Integral Boundary Conditions Research Articles

Applicability of Darbo’s Fixed Point Theorem on the Existence of a Solution to Fractional Differential Equations of Sequential Type

In this article, we study the existence of the solution for the fractional differential equations of sequential type with nonlocal integral boundary conditions. The main results are established with the aid of Darbo’s fixed point theorem and Hausdorff’s measure of noncompactness method. The stability of the proposed fractional differential equation is also investigated via the Ulam–Hyer technique. In addition, an applied example that supports the theoretical results reached through this study is included.

On Coupled System of Langevin Fractional Problems with Different Orders of μ-Caputo Fractional Derivatives

In this paper, we study coupled nonlinear Langevin fractional problems with different orders of μ-Caputo fractional derivatives on arbitrary domains with nonlocal integral boundary conditions. In order to ensure the existence and uniqueness of the solutions to the problem at hand, the tools of the fixed-point theory are applied. An overview of the main results of this study is presented through examples.

Leray–Schauder Alternative for the Existence of Solutions of a Modified Coupled System of Caputo Fractional Differential Equations with Two Point’s Integral Boundary Conditions

In this paper, a coupled system of differential equations involving fractional order with integral boundary conditions is discussed. In the problem at hand, three main aspects that are existence, uniqueness, and stability have been investigated. Firstly, the contraction mapping principle is used to discuss the uniqueness of solutions for the proposed fractional system, and secondly, the existence of solutions for the problem is investigated based on Leray–Schauder’s alternative. Thirdly, the stability of the presented coupled system is discussed based on the Hyers–Ulam stability method. Finally, some examples have been given to confirm and illustrate the conclusion. The comparison between the current symmetrical results and the existing literature is deemed satisfactory. It was found that the presented fractional coupled system with two with integral boundary conditions is existent, unique, and stable.

Existence and Hyers-Ulam stability of solutions to the implicit second-order differential equation via fractional integral boundary conditions

Existence and Hyers-Ulam stability of solutions to the implicit second-order differential equation via fractional integral boundary conditions

Nonlinear two conformable fractional differential equation with integral boundary condition

"This paper deals with a boundary value problem for a nonlinear differential equation with two conformable fractional derivatives and integral boundary conditions. The results of existence, uniqueness and stability of positive solutions are proved by using the Banach contraction principle, Guo-Krasnoselskii's fixed point theorem and Hyers-Ulam type stability. Two concrete examples are given to illustrate the main results."

An optimal control problem for the systems with integral boundary conditions

In this paper, we consider an optimal control problem with a «pure», integral boundary condition. The Green’s function is constructed. Using contracting Banach mappings, a sufficient condition for the existence and uniqueness of a solution to one class of integral boundary value problems for fixed admissible controls is established. Using the functional increment method, the Pontryagin‘s maximum principle is proved. The first and second variations of the functional are calculated. Further, various necessary conditions for optimality of the second order are obtained by using variations of controls.

On the Spectrum of the Non-Selfadjoint Differential Operator with an Integral Boundary Condition and Negative Weight Function

In this paper, we shall study the spectral properties of the non-selfadjoint operator in the space $L_{\varrho }^{2}\left(\mathbb{R}_{+}\right) $ generated by the Sturm-Liouville differential equation \begin{equation*} -y^{^{\prime \prime }}+q\left( x\right) y=\omega ^{2}\varrho \left( x\right) y, \quad x \in \mathbb{R}_{+} \end{equation*} with the integral type boundary condition \begin{equation*} \int \limits_{0}^{\infty }G\left( x \right) y\left( x\right) dx+ \gamma y^{\prime }\left( 0\right) -\theta y\left( 0\right) =0 \end{equation*} and the non-standard weight function \begin{equation*} \varrho \left( x\right) =-1 \end{equation*} where $\left \vert \gamma \right \vert +\left \vert \theta \right \vert \neq 0$. There are an enormous number of papers considering the positive values of $ \varrho \left( x\right) $ for both continuous and discontinuous cases. The structure of the weight function affects the analytical properties and representations of the solutions of the equation. Differently from the classical literature, we used the hyperbolic type representations of the fundamental solutions of the equation to obtain the spectrum of the operator. Moreover, the conditions for the finiteness of the eigenvalues and spectral singularities were presented. Hence, besides generalizing the recent results, Naimark's and Pavlov's conditions were adopted for the negative weight function case.

