Integral Boundary Research Articles

Integral Operators in b-Metric and Generalized b-Metric Spaces and Boundary Value Problems

We study fixed-point theorems of contractive mappings in b-metric space, cone b-metric space, and the newly introduced extended b-metric space. To generalize an existence and uniqueness result for the so-called Φs functions in the b-metric space to the extended b-metric space and the cone b-metric space, we introduce the class of ΦM functions and apply the Hölder continuous condition in the extended b-metric space. The obtained results are applied to prove the existence and uniqueness of solutions and positive solutions for nonlinear integral equations and fractional boundary value problems. Examples and numerical simulation are given to illustrate the applications.

On the Solutions of Nonlinear Implicit ω-Caputo Fractional Order Ordinary Differential Equations with Two-Point Fractional Derivatives and Integral Boundary Conditions in Banach Algebra

This article delves into the analysis of nonlinear implicit ω-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives and integral boundary conditions within the context of Banach algebra. The primary focus is on demonstrating the existence and uniqueness of solutions for these complex fractional differential equations by utilizing Banach's and Krasnoselskii's fixed point theorems. Furthermore, the study explores the stability of these solutions through the Ulam-Hyers and Ulam-Hyers-Rassias stability criteria, thereby assessing the robustness of the proposed model. To illustrate the versatility of the generalized model, several special cases are examined, showcasing its ability to encompass various classical models. The practical applicability of the theoretical findings is underscored through a numerical example, which demonstrates the feasibility and relevance of the proposed methodology. This thorough investigation advances the comprehension of nonlinear fractional differential equations with integral boundary conditions, highlighting the intricate relationship between fractional derivatives, nonlinearities, and integral terms. The results offer significant insights into the behavior and stability of solutions within this demanding mathematical framework.

Freestanding Penta-Twinned Pd-Ag Nanosheets.

2D metal nanosheets have attracted significant attention as efficient catalysts for various important chemical reactions. However, the development of metal nanosheets with controlled compositions and morphologies has been slow due to the challenges associated with synthesizing thermodynamically unfavorable 2D structures. Herein, we report a synthesis route of freestanding Pd-Ag penta-twinned nanosheets (Pd-Ag ptNSs) with distinct 5-fold twin boundaries. Through the coreduction of Ag and Pd precursors on presynthesized Pd ptNSs, Ag could be homogeneously alloyed with Pd, leading to the formation of well-defined Pd-Ag ptNSs. The promotional effects of the bimetallic composition, 2D structure, and twin boundaries on catalysis were studied by using Pd-Ag ptNS-catalyzed H2 production from formic acid decomposition as a model reaction. Notably, the catalytic activity of the Pd-Ag ptNSs drastically outperformed those of monometallic, bimetallic, and 3D counterparts, such as Pd ptNSs, Pd-Ag nanosheets without a TB, and Pd-Ag octahedral nanocrystals, demonstrating the promising potential of the integration of twin boundaries and multiple compositions in the development of high-performance 2D nanocatalysts.

Surface energy and elementary excitations of the XYZ spin chain with integrable open boundary fields

We study the thermodynamic limit of the anisotropic XYZ spin chain with non-diagonal integrable open boundary conditions. Although the U(1)-symmetry is broken, by using the new parametrization scheme, we exactly obtain the surface energy and the excitation energy of the system, which has solved the difficulty in the inhomogeneous T − Q relation. With the boundary parameters in the regions making the Hamiltonian Hermitian, we have obtained the distribution patterns of the zeros of the eigenvalue of the transfer matrix for the ground state and the excited ones. We find that the surface and excitation energies depend on the parities of sites number N, due to the long-range Neel order in the bulk. The easy-axis and thermodynamic limit for all the regions of boundary parameters are studied. We also obtain the physical quantities in the thermodynamic limit of boundary XXZ model by taking the trigonometric limit.

