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The conventional electrostatic solutions for two-dimensional (2D) electrodes possess edge singularities for the surface charge density σ and the normal component of the electric field E n. These singularities are generally non-physical because they admit infinite gradients of the concentration of free charge carriers. In particular, they are unacceptable in the studies of the local field sensitive effects, such as the electric breakdown and the ferroelectric domain reversal. We claim that account for diffusion of free charge carriers leads to the disappearance of the edge singularities. This generalization occurs consistently within the same basic concept of conduction. Specifically, we consider the case of U -biased circular disc electrode of radius a. Account for diffusion leads here to a strongly nonlinear integral 2D equation for the electrostatic potential φ (r). Numerical solution of this equation shows that the law σ (a) ∝ U 2 / a takes place. Outside a close vicinity of the disc edge, we stay close to the conventional electrostatic solution for φ and σ. [ABSTRACT FROM AUTHOR]

Abstract In this study, we manifest a new class of mappings that satisfy Geraghty–Ćirić-type contractive conditions in the context of b-metric spaces and prove a theorem on the existence and uniqueness of fixed points. Our results unify and generalize the results of Geraghty; Ćirić; Dukic, Kadelburg, and Radenović; and Shu-fang Li, Fei Hi, and Ning Lu in the setting of b-metric spaces. Furthermore, we provide examples to verify the correctness and applicability of our results. We also utilize our findings to show the existence of a unique solution for a nonlinear integral equation.

In this study, we manifest a new class of mappings that satisfy Geraghty–Ćirić-type contractive conditions in the context of b-metric spaces and prove a theorem on the existence and uniqueness of fixed points. Our results unify and generalize the results of Geraghty; Ćirić; Dukic, Kadelburg, and Radenović; and Shu-fang Li, Fei Hi, and Ning Lu in the setting of b-metric spaces. Furthermore, we provide examples to verify the correctness and applicability of our results. We also utilize our findings to show the existence of a unique solution for a nonlinear integral equation. [ABSTRACT FROM AUTHOR]

Surface complexation models (SCMs) based on Gouy-Chapman theory are often used to describe adsorption of ions onto mineral surfaces. To compensate for the buildup of charge at a solid surface, the composition of the electric diffuse layer next to the surface must balance the surface charge. To calculate the diffuse layer composition, several nonlinear equations and integrals must be solved, usually with an iterative approach. Before convergence, charge balance is typically not fulfilled. One numerical difficulty is that, because of these charge balance errors, the iterative solver may attempt to take the square root of a negative number. Herein, we show that for electrolytes containing only monovalent or divalent ions (i.e., most electrolytes encountered in practice), we can greatly simplify the integrals and eliminate the appearance of complex-valued integrands; it is even possible to derive explicit analytical formulas. Furthermore, using the new method prevents converging to non-physical roots of the Grahame equation, which links surface potential to surface charge. To the best of our knowledge, the presented formulation has not been implemented in geochemical modelling software before, although similar mathematical expressions have been presented in the literature. In Gouy-Chapman theory, ions can only be distinguished by their charges, but this is not consistent with all experimental findings. We present a model that allows for the preferential accumulation of ions in the diffuse layer. The model, which is implemented mathematically by including ion exchange sites with a variable exchange capacity, is flexible and more numerically tractable than the standard models for the diffuse layer. [ABSTRACT FROM AUTHOR]