Lyapunov inequality for a Caputo fractional differential equation with Riemann–Stieltjes integral boundary conditions

In this a Lyapunov‐type inequality is obtained for the fractional differential equation with Caputo derivative together with the boundary conditions where are functions of bounded variation, , and are nonnegative constants, and is a continuous function. We have divided the Greens function of this problem into two parts, and their upper bounds are obtained separately in order to obtain our results. The classical celebrated result of Lyapunov has become a consequence of our result.

A high-order reliable and efficient Haar wavelet collocation method for nonlinear problems with two point-integral boundary conditions

The primary goal of this study is to increase and improve the precision and order of convergence of the well-known Haar wavelet collocation method (HWCM) that is named as Higher order Haar wavelet collocation method (HHWCM). The HHWCM will then be applied to nonlinear ordinary differential equations with a variety of initial conditions, boundary conditions, periodic conditions, two-point conditions, integral conditions, and multi-point integral boundary conditions. The paper also contains important theorem about the convergence of the HHWCM with computational stability. The convergence of HHWCM is then compared to recently published works including the famous HWCM. In nonlinear case the quasilinearization process has been introduced to linearize the differential equation. Different orders of differential equations including homogeneous and non–homogeneous equations with constant and variable coefficients are also tested by HHWCM. The key advantages of the HHWCM include its easy to use, stability, convergence, high order accuracy, and efficiency under a range of boundary conditions. We have also implemented the HHWCM on nonlinear differential equation having no exact solution.

Control of age-structured population dynamics with intraspecific competition in context of bioreactors

Bioreactors are used for commercial manufacturing of products or as a simplified representation of wastewater treatment plants. If it is taken into account that individual organisms from the population in the bioreactor behave differently, an additional structure such as age can be added to the population. This leads to a nonlinear, hyperbolic, integro-partial differential equation (IPDE) with non-local integral boundary conditions (BC) in the modeling. The steady-states of this system can be uniquely determined by a substitution due to the non-local BC. Using the asymptotic properties of the system, the IPDE is split into finite-dimensional dynamics and infinite dimensional internal dynamics. The internal dynamics are globally exponentially stable. Therefore, a feedforward control based on the finite dimensional dynamics is designed, augmented with a feedback control to compensate for the remaining influence of the internal dynamics on the output. Finally, the control concept is supplemented by an observer. Global exponential attractiveness of certain reference trajectories is proven. The overall control concept is validated experimentally with a bioreactor. A system parameter is identified and it is shown that by applying the control concept, the biomass can track various reference trajectories with small errors.

CVEM-BEM Coupling for the Simulation of Time-Domain Wave Fields Scattered by Obstacles with Complex Geometries

Abstract In this paper, we present a numerical method based on the coupling between a Curved Virtual Element Method (CVEM) and a Boundary Element Method (BEM) for the simulation of wave fields scattered by obstacles immersed in homogeneous infinite media. In particular, we consider the 2D time-domain damped wave equation, endowed with a Dirichlet condition on the boundary (sound-soft scattering). To reduce the infinite domain to a finite computational one, we introduce an artificial boundary on which we impose a Boundary Integral Non-Reflecting Boundary Condition (BI-NRBC). We apply a CVEM combined with the Crank–Nicolson time integrator in the interior domain, and we discretize the BI-NRBC by a convolution quadrature formula in time and a collocation method in space. We present some numerical results to test the performance of the proposed approach and to highlight its effectiveness, especially when obstacles with complex geometries are considered.

Accelerated parameter-uniform numerical method for singularly perturbed parabolic convection-diffusion problems with a large negative shift and integral boundary condition

The singularly perturbed parabolic convection–diffusion equations with integral boundary conditions and a large negative shift are studied in this paper. The implicit Euler method for the temporal direction and the exponentially fitted finite difference scheme for the spatial direction are applied to formulate a parameter-uniform numerical method. The Simpson’s integration rule is used to handle the integral boundary condition. The Richardson extrapolation technique is applied to enhance the order of convergence of the method. The stability and uniform convergence analysis of the proposed method are studied. It is shown that the method is uniformly convergent with a convergence order of two in both temporal and spatial direction after Richardson extrapolation. Two test examples are considered to verify the validity of the proposed numerical scheme. The obtained numerical results confirm the theoretical estimates. The proposed method provide more accurate results and a higher order of convergence than methods available in the literature.

