function’s aid. Then, we apply Dhage’s nonlinear alternative to the Schaefer type to prove the existence results. Finally, two examples are provided to confirm the validity of our main results.

Existence Theorems for Hybrid Fractional Differential Equations with ψ -Weighted Caputo–Fabrizio Derivatives

In this study, two classes of hybrid boundary value problems involving ψ -weighted Caputo–Fabrizio fractional derivatives are considered. Based on the properties of the given operator, we construct the hybrid fractional integral equations corresponding to the hybrid fractional differential equations. Then, we establish and extend the existence theory for given problems in the class of continuous functions by Dhage’s fixed point theory. Furthermore, as special cases, we offer further analogous and comparable conclusions. Finally, we give two examples as applications to illustrate and validate the results.

Nonlocal Hybrid Integro-Differential Equations Involving Atangana–Baleanu Fractional Operators

In this study, we develop a theory for the nonlocal hybrid boundary value problem for the fractional integro-differential equations featuring Atangana–Baleanu derivatives. The corresponding hybrid fractional integral equation is presented. Then, we establish the existence results using Dhage’s hybrid fixed point theorem for a sum of three operators. We also offer additional exceptional cases and similar outcomes. In order to demonstrate and verify the results, we provide an example as an application.

On existence results for hybrid \(\psi-\)Caputo multi-fractional differential equations with hybrid conditions

In this paper, we study the existence and uniqueness results of a fractional hybrid boundary value problem with multiple fractional derivatives of \(\psi-\)Caputo with different orders. Using a useful generalization of Krasnoselskii’s fixed point theorem, we have established results of at least one solution, while the uniqueness of solution is derived by Banach's fixed point. The last section is devoted to an example that illustrates the applicability of our results.

Nonlocal Fractional Hybrid Boundary Value Problems Involving Mixed Fractional Derivatives and Integrals via a Generalization of Darbo’s Theorem

In this work, a new existence result is established for a nonlocal hybrid boundary value problem which contains one left Caputo and one right Riemann–Liouville fractional derivatives and integrals. The main result is proved by applying a new generalization of Darbo’s theorem associated with measures of noncompactness. Finally, an example to justify the theoretical result is also presented.

Existence of solutions for impulsive hybrid boundary value problems to fractional differential systems

In this paper, the existence of solutions for impulsive mixed boundary value problems involving Caputo fractional derivatives is obtained. And then, our conclusions are based on Krasnoselskii's fixed point theorem and Arzela-Ascoli theorem. Finally, some examples are given to illustrate the main results.

On the new fractional hybrid boundary value problems with three-point integral hybrid conditions

We investigate some new class of hybrid type fractional differential equations and inclusions via some nonlocal three-point boundary value conditions. Also, we provide some examples to illustrate our results.

Boundary Value Problems for Hybrid Caputo Fractional Differential Equations

In this paper, we discuss the existence of solutions for a hybrid boundary value problem of Caputo fractional differential equations. The main tool used in our study is associated with the technique of measures of noncompactness. As an application, we give an example to illustrate our results.

Lyapunov Type Inequality for a Nonlinear Fractional Hybrid Boundary Value Problem

In this paper, we present a Lyapunov type inequality for a nonlinear fractional hybrid boundary value problem. We illustrate the main result through a series of examples.

Combination Approach of Domain-Type and Boundary-Type Meshless Methods for Solving Hybrid Boundary-Value Problem of Homogeneous and Inhomogeneous Elliptic PDEs

The combination approach of the domain-type and boundary-type meshless methods has been applied to the hybrid boundary-value problem composed of the homogeneous and the inhomogeneous elliptic partial differential equations (PDEs). In addition, the performance of the proposed method has been investigated numerically. The results of computations show that the convergent rate is almost the same value regardless of the wave number. Moreover, the accuracy of the proposed method is degraded with an increase in the value of the wave number. Furthermore, the numerical solution of the proposed method has been obtained by using the GMRES method without the restart coefficient. Therefore, it is found that the proposed method can be used as one of the tools for solving the hybrid boundary-value problem composed of the homogeneous and the inhomogeneous elliptic PDEs.

A nonlocal hybrid boundary value problem of caputo fractional integro-differential equations

In this paper, we discuss the existence of solutions for a nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations. Our main result is based on a hybrid fixed point theorem for a sum of three operators due to Dhage, and is well illustrated with the aid of an example.