In this paper, a combination approach based on Bernoulli polynomials and Gauss-Jacobi quadrature formula is developed to solve the system of nonlinear variable-order fractional Volterra integral equations (V-O-FVIEs). For this, we extend the constant coefficient in the Gauss-Jacobi formula to the variable coefficient and used it in our method. The method converts the system of V-O-FVIEs into the corresponding nonlinear system of algebraic equations. In addition, we use Gronwall inequality and the collectively compact theory to prove the existence and uniqueness of the solution of the original equation and the approximate equation, respectively. The convergence analysis and the error estimation of proposed method are discussed. Finally, some numerical examples illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Volterra integral equation of the second kind with weakly singular kernel usually exhibits singular behavior at the origin, which deteriorates the accuracy of standard numerical methods. This paper develops a singularity separation Chebyshev collocation method to solve this kind of Volterra integral equation by splitting the interval into a singular subinterval and a regular one. In the singular subinterval, the general psi-series expansion for the solution about the origin or its Padé approximation is used to approximate the solution. In the regular subinterval, the Chebyshev collocation method is used to discretize the equation. The details of the implementation are also discussed. Specifically, a stable and fast recurrence procedure is derived to evaluate the singular weight integrals involving Chebyshev polynomials analytically. The convergence of the method is proved. We further extend the method to the nonlinear Volterra integral equation by using the Newton method. Three numerical examples are provided to show that the singularity separation Chebyshev collocation method in this paper can effectively solve linear and nonlinear weakly singular Volterra integral equations with high precision. [ABSTRACT FROM AUTHOR]

In this research, two systems of nonlinear Volterra integral equation and Volterra integro‐differential equation were considered. New results in sense of Ulam stabilities in relation to these two systems were proved on a finite interval. The proof of the results on the Ulam stabilities of that classes of the equations are based on the nonlinear alternative related to Banach's contraction principle. The outcomes of this research give new contribution to the theory of Ulam stabilities. [ABSTRACT FROM AUTHOR]

In many physical problems, it is important to capture exponentially small effects that lie beyond‐all‐orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans‐series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans‐series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans‐series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial‐over‐power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial‐over‐power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences. [ABSTRACT FROM AUTHOR]

From Newton's method, we construct a Newton-type iterative method that allows studying a class of nonlinear Hammerstein-type integral equations. This method is reduced to Newton's method if the kernel of the integral equation is separable and, unlike Newton's method, can be applied to approximate a solution if the kernel is nonseparable. In addition, from an analysis of the global convergence of the method, we can locate and separate solutions of the nonlinear Hammerstein-type integral equations involved. For this study of the global convergence, we use auxiliary functions and obtain restricted global convergence domains that are usually balls. [ABSTRACT FROM AUTHOR]

This article introduces a novel iterative process, denoted as F ★ , designed for the class of generalized α -Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results contribute to the theoretical foundation of iterative processes in functional analysis. The presented framework is applied to address nonlinear integral equations, showcasing the versatility and applicability of the proposed F * for the class of generalized iteration process. Additionally, the article includes numerical examples that not only validate the theoretical findings but also provide insights into the practical utility of the developed methodology. [ABSTRACT FROM AUTHOR]

This work studies time-dependent electromagnetic scattering from obstacles whose interaction with the wave is fully determined by a nonlinear boundary condition. In particular, the boundary condition studied in this work enforces a power law type relation between the electric and magnetic fields along the boundary. Based on time-dependent jump relations of classical boundary operators, we derive a nonlinear system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. These fields can subsequently be computed at arbitrary points in the exterior domain by evaluating a time-dependent representation formula. Fully discrete schemes are obtained by discretizing the nonlinear system of boundary integral equations with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Error bounds with explicitly stated convergence rates are proven, under the assumption of sufficient regularity of the exact solution. The error analysis is conducted through novel techniques based on time-discrete transmission problems and the use of a new discrete partial integration inequality. Numerical experiments illustrate the use of the proposed method and provide empirical convergence rates. [ABSTRACT FROM AUTHOR]

To study the stressed state and limit equilibrium of a closed elastoplastic cylindrical shell containing a plane longitudinal internal crack of any configuration, we use an analog of the δc-model and represent the resolving system of equations of the problem in the complex form. The obtained system of equations is reduced to a system of nonlinear singular integral equations whose solution is constructed by the method of mechanical quadratures with regard for the conditions of plasticity of thin shells, the conditions of boundedness of stresses, and the conditions of uniqueness of displacements. We also perform the numerical analysis of the dependences of the crack opening displacements and the sizes of plastic zones on the boundary conditions imposed on the shell edges, on the configuration of the crack, and on the geometric and mechanical parameters. [ABSTRACT FROM AUTHOR]