Characteristics of solution to singular fractional differential equation with two Riemann-Stieltjesdintegral boundary value conditions

The purpose of this paper is to study unique solution and iterative sequence of approximate solution for uniformly approaching unique solution to a new class of singular fractional differential equations with two kinds of Riemann-Stieltjes integral boundary value conditions by using some fixed point theorems. Because of different properties of the nonlinear terms and complexity of the boundary conditions in equations, we first probe several fixed point theorems of sum-type operators which expand many existing works in this research area. It is essential to point out that some conditions in our works greatly simplify the proof process of fixed point theorems. By applying the operator conclusions obtained in this paper, some sufficient conditions that guarantee the existence and uniqueness of solutions to singular differential equations are obtained, two iterative schemes that uniformly converges to the unique solution are given which provide computational methods of approximating solutions. As applications, some examples are provided to illustrate our main results.

Numerical solution for third order singularly perturbed turning point problems with integral boundary condition

Numerical solution for third order singularly perturbed turning point problems with integral boundary condition

Power-law fluid flow in the entrance region between two parallel plates

Flow of non-Newtonian fluids between parallel plates has many uses including heat transfer applications. Many non-Newtonian fluids can be categorized as pseudoplastic fluids. A broader category, power-law fluids, covers in addition dilatant and Newtonian fluids. Models for developing power-law fluid flow between parallel plates include computational and integral boundary layer methods. Most published studies include partial results and apply to one of the three categories. The present work addresses the three cases and exclusively shows the transition from developing to fully developed flow. In the inlet region, viscous effects are restricted to the boundary layer. The present approach is based on the inlet-filled region concept, and includes an additional zone, called filled region in which viscous effects are present in the whole region, with the flow asymptotically reaching the fully developed solution. The integral boundary layer method is used in both regions to solve for the flow. The results are found to be in good agreement with published Newtonian, pseudoplastic and dilatant fluid results for boundary layer thickness, centerline velocity, local friction factor, pressure drop profiles, and entrance region size. The present model is fundamentally analytical with minor computational work required.

Ulam–Hyers Stability and Simulation of a Delayed Fractional Differential Equation with Riemann–Stieltjes Integral Boundary Conditions and Fractional Impulses

In this article, we delve into delayed fractional differential equations with Riemann–Stieltjes integral boundary conditions and fractional impulses. By using differential inequality techniques and some fixed-point theorems, some novel sufficient assessments for convenient verification have been devised to ensure the existence and uniqueness of solutions. We further employ the nonlinear analysis to reveal that this problem is Ulam–Hyers (UH) stable. Finally, some examples and numerical simulations are presented to illustrate the reliability and validity of our main results.

Isogeometric boundary element method analysis for dielectric target shape optimization in electromagnetic scattering.

In order to optimize the overall form of electromagnetic scattering in two-dimensional dielectric media, this work offers a frequency-domain boundary element method based on isogeometric analysis. The Isogeometric boundary element method (IGABEM) is used to guarantee geometric correctness during optimization and prevent over-refinement of the mesh. Non-uniform rational B-splines are used to discretize the boundary integrals of the model, enabling rapid numerical computation while ensuring high accuracy. Furthermore, as an alternative model for electromagnetic scattering shape optimization issues, a gray wolf optimizer-based back-propagation neural network (GWO-ANN) is created, with radar cross-section (RCS) as the objective function. Finally, the GWO-ANN is used as a surrogate model for shape optimization in multi-frequency electromagnetic scattering problems with the RCS as the objective function. In computational examples, this algorithm efficiently and accurately solves electromagnetic scattering problems under multiple frequencies.

On the existence of a positive solution to a boundary value problem for a third-order nonlinear functional differential equation with an integral boundary condition at one of the ends of the segment

The article studies the existence of positive solutions on the segment [0,1] of a two-point boundary value problem for one nonlinear third-order functional differential equation with an integral boundary condition at one of the ends of the segment. Using the Go–Krasnoselsky fixed point theorem and some properties of the Green's function of the corresponding differential operator, sufficient conditions for the existence of at least one positive solution to the problem under consideration are obtained. An example is given to illustrate the results obtained.

