Solution Of Integral Equation Research Articles

Positivity results to iterative system of higher order boundary value problems

The present research explores the existence of positive solutions for the iterative system of higher-order differential equations with integral boundary conditions that include a non-homogeneous term. To address the boundary value problem, the solution is expressed as a solution of an equivalent integral equation involving kernels. Subsequently, bounds for these kernels are determined to facilitate further analysis. The primary tool employed in this study is the Guo-Krasnosel’skii fixed-point theorem, which is utilized to establish the existence of positive solutions within a cone of a Banach space. This approach enables a rigorous exploration of the existence of at least one positive solution and provides insights into the behavior of the differential equation under the given boundary conditions.

The influence of spatially anisotropic randomness on the solution of one-dimensional stochastic differential and integral equations

We demonstrate that the concept of a one-dimensional stochastic problem, in which only the statistical properties of the medium in one direction are used, is an unphysical situation. Even though the statistically averaged quantities, such as mean value and covariance, may depend only on one space dimension, the statistical properties of the medium in the other two directions must be included. A simple example, based on a second order differential equation, is used to illustrate the point and is supported by numerical calculations. The relevance of this matter to radiation and neutron transport in spatially stochastic media is made clear.

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 large time behaviour of the solutions of an evolutionary-epidemic system with spatial dispersal.

This paper investigates some properties of the large time behaviour of the solutions of a spatially distributed system of equations modelling the evolutionary epidemiology of a plant-pathogen system. The model takes into account the phenotypic trait and the mutation of the pathogen, which is described by a non-local operator. We roughly speaking prove that the solutions separate the phenotype trait from the spatio-temporal evolution in the large time asymptotic. This feature is obtained by investigating the positive and bounded entire solutions of the problem, that are shown to exhibit such a separation of the variables property, by reformulating them as the positive solutions of suitable integral equations in some ordered Banach space. In addition, some numerical simulations are performed to support our theoretical results.

Existence of solutions for fractional functional integral equations of Hadamard type via measure of noncompactness

Existence of solutions for fractional functional integral equations of Hadamard type via measure of noncompactness

Shape reconstruction of a cavity with impedance boundary condition via the reciprocity gap method

We consider an interior inverse scattering problem of reconstructing the shape of a cavity with impedance boundary condition from measured Cauchy data of the total field. The incident point sources and the measurements are distributed on two different manifolds inside the cavity. We first prove that the boundary of the cavity and the surface impedance can be uniquely determined by the scattered field data on the measurement manifold. Then we develop a reciprocity gap (RG) method to reconstruct the cavity. The theoretical analysis shows the uniquely solvability and existence of the approximate solution for the linear integral equation constructed in the RG method. We also prove that the shape of the cavity can be characterized by the blow-up property of the approximate solution of the proposed integral equation. Numerical examples are presented to verify the feasibility of the RG method.

On the series solutions of integral equations in scattering

On the series solutions of integral equations in scattering

A product integration method for nonlinear second kind Volterra integral equations with a weakly singular kernel (with application to fractional differential equations)

This paper presents a novel product integration method that provides an appropriate numerical solution for nonlinear weakly singular Volterra integral equations (WSVIEs). Extensive research in the literature has focused on studying the existence and uniqueness of solutions to these equations. However, when solving the WSVIEs, the solution may exhibit a singular behavior near the initial point of the integration interval, which can pose challenges for numerical computation. In these cases, we propose a smoothing change of variables that transforms the equation into one with a smooth solution, while still being weakly singular. We provide a convergence analysis and determine the order of convergence. The effectiveness of the proposed method is then demonstrated through the solution of various test problems.

Fluctuating hydrodynamics of an autophoretic particle near a permeable interface

We study the autophoretic motion of a spherical active particle interacting chemically and hydrodynamically with its fluctuating environment in the limit of rapid diffusion and slow viscous flow. Then, the chemical and hydrodynamic fields can be expressed in terms of integrals. The resulting boundary-domain integral equations provide a direct way of obtaining the traction on the particle, requiring the solution of linear integral equations. An exact solution for the chemical and hydrodynamic problems is obtained for a particle in an unbounded domain. For motion near boundaries, we provide corrections to the unbounded solutions in terms of chemical and hydrodynamic Green's functions, preserving the dissipative nature of autophoresis in a viscous fluid for all physical configurations. Using this, we give the fully stochastic update equations for the Brownian trajectory of an autophoretic particle in a complex environment. First, we analyse the Brownian dynamics of particles capable of complex motion in the bulk. We then introduce a chemically permeable planar surface of two immiscible liquids in the vicinity of the particle and provide explicit solutions to the chemo-hydrodynamics of this system. Finally, we study the case of an isotropically phoretic particle hovering above an interface as a function of interfacial solute permeability and viscosity contrast.

