Nonlocal Integral Boundary Conditions Research Articles

(k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional Langevin differential equation having multipoint boundary conditions

The primary objective of this manuscript is to investigate the existence and uniqueness of solutions for the Langevin (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation of different orders with multipoint nonlocal fractional integral boundary conditions. We consider the generalized version of the Hilfer fractional diferential equation called as (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation. We provide some significant outcomes about (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional Langevin differential equation that requires deriving equivalent fractional integral equation to (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer Langevin fractional differential equation. The existence result is established using the Krasnoselskii’s fixed-point theorem, while the uniqueness is addressed with the help of Banach contraction principle. Additionally, we investigate the different forms of Ulam stability for the solution of the mentioned problem, under specific conditions. To validate our main outcomes, we present a detailed example at the end of the manuscript.

Solutions of Second-Order Nonlinear Implicit ψ-Conformable Fractional Integro-Differential Equations with Nonlocal Fractional Integral Boundary Conditions in Banach Algebra

In this paper, we introduce and thoroughly examine new generalized ψ-conformable fractional integral and derivative operators associated with the auxiliary function ψ(t). We rigorously analyze and confirm the essential properties of these operators, including their semigroup behavior, linearity, boundedness, and specific symmetry characteristics, particularly their invariance under time reversal. These operators not only encompass the well-established Riemann–Liouville and Hadamard operators but also extend their applicability. Our primary focus is on addressing complex fractional boundary value problems, specifically second-order nonlinear implicit ψ-conformable fractional integro-differential equations with nonlocal fractional integral boundary conditions within Banach algebra. We assess the effectiveness of these operators in solving such problems and investigate the existence, uniqueness, and Ulam–Hyers stability of their solutions. A numerical example is presented to demonstrate the theoretical advancements and practical implications of our approach. Through this work, we aim to contribute to the development of fractional calculus methodologies and their applications.

Existence and stability results for a Langevin system with Caputo–Hadamard fractional operators

This paper is devoted to analyzing a new model of a coupled Langevin system with fractional operators under nonlocal antiperiodic integral boundary conditions. This model involves nonlinear Langevin fractional equations with Caputo–Hadamard and Caputo fractional operators. Also, the existence and uniqueness of solutions to the suggested model have been investigated by the fixed-point technique. Moreover, the Hyers–Ulam stability of the solutions has been discussed. Finally, we provide an illustrative example to support the theoretical results.

On the solutions of a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions

We consider a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions and we obtain an existence result for this problem in the case when the set-valued maps have nonconvex values.

Existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions with nonlocal integral boundary conditions

Existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions with nonlocal integral boundary conditions

On the reconstruction of an integro-differential Dirac operator with parameter-dependent nonlocal integral boundary conditions from the nodal data

On the reconstruction of an integro-differential Dirac operator with parameter-dependent nonlocal integral boundary conditions from the nodal data

Existence results for a coupled system of nonlinear fractional functional differential equations with infinite delay and nonlocal integral boundary conditions

<abstract><p>This article is devoted to studying a new class of nonlinear coupled systems of fractional differential equations supplemented with nonlocal integro-coupled boundary conditions and affected by infinite delay. We first transform the boundary value problem into a fixed-point problem, and, with the aid of the theory of infinite delay, we assume an appropriate phase space to deal with the obtained problem. Then, the existence result of solutions to the given system is investigated by employing Schaefer's fixed-point theorem, while the uniqueness result is established in view of the Banach contraction mapping principle. The illustrative examples are constructed to ensure the availability of the main results.</p></abstract>

A higher-order collocation technique based on Haar wavelets for fourth-order nonlinear differential equations having nonlocal integral boundary conditions