We study the problem of nonuniqueness of solutions for one class of Hammerstein-type nonlinear integral equations appearing in the problems of approximation of a finite function of two variables by the modulus of the double discrete Fourier transform. We establish the existence and properties of real (primary) solutions of four types and perform numerical investigations of branching of primary solutions of the second type. We also determine the efficiency of real and branched complex solutions. [ABSTRACT FROM AUTHOR]

We consider the steady-state limiting current that can be carried by an infinite periodic array of thin electron sheets spaced by period p in a planar diode of gap voltage V and gap separation d. Our primary assumptions are as follows: (1) electron motion is restricted by an infinite magnetic field to the direction normal to the electrode surfaces, (2) all electrons are emitted from the cathode with initial kinetic energy Ein, and (3) electron motion is non-relativistic. The limiting current density, averaged over a period and normalized to the classical 1D Child–Langmuir (CL) current density (including a factor that accounts for nonzero Ein), is found to depend only on the two dimensionless parameters p/d and Ein/eV. This average limiting current density is computed from the maximum current density for which the iterative solution of a non-linear integral equation converges. Numerical results and empirical curve fits for the limiting current are presented, together with an analysis as p/d and Ein/eV approach zero or infinity, in which cases previously published results are recovered. Our main finding is that, while the local anode current density within each electron sheet is infinite in our model (that is, it exceeds the classical 1D CL value by an "infinite" factor), the period average anode current density is in fact still bounded by the classical 1D CL value. This study therefore provides further evidence that the classical 1D Child–Langmuir current density is truly a fundamental limit that cannot be circumvented. [ABSTRACT FROM AUTHOR]

Piezoelectric thin sheets/films hold a key position in the design of some advanced electromechanical structures, such as integrated actuators, energy harvesting devices and programmable structures. The development of new advanced materials, such as metamaterials, has generated a growing interest in utilizing the nonlinear response of piezoelectric elements to achieve new desired features. Nonetheless, there has been a lack of exploration into the modelling and analysis of thin-sheet piezoelectric actuators undergoing nonlinear deformation. The current paper presents a nonlinear model of a thin-sheet piezoelectric actuator partially bonded to a linear elastic, electrically insulative host. The formulation of the problem takes into account the associated effects of the debonding and the adhesive layer on the response of the piezoelectric structure. The thin-sheet actuator is assumed as a generalized electro-elastic Euler–Bernoulli beam, characterized by the piezoelectric effect, axial and bending deformation and nonlinear deflection in the debonded part. The developed nonlinear integral equations in terms of interfacial stresses are solved using Chebyshev polynomials to determine the deformation of the actuator and the local stress fields. This nonlinear model enables the analysis of the electromechanical behaviour of actuators subjected to different in-plane electric loading, considering a range of geometries. The bending moment, axial force, displacement and interfacial stress distribution along the actuator are carefully evaluated. The current study creates valuable insights into the behaviour of such actuators with nonlinear deformation, facilitating the enhancement of actuator performance for applications in specific metamaterials. [ABSTRACT FROM AUTHOR]

Two weakly nonlinear integral equations of motion commonly used in the literature to study the nonlinear dynamics of straight and initially not rectilinear Euler‐Bernoulli beams, respectively, are further investigated. Attention is focused on the process known as "static condensation", which consists of neglecting the axial inertia in the exact, fully nonlinear system of equations of motion to determine the axial displacement as a function of the transversal one. The novelty of the paper relies on showing that, contrarily to expectation and somehow surprisingly, the integral equation for beams with a not rectilinear initial configuration cannot be obtained by the static condensation process starting from the exact, fully nonlinear, equations of motion, apart from a very particular and specific case. On the contrary, it is confirmed the well‐known result that for rectilinear beams the integral equation can be obtained by the static condensation. This highlights a major difference between the two integral equations in terms of reliability and allows us a better understanding of the integral equation of rectilinear beams, underlying its stronger mathematical background than the classical counterpart for not straight beams (i.e., its being obtainable from the exact, fully nonlinear, equations of motion via static condensation), which provides it with a "special" behavior and makes it more trustworthy. [ABSTRACT FROM AUTHOR]