Discretization of Continuous Spaces using Barycentric Subdivision Method with Metric Space Constraints on the Nearest Neighbour Edges

The Rao-Wilton-Glisson (RWG) is a commonly used basis function in the numerical solution of the electric field integral equation (EFIE) using the Method of Moments (MoM) and Galerkin approach. This method relies on triangular patches to approximate the surface current. Traditionally, barycentric subdivision of a primary triangle into n-sub-triangles has been used with RWG basis function to solve the EFIE using MoM. This paper presents a method of approximating a surface using triangular patches by sub-dividing a primary triangle into (2n-1) subtriangles. which creates a denser mesh than the widely used nine (9) points quadrature method. The structure is approximated by small square patches, which are further sub-divided into two primary triangles. By applying barycentric sub-division, the primary triangles are decomposed into sub-triangles to create a mesh over the surface of the structure. Using graph theory, the triangular meshes are defined by the function, G (V, E), where V and E are the vertices and edges of the triangles in the mesh space, respectively. The connectivity matrix of shared edges is found by imposing a constraint on the edge length using metric spaces. This method approximates a square patch with 32 scalene triangles and shows that it can be used to reconstruct equilateral patches and doubly split double ring into mesh structures.

Q-Learning-Based Dumbo Octopus Algorithm for Parameter Tuning of Fractional-Order PID Controller for AVR Systems

The tuning of fractional-order proportional-integral-derivative (FOPID) controllers for automatic voltage regulator (AVR) systems presents a complex challenge, necessitating the solution of real-order integral and differential equations. This study introduces the Dumbo Octopus Algorithm (DOA), a novel metaheuristic inspired by machine learning with animal behaviors, as an innovative approach to address this issue. For the first time, the DOA is invented and employed to optimize FOPID parameters, and its performance is rigorously evaluated against 11 existing metaheuristic algorithms using 23 classical benchmark functions and CEC2019 test sets. The evaluation includes a comprehensive quantitative analysis and qualitative analysis. Statistical significance was assessed using the Friedman’s test, highlighting the superior performance of the DOA compared to competing algorithms. To further validate its effectiveness, the DOA was applied to the FOPID parameter tuning of an AVR system, demonstrating exceptional performance in practical engineering applications. The results indicate that the DOA outperforms other algorithms in terms of convergence accuracy, robustness, and practical problem-solving capability. This establishes the DOA as a superior and promising solution for complex optimization problems, offering significant advancements in the tuning of FOPID for AVR systems.

Numerical Solutions of Some Weak Singular Nonlinear Integral Equations of The First Type Using The Spectral Collocation Method

In this paper, a collocation-spectral approximation is proposed for weakly nonlinear and neutral singular Volterra integral-differential equations with rough solutions. We used some appropriate transformations to transform the equation of the equation into a new equation, so that the solution of the new equation has a better order (smoothness) and Jacobi's orthogonal polynomial theory can be easily used. Under appropriate assumptions on the nonlinear part, we were able to perform an acceptable error analysis on the soft and the weighted soft. To obtain a numerical approximation, some numerical examples (linear and non-linear) with uneven solutions are considered and numerical results are also presented. Also, a comparison between the proposed method and some existing numerical methods is also provided.

On Cauchy Problem Solution for a Harmonic Function in a Simply Connected Domain with Multi-Component Boundary

Here, we present an investigation of the Cauchy problem solvability for the Laplace equation in a simply connected plane domain with a multi-component boundary. We reduce our investigation to solution of two singular integral equations. If the problem is resolvable, then we restore its solution via the integral Cauchy formula. We present the examples of the solvable problems. The construction involves an auxiliary approximate conformal mapping or the solution of certain singular integral equations.

A novel approach for the numerical solution of nonlinear Fredholm integral equations using Hosoya polynomial method

Abstract In this paper, we study the graph theoretical polynomial known as the Hosoya polynomial obtained from one of the standard classes of graphs called path. Using this polynomial applied for the numerical solution of the nonlinear Fredholm integral equation, which reduces in the algebraic system of equation with collocation points, then solving this system using Newton’s iterative with the help of MATLAB, we obtain the required approximate solution. The desired results in terms of a set of continuous polynomials over a closed interval [0, 1]. Illustrative applications show the efficiency, accuracy and validity of the proposed technique.