On the Existence Results for a Mixed Hybrid Fractional Differential Equations of Sequential Type

In this article, we study the existence of a solution to the mixed hybrid fractional differential equations of sequential type with nonlocal integral hybrid boundary conditions. The main results are established with the aid of Darbo’s fixed point theorem and Hausdorff’s measure of noncompactness method. The stability of the proposed fractional differential equation is also investigated using the Ulam–Hyer technique. In addition, an applied example that supports the theoretical results reached through this study is included.

Functional Impulsive Fractional Differential Equations Involving the Caputo-Hadamard Derivative and Integral Boundary Conditions

In this paper, we investigate the existence and uniqueness of solutions for functional impulsive fractional differential equations and integral boundary conditions. Our results are based on some fixed point theorems. Finally, we provide an example to illustrate the validity of our main results.

ON AN INVERSE PROBLEM WITH AN INTEGRAL OVERDETERMINATION CONDITION FOR THE BURGERS EQUATION

In this paper we consider one inverse problem for the Burgers equation with anintegral overdetermination and periodic boundary conditions in a domain that is trapezoid. Using an integral overdetermination, boundary and initial conditions, we reduce the inverse problem to the study of an already direct initial boundary value problemfor the loaded Burgers equation. Next, we use a one-to-one transformation of independent variables to move from a trapezoid to a rectangular domain, where we study an auxiliary problem, for which the methods of Faedo-Galerkin, a priori estimates and functional analysis have been proveda theorem on its unique solvability in Sobolev classes. Note that the obtained a priori estimates are uniform with respect to the summation index of the approximate solution and do not depend on time.Further, on the basis of this theorem, due to the correspondence of spaces, theorems on the unique solvability of the original inverse problem are proved. Also, for the selected initial data, the paper presents graphs of the initial-boundary problem for the loaded Burgers equation and the desired function of the inverse problem, which together constitute the solution of the original inverse problem.

Atangana–Baleanu–Caputo differential equations with mixed delay terms and integral boundary conditions

In this research work, we investigate a class of Atangana–Baleanu–Caputo (ABC) differential equations (DEs) having proportional and discrete delay terms accompanied by integral boundary conditions. We study the two important aspects, that is, the existence of the solution and stability analysis for the problem mentioned above. We confirm the existence of at least one solution via Schaefer's fixed‐point theorem and its uniqueness by Banach's fixed‐point theorem. Moreover, for stability analysis, we use the concept of Hyers–Ulam stability. For application purposes, we apply the main results to a numerical problem for specific values of the parameters involved.

Asymptotic Properties of a General Model of Immune Status

We consider a model of dynamics of the immune system. The model is based on three factors: occasional boosting and continuous waning of immunity and a general description of the period between subsequent boosting events. The antibody concentration changes according to a non-Markovian process. The density of the distribution of this concentration satisfies some partial differential equation with an integral boundary condition. We check whether this system generates a stochastic semigroup and we study the long-time behavior of this semigroup. In particular we prove a theorem on its asymptotic stability.

A HDG Method for Elliptic Problems with Integral Boundary Condition: Theory and Applications

A HDG Method for Elliptic Problems with Integral Boundary Condition: Theory and Applications

Uniqueness of solutions for aψ-Hilfer fractional integral boundary value problem with thep-Laplacian operator

AbstractIn this article, we discuss the existence of a unique solution to aψ\psi-Hilfer fractional differential equation involving thepp-Laplacian operator subject to nonlocalψ\psi-Riemann-Liouville fractional integral boundary conditions. Banach’s fixed point theorem is the main tool of our study. Examples are given for illustrating the obtained results.

Existence Results for Caputo Tripled Fractional Differential Inclusions with Integral and Multi-Point Boundary Conditions

In this study, based on Coitz and Nadler’s fixed point theorem and the non-linear alternative for Kakutani maps, existence results for a tripled system of sequential fractional differential inclusions (SFDIs) with integral and multi-point boundary conditions (BCs) in investigated. A practical examples are given to illustrate the obtained the theoretical results.