In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literature to their more general forms. The qualitative and quantitative characteristics of solutions to various dynamic equations on time scales involving retarded integrals can be studied using these inequalities. The results presented in this manuscript furnish a powerful tool to analyze the boundedness of nonlinear integral equations with retarded integrals on several time scales. In the end, we also include numerical illustrations to signify the applicability of these results to power nonlinear retarded integral equations on real and quantum time scales. [ABSTRACT FROM AUTHOR]

The blow-up phenomenon may occur in nonlinear integral equations and differential equations, which has important significance for the simulation of some practical problems. This paper is devoted to predicting the blow-up time for convolution Volterra-Hammerstein integral equations using the series expansion of the solution about the origin. First, the finite-term series expansion for the solution about the origin is obtained via Picard iteration and symbolic computation, which has high accuracy near the origin. Second, the Padé approximation of the series expansion is made to extend the convergence region of the series. The blow-up time of the solution is estimated by calculating the smallest positive root of the denominator of the rational function. Third, an integral transform is performed to the series expansion to convert the branch point to a pole of order one at the blow-up time, so the predicted accuracy of the blow-up time is improved remarkably. Numerical examples illustrate that the proposed method is very efficient for estimating the blow-up time of Volterra integral equations and some kinds of differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

This paper is devoted to the explication of the introduction of rational-type soft fuzzy contraction in soft fuzzy metric space. New fixed point results for these new contractions along with relevant and valid examples have been presented and generalized. We have also presented an application to our new formed results in finding the solution of a nonlinear integral equation. [ABSTRACT FROM AUTHOR]

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. [ABSTRACT FROM AUTHOR]

In this article, we consider Lucas series for N = 2, 5, 6 in numerical solutions of linear and nonlinear Volterra integral equations of the second kind. The series is used to transform the equation into a system of nonlinear algebraic equations, and the unknown parameters are determined. The application of this method has shown that the Lucas series is a powerful and active candidate that can be used to approximate the solution of linear and nonlinear Volterra integral equations of the second kind. The method gave a good performance, as can be seen from the four sample problems considered. The numerical results reveal computational efficiency of the method and it is also seen to be highly accurate and converge to the exact solution in some cases. Approximate and exact solutions are plotted to further confirm the accuracy of the method. [ABSTRACT FROM AUTHOR]

Introduction/purpose: This paper introduces the concept of β − F-weak contraction by using the concepts of F−weak contraction and α − ψ− contraction. Methods: The use of the β − F-weak contraction proves some fixed points theorems in the framework of S−metric spaces. Results: The obtained results on fixed points in S−metric spaces generalize some known results in the literature. Conclusions: The β − F−weak contraction generalizes some important contraction types and examines the existence of a fixed point in S−metric spaces. The results are used to solve a non-linear Fredholm integral equation. [ABSTRACT FROM AUTHOR]

The paper contains a brief summary of the generalization of metrical structure regarding the fixed point theorem and corresponding operator equation problems. We observed that many researcher either tried to weaken the metrical structure, the contraction condition, or both. The idea behind this paper is to look for a minimal metrical structure to establish fixed point theorems. In this connection, we present new variants of the known fixed point theorem under non-triangular metric space (namely F-contraction, (A; S)-contraction, (A; S)-contraction). We also apply the obtain result in solving various types of operator equation problems. e.g., high-order fractional differential equation with non-local boundary conditions and non-linear integral equation problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a new fixed-point result to draw conclusions about the existence and uniqueness of the solution for a nonlinear Fredholm integral equation of the second kind with non-differentiable Nemytskii operator. To do this, we will transform the problem of locating a fixed point for an integral operator into the problem of locating a solution of an integral equation. Thus, assuming conditions on the Nemytskii operator, we will obtain a global convergence domain for the solution of the considered integral equation, taking for this a uniparametric family of derivativefree iterative processes with quadratic convergence. This result provides us a new fixed-point result for the integral operator considered. [ABSTRACT FROM AUTHOR]