Full field crack solutions in anti-plane flexoelectricity

In flexoelectric materials, strain gradients can induce electrical polarization. However, internal defects such as cracks profoundly affect the electromechanical coupling properties of flexoelectric solids. In particular, anti-plane cracks involve less physical fields, which are easier to study. In this study, we present a comprehensive and innovative investigation of the anti-plane crack problems in flexoelectric materials, including semi-infinite and finite-length anti-plane cracks. For the first time, we formulate a full-field solution for semi-infinite anti-plane cracks in flexoelectric media by applying the Wiener–Hopf technique. Furthermore, the collocation method and the Chebyshev polynomial expansion are used for the first time to derive the full-field hypersingular integral equation solution for finite-length anti-plane cracks in flexoelectric solids. In addition, a comparative analysis between the full-field and asymptotic solutions for semi-infinite cracks is performed, shedding light on the discrepancies in the representation of the electromechanical coupling behavior near the crack tip. The mixed finite element method is used to compare with the full-field solutions of finite-length cracks. The agreement between the numerical results and the full-field solutions demonstrates the rigor of our study. This research advances the knowledge of defects in flexoelectricity and provides significant insight into relevant failure mechanisms.

Weak ψ-Contractions on Directed Graphs with Applications to Integral Equations

This article deals with a few outcomes ensuring the fixed points of a weak (G,ψ)-contraction map of metric spaces comprised with a reflexive and transitive digraph G. To validate our findings, we furnish several examples. The findings we obtain enable us to seek out the unique solution of a nonlinear integral equation. The outcomes presented herewith sharpen, subsume, unify, improve, enrich, and compile a number of existing theorems.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations

In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of β-Banach spaces. Moreover, we examine the β–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained.

Uniqueness, regularity, continuity, and stability of solutions of integral equations in irregular domains

This study explores the uniqueness, regularity, continuity, and stability of solutions to integral equations related to elliptical boundary value problems in irregular domains. Traditional methods usually consider soft boundaries, but this work extends these results to include domains with rough boundaries. By employing Sobolev spaces, especially fractional Sobolev spaces, we develop a comprehensive theoretical framework. Under appropriate conditions, the studied integral equation has a unique solution in fractional Sobolev spaces, maintaining the regularity properties of the input function. Two new theorems are introduced to strengthen the findings. The first theorem establishes that solutions are continuous with respect to perturbations in the input data. This means small changes in the input do not cause significant deviations in the solution, ensuring robustness in practical applications. The second theorem demonstrates the stability of the solutions when the domain geometry undergoes small changes. This stability is crucial for real-world scenarios where the exact shapes of domains may not be perfectly known or may vary slightly. These results significantly improve the mathematical understanding of boundary value problems in non-smooth domains. The proposed extended framework allows for more generalized applications, addressing challenges in various fields of physics and engineering. The ability to handle irregular domains with rough boundaries opens up new possibilities for modeling and solving complex problems where traditional soft boundary assumptions are not valid. This research provides a robust basis for future studies and applications, highlighting the importance of considering fractional Sobolev spaces in the analysis of integral equations.

Numerical solution of fuzzy stochastic Volterra integral equations with constant delay

This paper considers the nonlinear fuzzy stochastic Volterra integral equations with constant delay, which are general and include many fuzzy stochastic integral and differential equations discussed in literature. Since Doob's martingale inequality is no longer applicable to such equations, a new maximum inequality is obtained. Combining with the Picard approximation method, the existence and uniqueness of solutions to nonlinear fuzzy stochastic Volterra integral equations with constant delay are given. Moreover we prove that the solution behaves stably in the case of small changes of initial values, kernels and nonlinearities. We further develop a Euler-Maruyama (EM) scheme and prove the strong convergence of the scheme. It is shown that the strong convergence order of the EM method is 0.5 under Lipschitz condition. Moreover, the strong superconvergence order is 1 if further, the kernel h(t,s) of the stochastic term satisfies h(t,t)=0. Numerical examples demonstrated that the numerical results are consistent with the theoretical research conclusions. Furthermore, the application model of the fuzzy stochastic Volterra integral equation with constant delay in population dynamics is considered, and the exact solution of the numerical example is given in explicit form.