System Of Integral Equations Research Articles

Solvability of infinite system of implicit fractional integral equations of two variables in tempered sequence spaces via generalized Darbo’s fixed point theorem

Solvability of infinite system of implicit fractional integral equations of two variables in tempered sequence spaces via generalized Darbo’s fixed point theorem

Обобщённая математическая модель основных задач теории упругости для анизотропных тел

Introduction. The assessment of the strength of underground structures is a complex task that requires a comprehensive approach. In many cases empirical methods based on experience are not sufficient. For effective application of analytical methods, it is necessary to use mathematical modelling. This allows to simplify the problem considerably, to consider different variants, to identify cases to obtain a solution in a closed form. Materials and methods. The model is based on a system of singular integral equations with a shift and complex conjugate terms. The solution of the system is the complex elastic potential. Systems of singular equations with a Carleman shift are considered on the space of random functions converging in the mean square to expand the possibility of practical use of the work results. Results. The results of the study are: a mathematical model formulation, the development of an algorithm for reducing a system of singular equations to a system of known Fredholm integral equations of the second kind, a method for determining the solvability conditions and the number of solutions. Discussion. As a result of the study: (1) a general algorithm for finding a solution to a mathematical model is obtained; (2) the methodology for determining the number of solutions and the type of solvability conditions is given; (3) The solution algorithm is extended for the stochastic case; (4) The possibility of wider application of the mathematical apparatus of functions of complex variable for research of strength properties of rocks is proved. Conclusion. The proposed comprehensive approach to assessing the strength characteristics of underground structures, combining a set of analytical methods in conjunction with mathematical modelling, expands the variability of calculation and allows to simplify the problem significantly and obtain a solution in closed form. Resume. The paper develops a mathematical model of the stress state of an elastic body, which allows solving the basic problems of elasticity theory for a variety of media. Suggestions for practical applications and directions for future research. The results of the study make it possible to improve the efficiency of analytical methods in the study of the strength of underground structures. The obtained results are applicable for predicting the strength and safety of underground structures.

Applications of a new measure of noncompactness to the solvability of systems of nonlinear and fractional integral equations in the generalized Morrey spaces

Applications of a new measure of noncompactness to the solvability of systems of nonlinear and fractional integral equations in the generalized Morrey spaces

An alternative potential method for mixed steady‐state elastic oscillation problems

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

Impulsive impact of a twin hull

An impulsively starting motion of two cylindrical bodies floating on a free liquid surface is considered. The shape of the cross-section of each body and the distance between them are arbitrary. The integral hodograph method is advanced to derive the complex velocity potential defined in a rectangular parameter region in terms of the elliptic quasi-doubly periodic Jacobi theta functions. A system of singular integral equations in the velocity magnitude on the free surface and in the slope of the wetted part of each body is derived using the kinematic boundary condition, which is then solved numerically. The velocity field, the pressure impulse on the bodies and the added mass coefficients of each body immediately after the impact are determined in a wide range of distances between the bodies and for cross-sectional shapes such as the flat plate and half-circle.

On infinite systems of nonlinear integral equations in two variables in Banach Space $BC(\mathbb{R_+}\times \mathbb{R_+},c_0$)

In this paper, the solvability of an infinite system of integral equations of the Volterra-Hammerstein type in Banach space $BC(\mathbb{R_+}\times \mathbb{R_+},c_0$) is examined. Technique associated with the measure of noncompactness plays the most important role in adopted analysis and authors present an example to validate the applicability of the result.

On solutions vanishing at infinity of infinite systems of quadratic Urysohn integral equations

The paper is devoted to present a result on the existence of solutions of an infinite system of quadratic integral equations of the Urysohn type considered on the real half-axis. Our investigations are conducted in the Banach space consisting of bounded and continuous functions defined on the real half-axis with values in the space of real sequences converging to zero. That space is equipped with the standard supremum norm. The main tools used in our study is the technique of measures of noncompactness and the Schauder fixed point principle. We illustrate our result by a suitable example.

Spectral-Galerkin method for second kind VIEs with highly oscillatory kernels of the stationary point

In this paper, we discuss an efficient spectral approach for solving a class of second kind VIEs with highly oscillatory kernels possessing the stationary point. First, we use one variable transform to convert the highly oscillatory problem into the long-time one, and then split the long-time problem into a linear system of integral equations by using a dilation approach. Next on each interval we study the characteristic of the original solution and then propose an efficient spectral-Galerkin method for solving the linear system of integral equations. We prove that the proposed algorithm requires to solve the number O(log2⁡ω) of linear algebra equations of order n+1 and reaches the convergence order O(n−r), where r denotes the degree of the regularity of the original solution and ω denotes the wave number. Moreover, we give the approach of computing the highly oscillatory integrals produced in the spectral-Galerkin method. At last, two numerical examples are provided to verify the efficiency of our proposed method.

Piecewise polynomial numerical method for Volterra integral equations of the fourth-kind with constant delay

This work studies the fourth-kind integral equation as a mixed system of first and second-kind Volterra integral equations (VIEs) with constant delay. Regularity, smoothing properties and uniqueness of the solution of this mixed system are obtained by using theorems which give the relevant conditions related to first and second-kind VIEs with delays. A numerical collocation algorithm making use of piecewise polynomials is designed and the global convergence of the proposed numerical method is established. Some typical numerical experiments are also performed which confirm our theoretical result.