To understand the statistics of magnetic field lines in turbulence is complicated. Previous descriptions have been based on either pure numerical treatments, often called simulations, or analytical theory. In the latter case one operates with nonlinear integral or differential equations which are difficult to solve. In order to achieve a complete analytical description, one needs to employ a set of approximations which are often difficult to justify. Although, previously developed numerical and analytical treatments of field line random walk agree with each other, they do not provide simple arguments for the analytical form of field line diffusion coefficients. It is the purpose of the current article to derive field line random walk coefficients entirely based on simple heuristic arguments. Special cases such as slab, two-dimensional, three-dimensional, and two-component turbulence are considered. Heuristically constructed diffusion parameters are compared with previously obtained analytical results. It is shown that the heuristic arguments provide the same scaling behavior as the formulas derived by employing systematic analytical theories. [ABSTRACT FROM AUTHOR]

In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of 2m × 2m equations, we have simplified the problem to finding the solution for a set of 2 × 2 nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam-Hyers stability and generalized Ulam-Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method. [ABSTRACT FROM AUTHOR]

In this article, two types of contractive conditions are introduced, namely extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction. For the case of two mappings and their coincidence point theorems, a variant of (ϰ , Ω - Ϝ) -contraction has been introduced, which is called (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction. In the end, the applications of an extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required. [ABSTRACT FROM AUTHOR]

In the current paper, an iterative numerical scheme based upon the quasilinearization technique and Nyström method to find the approximate solution of the nonlinear Fredholm integral equations of the second kind is provided. First, using the quasilinearization method, the nonlinear Fredholm integral equation is reduced to a sequence of linear Fredholm integral equations. Under some suitable assumptions, the exact solutions of this sequence of linear integral equations converge to the unique solution of the original problem quadratically. Then in each iteration, utilizing the linear barycentric rational quadrature, the linear Fredholm integral equation is approximated with the Nyström method. Convergence analysis of the method is investigated thoroughly. Numerical examples and comparisons with other existing methods are provided to confirm the theoretical results and demonstrate the performance and validity of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we use a method based on radial basis functions and the collocation method for the numerical solution of a class of Volterra integral equations of the third kind, using zeros of the shifted Legendre polynomial as the collocation points. The principal benefit of this scheme is that it does not require any discretization and so it is independent of the geometry of the domains and can thus be applied to the solution of various kinds of integral equations. The procedure is more flexible for the majority of classes of Volterra integral equations of the third kind. The construction of the suggested technique has been introduced. The convergence analysis of the presented method is investigated. Finally, certain numerical examples are included to show the accuracy and efficiency of the new technique. The numerical results obtained and their comparison with other methods demonstrate the reliability of this method. Our proposed method gives acceptable accuracy with a small use of data, which also reduces the computational costs. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a study of α-η-type generalized F-proximal contraction mappings in the framework of modular metric spaces and to prove some best proximity point theorems for these types of mappings. Some examples are given to show the validity of our results. We also apply our results to establish the existence of solutions for a certain type of non-linear integral equation. [ABSTRACT FROM AUTHOR]