A Two-Dimensional Nonlocal Fractional Parabolic Initial Boundary Value Problem

In this paper, we investigate a two-dimensional singular fractional-order parabolic partial differential equation in the Caputo sense. The partial differential equation is supplemented with Dirichlet and weighted integral boundary conditions. By employing a functional analysis method based on operator theory techniques, we prove the existence and uniqueness of the solution to the posed nonlocal initial boundary value problem. More precisely, we establish an a priori bound for the solution from which we deduce the uniqueness of the solution. For proof of its existence, we use various density arguments.

ĐÁNH GIÁỔNĐỊNHCHOPHƯƠNGTRÌNH TRUYỀNNHIỆTNGƯỢCTHỜIGIAN VỚICÁCĐIỀUKIỆNBIÊNNEUMANNVÀTÍCHPHÂN

In this paper, we first prove that the heat equation backward in time with Neumann and integral boundary conditions is an ill-posed problem. Then, we establish a stability estimate of Hölder type for this ill-posed problem. Keywords: Heat equation backward; ill-posed problem; stability estimate.

Solvability of a sixth‐order boundary value problem with multi‐point and multi‐term integral boundary conditions

This paper aims to investigate the existence and uniqueness of solutions for a sixth‐order differential equation involving nonlocal and integral boundary conditions. Firstly, we obtain the properties of the relevant Green's functions. The existence result of at least one nontrivial solution is obtained by applying the Krasnoselskii–Zabreiko fixed point theorem. Moreover, we also establish the existence of unique solution to the considered problem via Hölder and Minkowski inequalities and Rus's theorem. Finally, two numerical examples are included to show the applicability of our main results.

A comparative study on numerical methods for Fredholm integro-differential equations of convection-diffusion problem with integral boundary conditions

This paper numerically solves Fredholm integro-differential equations with small parameters and integral boundary conditions. The solution of these equations has a boundary layer at the right boundary. A central difference scheme approximates the second-order derivative, a backward difference (upwind scheme) approximates the first-order derivative, and the trapezoidal rule is used for the integral term with a Shishkin mesh. It is shown that theoretically, the proposed scheme is uniformly convergent with almost first-order convergence. Further to improve the order of convergence from first order to second order, we use the post-processing and the hybrid scheme. Two numerical examples are computed to support the theoretical results.

(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.

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Solvability of a Hadamard fractional boundary value problem with multi‐term integral and Hadamard fractional derivative boundary conditions

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results.

Study of Caputo fractional derivative and Riemann–Liouville integral with different orders and its application in multi‐term differential equations

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of ‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration.

A deep learning classification framework for research methods of marine protected area management

The latest emerging transdisciplinary marine protected area (MPA) research scheme requires efficient approaches for theoretically based and data-driven method integration. However, due to the rapid development and diversification of research methods, it is growingly difficult to locate new methods in methodological dimensions and integrate them to the utmost utility. This study proposes a deep learning-based classification framework for MPA management methods focused particularly on data and theory capabilities using natural language processing (NLP). It extracted keywords from academic sources and performed clustering based on semantic similarity, generating benchmark texts for abstract labeling. By training the deep learning NLP model and analyzing the abstracts of 9049 MPA management empirical research articles from 1986 to 2024, the data and theory scores were attributed to each article, and a total of 19 major method categories and 110 segment branches were identified in qualitative, quantitative, and mixed genres. Combination types of research methods were summarized, yielding the data-theory neutralization principle where the average data and theory scores tend to approximate 0.50. Applying the principle broadens traditional boundaries for method integration and extends method synthesis to higher numbers, generating a practical research 2paradigm for future MPA research. Implications include bridging social and ecological data, theorizing emergent challenges in complex systems and integrating theory construction and data science. The framework is applicable to quantification of other environmental management disciplines and can serve as guidance for multidisciplinary method integration.© 2017 Elsevier Inc. All rights reserved.

Existence of Positive Solutions of a Fractional Dynamic Equation Involving Integral Boundary Conditions on Time Scales

Existence of Positive Solutions of a Fractional Dynamic Equation Involving Integral Boundary Conditions on Time Scales