This article encounters the use of two wavelet methods, namely the collocation method based on Haar wavelets (CMHW) and the higher-order collocation method based on Haar wavelets (HCMHW), to solve linear and nonlinear fourth-order differential equations with different forms of given data such as two-point boundary conditions and two-point integral boundary conditions. Managing these types of boundary conditions can be challenging in numerical methods. However, in this study, these types of equations are handled in a simple manner using the Haar wavelet expressions, as provided in the given information. In the case of nonlinear problems, the quasi-linearization technique is introduced to linearize the equation. Nonlinear fourth-order differential equations are transformed into a simple linear system of algebraic equations using the quasi-linearization technique and Haar wavelets. These equations are then solved very easily to find the solution of the differential equations. The convergence rate and stability of both the methods are studied in details. The convergence rate of the proposed HCMHW is faster than the CMHW (2+2s>2,s=1,2…). Some of the examples are given to indicate the better performance and accuracy of the proposed HCMHW.

Hybrid differential inclusion with nonlocal, infinite-point or Riemann-Stieltjes integral boundary conditions

Hybrid differential inclusion with nonlocal, infinite-point or Riemann-Stieltjes integral boundary conditions

Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator

In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented in this study establish the existence, uniqueness, and Hyers–Ulam (HU) stability of the solution to the proposed equation. We achieved our main result by using the Banach contraction mapping principle and Krasonoselskii’s fixed point theorem. Furthermore, we introduce an application to demonstrate the validity of the results of our findings.

Solving 2D-integro-differential problems with nonlocal boundary conditions via a matrix formulated approach

A new operational matrix based approach is studied for numerical solution of 2D-integro-differential equations with non-local (integral) boundary conditions whose arise in some physical problems. Some important theoretical results are presented to reduce complexity and computational costs of the proposed method. We also give an error estimation which will be useful in estimating the error of approximate solution for the problems that we do not have any information about their exact solution. Illustrative numerical examples are also given to clarify the performance and accuracy of the new method.

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.

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.

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.

Investigation of a Coupled System of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Hadamard Fractional Integral Boundary Conditions

We investigate the existence criteria for solutions of a nonlinear coupled system of Hilfer–Hadamard fractional differential equations of different orders complemented with nonlocal coupled Hadamard fractional integral boundary conditions. The desired results are accomplished with the aid of standard fixed-point theorems. We emphasize that the fixed point approach is one of the effective methods to establish the existence results for boundary value problems. Examples illustrating the obtained results are constructed.

Existence and Uniqueness Results for Different Orders Coupled System of Fractional Integro-Differential Equations with Anti-Periodic Nonlocal Integral Boundary Conditions

This paper presents a new class of boundary value problems of integrodifferential fractional equations of different order equipped with coupled anti-periodic and nonlocal integral boundary conditions. We prove the existence and uniqueness criteria of the solutions by using the Leray-Schauder alternative and Banach contraction mapping principle. Examples are constructed for the illustration of our results.

Устойчивость и сходимость разностной схемы, аппроксимирующей нелокальную краевую задачу для нагруженного уравнения влагопереноса с производными дробного порядка

We study a boundary value problem for a loaded moisture transfer equation with two Gerasimov - Caputo fractional derivatives of different orders (α, β), variable coefficients, and a nonlocal integral boundary condition. On a uniform grid a difference scheme of order of approximation 𝑂(ℎ2 + 𝜏2) for 𝛼 = 𝛽 and 𝑂(ℎ2 + 𝜏2−𝑚𝑎𝑥⁡{𝛼,𝛽}) for 𝛼 ≠ 𝛽 is constructed. The study is carried out by the method of energy inequalities. For different ratios between the orders of fractional derivatives α and β, for solving the posed nonlocal boundary value problem, a priori estimates in the difference interpretation are obtained, from which the uniqueness of the solution of the posed problem, the continuous and uniform dependence of the solution on the input data, and the conv ergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.

Existence of solutions for fractional differential equations with Ψ-Hilfer fractional derivative with nonlocal integral boundary conditions

Existence of solutions for fractional differential equations with Ψ-Hilfer fractional derivative with nonlocal integral boundary conditions

Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρk,ϕk)-Hilfer Fractional Integro-Differential Equations

In this paper, we establish the existence and stability results for the (ρk,ϕk)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρk,ϕk)-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results.