Improved modeling of permafrost active layer freeze‐thaw plays a crucial role in understanding the response of the Arctic ecosystem to the accelerating warming trend in the region over the past decades. However, modeling the dynamics of the active layer at diurnal time scale remains challenging using the traditional models of freeze‐thaw processes. In this study, a physically based analytical model is formulated to simulate the thaw depth of the active layer under changing boundary conditions of soil heat flux. Conservation of energy for the active layer leads to a nonlinear integral equation of the thaw depth using a temperature profile approximated from the analytical solution of the heat transfer equation forced by ground heat flux. Temporally variable ground heat flux is estimated using non‐gradient models when field observations are not available. Validation of the proposed model conducted against field data obtained from three Arctic forest and tundra sites demonstrates that the model is able to simulate both thaw depth and soil temperature profiles accurately. The model has the potential to estimate regional variability of the thaw depth for permafrost related applications. Plain Language Summary: The seasonally thawed layer on top of the permafrost (active layer) is a key component of the Arctic system affected by the strong warming trend over the past decades. This soil layer experiences a pronounced seasonal cycle of freezing and thawing processes caused by the availability of Sun's energy. Mathematical modeling of the thaw depth of the active layer has remained challenging. This study formulates a novel model for the simulation of the diurnal cycle of thawing process. The formulation is developed using innovative models of heat flux that goes into the soil and soil temperature profile. Ground heat flux is derived from available energy at the land surface using a theory of surface heat flux partition. The soil temperature profile is expressed using ground heat flux within the active layer. The proposed model has been validated against field observations during thawing season. The model simulation and field observations of the thaw depth are in a good agreement at three Arctic study sites with forest and tundra surface conditions. The proposed formulation can be used for modeling freeze‐thaw cycles of the active layer at the regional scales since data on surface available energy can be obtained from remote sensing observations. Key Points: The proposed model is highly effective in modeling thawing depth at higher time resolution and representing the soil energy budgetNon‐gradient models demonstrate a strong capability to model soil energy budget in data‐sparse harsh environments [ABSTRACT FROM AUTHOR]

This document is an editorial for a special issue of the journal AIMS Mathematics on the topic of discrete optimization. The issue includes 21 papers covering a range of subjects, including molecular trees, network systems, variational inequality problems, scheduling, image restoration, spectral clustering, integral equations, convex functions, graph products, optimization algorithms, air quality prediction, humanitarian planning, inertial methods, neural networks, transportation problems, emotion identification, fixed-point problems, structural engineering design, single machine scheduling, and ensemble learning. The papers present new theoretical results, algorithms, and applications in these areas. The guest editor expresses gratitude to the journal staff and reviewers and hopes that readers will find inspiration for their own research. [Extracted from the article]

In this paper, we investigated several new fixed points theorems for multivalued mappings in the framework b -metric spaces. We first generalized S -iterative schemes for multivalued mappings to above spaces by means of a convex structure and then we developed the existence of fixed points and approximate endpoints of the multivalued contraction mappings using iteration techniques. Furthermore, we introduced the modified S -iteration process for approximating a common endpoint of a multivalued α s -nonexpansive mapping and a multivalued mapping satisfying conditon (E ′ ) . We also showed that this new iteration process converges faster than the S -iteration process in the sense of Berinde. Some convergence results for this iterative procedure to a common endpoint under some certain additional hypotheses were proved. As an application, we applied the S -iteration process in finding the solution to a class of nonlinear quadratic integral equations. In this paper, we investigated several new fixed points theorems for multivalued mappings in the framework -metric spaces. We first generalized -iterative schemes for multivalued mappings to above spaces by means of a convex structure and then we developed the existence of fixed points and approximate endpoints of the multivalued contraction mappings using iteration techniques. Furthermore, we introduced the modified -iteration process for approximating a common endpoint of a multivalued -nonexpansive mapping and a multivalued mapping satisfying conditon . We also showed that this new iteration process converges faster than the -iteration process in the sense of Berinde. Some convergence results for this iterative procedure to a common endpoint under some certain additional hypotheses were proved. As an application, we applied the -iteration process in finding the solution to a class of nonlinear quadratic integral equations. [ABSTRACT FROM AUTHOR]

