We revisit the analogy between feed-forward deep neural networks (DNNs) and discrete dynamical systems derived from neural integral equations and their corresponding partial differential equation forms. A comparative analysis between the numerical/exact solutions of the Burgers' and Eikonal equations and those obtained via physics-informed neural networks (PINNs) is presented. We show that PINN learning provides a different computational pathway compared to standard numerical discretization in approximating essentially the same underlying dynamics of the system. Within this framework, DNNs can be interpreted as discrete dynamical systems whose layerwise evolution approaches attractors, and multiple parameter configurations may yield comparable solutions, reflecting the degeneracy of the inverse mapping. In contrast to the structured operators associated with finite-difference procedures, PINNs learn dense parameter representations that are not directly associated with classical discretization stencils. This distributed representation generally involves a larger number of parameters, leading to reduced interpretability and increased computational cost. However, the additional flexibility of such representations may offer advantages in high-dimensional settings where classical grid-based methods become impractical.

Global and blow-up solutions for a non-local integrable equation with applications to geometry

We present a classical density functional theory (DFT) study of liquid water in which bridge functions arising from triplet correlations are explicitly incorporated beyond the hypernetted-chain (HNC) approximation. Starting from a third-order density expansion of the Helmholtz free energy functional, we construct a tractable DFT-RISM (reference interaction site model) framework in which three-body direct correlations are included through a physically motivated factorization scheme. Particular attention is paid to the oxygen-oxygen (O-O) radial distribution function, whose second peak at around 4.5 Å is a hallmark of local tetrahedral ordering. We show that this structural feature can be generated within an integral equation framework by introducing an effective bridge function with appropriate inter-particle correlation contributions. Comparison with molecular dynamics data demonstrates that the inclusion of triplet correlations significantly improves the description of the O-O correlations beyond the HNC level, while maintaining reasonable agreement for O-H and H-H correlations. These results provide a physically transparent route to understanding the origin of tetrahedral ordering in liquid water and highlight the essential role of three-body correlations in molecular liquids, particularly in hydrogen-bonded systems.

Existence and uniqueness results for non-linear fractional integral equation with reciprocal singularities

Fast hybrid integral equation solver for acoustic scattering

An epidemiological model with arbitrary distributions for infection and relapse stages.

This paper studies magnetic insulation in a space-charge-limited vacuum diode through a stationary self-consistent model derived from a singularly perturbed 1.5-dimensional Vlasov-Maxwell system. The central objective is to characterize the transition to the insulated regime, in which electrons are reflected toward the cathode at a free boundary point [Formula: see text]. The analysis is developed in two stages. First, the original kinetic model is reduced to a nonlinear singular system for the electric and magnetic potentials, and then to a nonlinear singular equation for the effective potential [Formula: see text]. For the region [Formula: see text], where [Formula: see text], we prove the existence of physically admissible nonnegative solutions by reformulating the problem as a coupled system of nonlinear Fredholm integral equations and establishing fixed-point existence. Second, for the fully insulated regime [Formula: see text], where [Formula: see text], we perform a bifurcation analysis of complex solutions and their dependence on system parameters and boundary conditions. The resulting bifurcation diagrams identify critical parameter thresholds, describe regime transitions, and provide a quantitative estimate of the insulated diode spacing. These results provide an integrated analytical-computational approach for predicting magnetic-insulation behavior in high-power vacuum diodes for the reduced model studied, combining rigorous existence results with computational bifurcation validation and parameter-space exploration.

In this paper, few common fixed point theorems for Geraghty-type contractions are established in $G_r$-complete fuzzy $b$-metric spaces. Moreover, an example is constructed to authenticate the main result. Ultimately, an application of the results is obtained by exhibiting the existence of a solution for an integral equation.

In this paper, we solve the Cauchy problem for a loaded fractional diffusion equation in an infinite strip. The loaded term is defined as the trace of the fractional derivative of the desired solution on a continuous curve lying inside the domain. We consider all three cases of possible distribution of the order of differentiation in the loaded term (µ) and the order of the time-fractional derivative in the principal differential part of the equation (α). In the first case considered (α > µ), the problem under study is reduced to an integral equation. In the second case (α = µ), we obtain a functional equation. In the third case (α < µ), we are dealing with a differential equation. We show that the condition α > µ ensures the unique solvability of the problem under consideration. In the case of an essentially loaded equation (α ≤ µ), the problem may lose both uniqueness and solvability. In particular, it is shown that if α < µ, then the problem under consideration ceases to be uniquely solvable, and the corresponding homogeneous problem has infinitely many nontrivial solutions. Moreover, in this case, the solvability requires additional conditions that narrow the set of admissible input data.