Response Time of Queueing Mechanisms

We study response time, a key performance characteristic of queueing mechanisms. The studied model incorporates both active and passive queue management, arbitrary service time distribution, as well as a complex model of arrivals. Therefore, the obtained formulas can be used to calculate the response time of many real queueing mechanisms with different features, by parameterizing adequately the general model considered here. The paper consists of two parts. In the first, mathematical part, we derive the distribution function for the response time, its density, and the mean value. This is done by constructing two systems of integral equations, for the distribution function and the mean value, respectively, and solving these systems with transform techniques. All the characteristics are derived both in the time-dependent and steady-state cases. In the second part, we present numerical values of the response time for a few system parameterizations and point out several of its properties, some rather counterintuitive.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

A novel approach to solving system of integral partial differential equations based on hybrid modified block‐pulse functions

In nonlinear this paper, we present a new approach to solve system of two‐dimensionalFredholm–Volterra integral partial differential equations. This approach is based on a new hybrid of two‐dimensional Bernoulli polynomials and modified block‐pulse functions. Different from the traditional methods, in this new approach, the operational matrices are considered. Furthermore, by solving this matrix equation and using the collocation method, approximate solutions for the problem are obtained. Finally, error analysis and three practical examples are provided accuracy and efficiency illustrate our approach.

Solvability Analysis of the Nonlinear Integral Equations System Arising in the Logistic Dynamics Model in the Case of Piecewise Constant Kernels

Solvability Analysis of the Nonlinear Integral Equations System Arising in the Logistic Dynamics Model in the Case of Piecewise Constant Kernels

Numerical Simulation of Initial Value Problem of Integro-differential Equation Models

In this article, the Volterra-Fredholm integral equation (V-FIE) is derived from an initial value problem of kind integro-differential equation (IVP). We discuss the existence and uniqueness of the solution to the problem in Hilbert space. A numerical method is used to reduce this type of equation to System of Fredholm integral equations of the second kind(SFIEs). In light of this, the clustering method and the Galerkin method to solve the system of second-order Fredholm integral equations(SFIEs) and calculate the error in each case. Finally, the approximate and exact solutions are plotted on the same coordinate plane Using MATLAB code (2022).

Some multidimensional fixed point theorems for nonlinear contractions in C-distance spaces with applications

In this manuscript, we use the concept of multidimensional fixed point in a generalized space, namely, C-distance space with some nonlinear contraction conditions, such as Jaggi- and Dass-Gupta-type contractions. We provide results with a Jaggi-type hybrid contraction for the mentioned space. Moreover, we use control functions to get the desired results. After each theorem, we compare our results with previous ones to show that they are generalized. We provide examples to support our results. An application is also performed to solve the system of integral equations.

Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper.

3-Dimensional computational analysis of [formula omitted]-contraction in GV-fuzzy metric spaces with applications

It is a well known fact that contraction conditions play a major role in composing coincidence point results. In present work, a new ϕ-contraction is being corroborated to yield coupled coincidence points in partially ordered GV-fuzzy metric spaces. Due to the significant feature of H-type t-norm, in this communication, we endue GV-fuzzy metric spaces with this t-norm. We use the mixed monotone property of mappings with regard to partial ordering. Present work generalizes some already existing results. An applied example is also formulated to cast the 3-dimensional analysis of ϕ-contraction utilized in the main result. Computational analysis of the illustrative example has been done using the software MATLAB version R2022b which shows the experimental verification of our work. Furthermore, applications to the solution of the system of Fredholm type integral equations and solution of the system of equations in dynamic programming accords the validity of the present work. Endmost, the concluding remark projects the significance of current work.

The problem of axisymmetric torsion of a rigid disk parallel to an internally penny crack and interface in a bilayer medium

The aim of this work was to investigate an axisymmetric problem involving a penny-shaped crack embedded in a bi-material with finite thickness under torsion. This torsion is caused by a circular rigid disk attached to the upper layer and fixed by an undeformable support at the lower layer. The location of the crack can be found in the upper layer (case 1) or the lower layer (case 2), both cases are considered. The problem was solved by applying the Hankel transform to reduce it to a system of dual integral equations, which further simplified into Fredholm integral equations of the second-kind, These equations were then solved numerically using the Gaussian quadrature rule. Dimensionless plots were presented to demonstrate the influence of layers thickness, the distance between the support and the crack, and the shear modulus ratio between layers on the stress intensity factor of the crack. The obtained results illustrated the agreement between the analytical solution and the numerical results.

Non-classical two-phase Stefan problem with variable thermal coefficients

We study a one-dimensional two-phase Stefan problem governed by diffusion-convection equations with a Dirichlet boundary condition at the fixed face, variable thermal coefficients and particular heat sources. A similarity solution is obtained through the solution to a system of integral equations. Some particular cases are analyzed, providing also computational examples.

ENHANCING CONVERGENCE AND ACCURACY IN SOLVING SYSTEM OF NONLINEAR INTEGRAL EQUATIONS USING MULTILAYER NEURAL NETWORKS

Abstract. This paper presents a novel approach for solving nonlinear integral equations of Fredholm and Volterra types using a multi-layer neural network (MLNN) trained through unrestricted optimization. The learning process involves minimizing the integral equation error function using the Nelder-Mead simplex method (NMSM), enabling efficient convergence in both Fredholm and Volterra equations. Our method ensures high accuracy despite potentially time-consuming computations for complex problems, maintaining convergence and approximation precision. Furthermore, the proposed approach demonstrates robustness against high-dimensional and variant equations. Numerical experiments employing the NMSM showcase its superior performance compared to existing methods. This research offers a promising avenue for solving a system of nonlinear integral equations efficiently and accurately, with broad applicability across diverse domains.