Lyapunov’s type inequalities for hybrid fractional differential equations

We investigate new results about Lyapunov-type inequalities by considering hybrid fractional boundary value problems. We give necessary conditions for the existence of nontrivial solutions for a class of hybrid boundary value problems involving Riemann-Liouville fractional derivative of order $2<\alpha\le3$ . The investigation is based on a construction of Green’s functions and on finding its corresponding maximum value. In order to illustrate the results, we provide numerical examples.

Existence of solutions for a fractional hybrid boundary value problem via measures of noncompactness in Banach algebras

We study the existence of solutions for the following fractional hybrid boundary value problem $$ \cases \displaystyle D_{0^+}^{\alpha}\bigg[\frac{x(t)}{f(t,x(t))}\bigg]+g(t,x(t))=0, &0< t< 1,\\ x(0)=x(1)=0, \endcases $$ where $1< \alpha\leq 2$ and $D_{0^+}^{\alpha}$ denotes the Riemann-Liouville fractional derivative. The main tool is our study is the technique of measures of noncompactness in the Banach algebras. Some examples are presented to illustrate our results. Finally, we compare the results of paper with the ones obtained by other authors.

On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions

We investigate sufficient conditions for existence and uniqueness of solutions for a coupled system of fractional order hybrid differential equations (HDEs) with multi-point hybrid boundary conditions given by $$\begin{aligned}& \mathcal{D}^{\omega}\biggl(\frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\biggr)=-\mathcal {K}_{1}\bigl(t,x(t),z(t)\bigr), \quad\omega\in(2,3], \\& \mathcal{D}^{\epsilon}\biggl(\frac{z(t)}{\mathcal {G}(t,x(t),z(t))}\biggr)=-\mathcal{K}_{2} \bigl(t,x(t),z(t)\bigr), \quad\epsilon\in (2,3],\\& \frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\bigg|_{t=1}=0,\qquad \mathcal{D}^{\mu}\biggl( \frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\biggr)\bigg|_{t= \delta_{1} }=0,\qquad x^{(2)}(0)=0, \\& \frac{z(t)}{\mathcal{G}(t,x(t),z(t))}\bigg|_{t=1}=0, \qquad\mathcal{D}^{\nu}\biggl( \frac{z(t)}{\mathcal{G}(t,x(t),z(t))}\biggr)\bigg|_{t= \delta_{2}}=0,\qquad z^{(2)}(0)=0, \end{aligned}$$ where $t\in[0,1]$ , $\delta_{1}, \delta_{2}, \mu, \nu\in(0,1)$ , and $\mathcal{D}^{\omega}$ , $\mathcal{D}^{\epsilon}$ , $\mathcal{D}^{\mu}$ and $\mathcal{D}^{\nu}$ are Caputo’s fractional derivatives of order ω, ϵ, μ and ν, respectively, $\mathcal{K}_{1}, \mathcal{K}_{2}\in C([0,1]\times\mathcal{R}\times \mathcal{R},\mathcal{R} )$ and $\mathcal{G},\mathcal{H}\in C([0,1]\times\mathcal{R}\times\mathcal {R},\mathcal{R} - \{0\} )$ . We use classical results due to Dhage and Banach’s contraction principle (BCP) for the existence and uniqueness of solutions. For applications of our results, we include examples.

Results for Mild solution of fractional coupled hybrid boundary value problems

Abstract The study of coupled system of hybrid fractional differential equations (HFDEs) needs the attention of scientists for the exploration of its different important aspects. Our aim in this paper is to study the existence and uniqueness of mild solution (EUMS) of a coupled system of HFDEs. The novelty of this work is the study of a coupled system of fractional order hybrid boundary value problems (HBVP) with n initial and boundary hybrid conditions. For this purpose, we are utilizing some classical results, Leray–Schauder Alternative (LSA) and Banach Contraction Principle (BCP). Some examples are given for the illustration of applications of our results.

Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations

In this paper, we discuss some existence results for a two-point boundary value problem involving nonlinear impulsive hybrid differential equation of fractional order q ∈ ( 1 , 2 ] . Our results are based on contraction mapping principle and Krasnoselskii’s fixed point theorem.

Diffraction of surface waves by floating elastic plates

Abstract Based on the dynamical theories of water waves and Mindlin thick plates, the diffraction of surface waves by a floating elastic plate is presented by using the Wiener–Hopf technique. Firstly, the problem is related to a wave guide in water of finite depth, which is analysed to determine the poles. The resulting hybrid boundary value problem is reduced to solving an infinite system of linear algebraic equations. The results obtained are compared with those calculated by an alternative analysis, and with experimental data. Finally, the effects of the geometric and physical parameters on the distribution of deflection and bending moments in plates are analysed and discussed.