In the present paper, a new efficient technique is described for solving nonlinear mixed partial integrodifferential equations with continuous kernels. Using the separation of variables, the nonlinear mixed partial integrodifferential equation is converted to a nonlinear Fredholm integral equation. Then, using different numerical methods, the Bernoulli polynomial method and the Chebyshev polynomials of the sixth kind, the nonlinear Fredholm integral equation has been reduced into a system of nonlinear algebraic equations. The Banach fixed-point theory is utilized in order to have a conversation about the nonlinear mixed integral equation's solution, namely, its existence and uniqueness. In addition, we talk about the convergence and stability of the solution. Finally, a comparison between the two different methods and some other famous methods is presented through various examples. All the numerical results are calculated and obtained using the Maple software. [ABSTRACT FROM AUTHOR]

The exploration of collisional fragmentation pheno-mena remains largely unexplored, yet it holds considerable importance in numerous engineering and physical processes. Given the nonlinear nature of the governing equation, only a limited number of analytical solutions for the number density function corresponding to empirical kernels are available in the literature. This article introduces a semi-analytical approach using the homotopy perturbation method to obtain series solutions for the nonlinear collisional fragmentation equation. The method presented here can be readily adapted to solve both linear and nonlinear integral equations, eliminating the need for domain discretization. To gain deeper insights intothe accuracy of the proposed method, a convergence analysis is conducted. This analysis employs the concept of contractive mapping within the Banach space, a well-established technique universally acknowledged for ensuring convergence. Various collisional kernels (product and polymerization kernels), breakage distribution functions (binary and multiple breakage) and various initial particle distributions are considered to obtain the new series solutions. The obtained results are successfully compared against finite volume method [26] solutions in terms of number density functions and their moments. The error between the exact and obtained series solutions is shown in plots and tables to confirm the applicability and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

We study the evolution of a system of two species with nonlinear mobility and nonlocal interactions on a graph whose vertices are given by an arbitrary, positive measure. To this end, we extend a recently introduced 2-Wasserstein-type quasi-metric on generalized graphs, which is based on an upwind-interpolation, to the case of two-species systems; concave, nonlinear mobilities; and p \not= 2. We provide a rigorous interpretation of the interaction system as a gradient flow in the Finslerian setting, arising from the new quasi-metric. [ABSTRACT FROM AUTHOR]

We consider a special system of integral equations of convolution type with a monotone convex nonlinearity naturally arising when searching for stationary or limit states in various dynamic models of applied nature, for example, in models of the spread of epidemics, and prove theorems stating the existence or absence of a nontrivial bounded solution with limits at depending on the values of these limits and on the structure of the matrix kernel of the system. We also study the uniqueness of such a solution assuming that it exists. Specific examples of systems whose parameters satisfy the conditions stated in our theorems are given. [ABSTRACT FROM AUTHOR]

A numerical method to solve the nonlinear Volterra integral equations of the first kind with discontinuous kernels is proposed. Usage of operational matrices for this kind of equation is a cost-efficient scheme. Shifted Legendre polynomials are applied for solving Volterra integral equations with discontinuous kernels by converting the equation to a system of nonlinear algebraic equations. The convergence analysis is given for the approximated solution and numerical examples are demonstrated to denote the precision of the proposed method. [ABSTRACT FROM AUTHOR]

We present the inverse problem of successive determination of the two unknowns that are a coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of an integral operator describing the memory of the medium. The direct initial-boundary value problem involves the zero data and the Neumann boundary condition. The trace of the Fourier image of a solution to the direct problem on the boundary of the medium serves as additional information. Studying inverse problems, we assume that the unknown coefficient is expanded into an asymptotic series in powers of a small parameter. Also, we construct some method for finding the coefficient that accounts for the memory of the environment to within an error of order . At the first stage, we determine a solution to the direct problem in the zero approximation and the kernel of the integral operator, while the inverse problem reduces to an equivalent problem of solving the system of nonlinear Volterra integral equations of the second kind. At the second stage, we consider the kernel given and recover a solution to the direct problem in the first approximation and the unknown coefficient. In this case, the solution to the equivalent inverse problem agrees with a solution to the linear system of Volterra integral equations of the second kind. We prove some theorems on the unique local solvability of the inverse problems and present the results of numerical calculations of the kernel and the coefficient. [ABSTRACT FROM AUTHOR]

Many physical problems represented as initial and boundary value problems are usually solved by transforming them into integral equations on the half-line. Therefore, this paper discusses the Galerkin and collocation methods along with the multi-Galerkin and multi-collocation methods for non-linear integral equations with compact and Wiener-Hopf kernel operators on the half-line in the space of piecewise polynomial subspaces. As Hammerstein integral equation on half-line has the unbounded domain, the finite section approximation method is applied; and afterward, we find an approximate solution for the finite section integral equation. In order to obtain improved superconvergence rates, we apply Sunil Kumar's method and then compare proposed convergence rates with the existing results of Nahid et al. [22] both theoretically and numerically. These results are further improved by applying multi-projection methods. In addition, it has shown that the proposed theory enhances the results of [22]. Finally, numerical examples are presented to demonstrate the given theoretical framework. [ABSTRACT FROM AUTHOR]

The objective of the present work is to develop a numerical method for solving a nonlinear integral equation given by y (t) = f (t) + ∫ 0 1 k (t , s) 1 [ y (s) ] α d s , where y(t)∈C2[0,1] with y(t)>0 , f(t)∈C[0,1] with f(t)≥0 , the kernel function k(t,s) is non-negative and continuous on [0,1]×[0,1] and α>0. The existence of a continuous positive solution of this equation is well-established in the literature. However, there is no reported method to solve it numerically for any α>0. To attain the desired objective, the renowned Chebyshev collocation method is used. Having unknown Chebyshev coefficients, this method converts the integral equation into a matrix equation that produces a set of nonlinear algebraic equations. To solve these equations, computationally, the well-established Newton's method is employed. To validate the effectiveness and precision of the method, various numerical examples with well-defined exact solutions are examined. Obtained numerical solutions confirm the accuracy and validity of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, motivated by Reich contraction and tool of measure of noncompactness, some generalizations of Reich, Kannan, Darbo, Sadovskii, and Krasnoselskii type fixed point results are presented by considering a pair of maps A, B on a nonempty closed subset M of a Banach space X into X. The existence of a solution to the equation A x + B x = x , where A is k-set contractive and B is a generalized Reich contraction, is established. As applications, it is established that the main result of this paper can be applied to learn conditions under which a solution of a nonlinear integral equation exists. Further we explain this phenomenon with the help of a practical example to approximate such solutions by using fixed point techniques. The graphs of exact and approximate solutions are also given to attract readers for further research activities. [ABSTRACT FROM AUTHOR]

Analysis of turbulence-induced scintillation on a round-trip propagation path was performed. A simplified equation for the cross-covariance of intensity fluctuations on a round-trip path was derived. This equation was validated both experimentally over a 1 km horizontal path and using a wave-optics simulation. An analytical solution was obtained for a non-linear integral equation relating the crosswind velocity profile to a spatial–temporal cross-covariance of intensity fluctuations observed in the laser return. A method for crosswind profile reconstruction using an optical sensor with varying beam divergence was developed and experimentally demonstrated. A portable, battery-powered optical wind sensor with fixed beam divergence suitable for mounting on a riflescope was constructed and field-demonstrated in live fire tests. Results of the live fire tests validated the proposed method. [ABSTRACT FROM AUTHOR]

In this manuscript, we present several types of interpolative proximal contraction mappings including Reich–Rus–Ciric-type interpolative-type contractions and Kannan-type interpolative-type contractions in the setting of bipolar metric spaces. Further, taking into account the aforementioned mappings, we prove best proximity point results. These results are an extension and generalization of existing ones in the literature. Furthermore, we provide several nontrivial examples, an application to find the solution of an integral equation, and a nonlinear fractional differential equation to show the validity of the main results. [ABSTRACT FROM AUTHOR]