Weak convergence of the split-step backward Euler method for stochastic Volterra integral equations

In the present work, the diffraction of P-waves is examined in two bonded, dissimilar infinite strips containing an array of collinear Griffith cracks along their interface to stop this type crack expansion. The presence of collinear cracks causes diffraction behavior similar to that observed when vertical clamped boundaries exist at the midpoint between adjacent cracks. To address this, a mixed boundary value problem with appropriate boundary conditions are considered here. By applying suitable integral transform techniques, the problem is reduced to a Fredholm integral equation of the second kind. This integral equation is then solved numerically by using the method of Fox and Goodwin to evaluate the stress intensity factor and other relevant physical quantities near the crack tips. The analysis of these quantities demonstrates that damage detection is feasible within a specific range of wave frequencies by controlling the applied load. Graphical results illustrate the stress intensity factor, normal stress distribution within the crack region, and crack opening displacement at low frequencies for different isotropic materials. Our findings indicate that crack propagation can be identified within a certain frequency band, which is crucial for predicting material failure and assessing structural integrity in applications such as aircraft, bridges, and rotors.

Special solutions to five autonomous integrable partial difference equations via the third and sixth Painlevé equations and the Garnier system in two variables

The embedded cluster reference interaction site model (EC-RISM) employs statistical-mechanical integral equation theory to predict solvent site distributions and their interaction with a solute using quantum-mechanical electronic structure methods. In contrast to apparent surface charge models such as the polarizable continuum (PCM) or the conductor-like screening (COSMO) model, EC-RISM can account for directional solvent-solute interactions due to the description of the solvent based on conventional molecular force field models. Here we present an implementation of EC-RISM combined with correlated wave function methods for ground- and excited-state energies and, for the first time, also for ground- and excited-state energy gradients. This is achieved by self-consistent equilibrating the solvent reaction field with the solute charge density in the correlated and, possibly, electronically excited state. To account for excitonic coupling and nonequilibrium contributions to electronic transition energies, EC-RISM is further combined with COSMO. We present applications to the molecular structures and the absorption and emission energies of 4-(N,N-dimethylamino)benzonitrile (DMABN), the photobase 7-amino-4-methylcoumarin, and the photoacids phenol and 3-cyanophenol in aqueous solution. As expected, for systems without strong directional solvent interactions, such as DMABN, EC-RISM yields results similar to those obtained with COSMO whereas, for molecules or ions that form strong hydrogen bonds to the solvent (particularly the deprotonated photoacids) EC-RISM provides substantial improvements.

Integrable Z22-graded super-Liouville equation and induced Z22-graded super-Virasoro algebra

In this study, we address the problem of locating and approximating solutions of nonlinear Hammerstein-type integral equations with non-separable kernels by employing a higher-order iterative method. To facilitate this process, in the first place, we approximate the non-separable kernel by a separable one and to continue we modify the fifth order iterative method in Arroyo et al. [Approximation of artificial satellites' preliminary orbits: the efficiency challenge. Mathematical and Computer Modelling. 2011;54(7–8):1802–1807] by approximating a solution of a nonlinear Hammerstein-type integral equation. Next, we establish the convergence analysis of a fifth order iterative method, particularly focused on restricted global convergence. After that, we presented the theoretical domains of existence and uniqueness of the solution, by which we are able to find the best ball of location, separation, and uniqueness. Moreover, we examine the effectiveness of approximation of the inverse operator, especially as the number of terms increases in the separable kernel and outline the procedure used to construct the respective operator. We consider a nonlinear Hammerstein-type integral equation to validate the theoretical results.

Many acidic aqueous solutions have lower surface tension than that of pure water, mainly due to surface adsorption of hydronium ions. However, an analytical theory for the surface tension of these acid solutions is yet to be developed. In this work, we propose an analytical theory to explain this phenomenon. We map an acid solution to a restricted primitive model (RPM) and a spherical bubble, serving as the detector of surface tension, to a neutral hard sphere solute in the acid solution. The surface adhesive interaction is described by a length parameter of nonadditivity. The cavity formation energy of the spherical solute is determined analytically using integral equation theory, which, combined with the morphological thermodynamics theory, leads to a formula for the surface tension of acid solutions. The theory is applied to four 1:1 acid solutions (HCl, HBr, HNO3, and HClO4), and good agreement with experimental data is found for concentrations up to 1 mol/L. This work completes the final piece of our series of studies on the influence of ions on the surface tension of electrolyte solutions. It demonstrates that the analytical theory based on cavity formation energy and morphological thermodynamics theory can quantitatively explain most ion-specific effects on surface tension.

In this paper, we investigate the existence of common fixed points for nonexpansive mappings. We propose a novel four-step iterative scheme, referred to as the Z-iteration, which is specifically developed for handling pairs of such mappings. Using this algorithm, we establish several weak and strong convergence results that guarantee the existence of common fixed points. To substantiate the theoretical results, we present constructive examples. Moreover, the practical utility of the proposed method is demonstrated by applying it to approximate solutions of a specific class of nonlinear integral equations in Banach spaces, with a representative example provided to validate its effectiveness.

ABSTRACT This paper presents an accelerated finite element boundary integral (FEBI) method enhanced by kernel-independent and truncation (KIT) techniques for efficient simulation of 3D arbitrary-shaped eddy current non-destructive testing (ECNDT) in B-scan with frequency sweeps. In FEBI, the boundary integral equation (BIE) generates a full matrix after discretisation, while the matrix generated by FEM is sparse; thus, the one by BIE is dominant. To alleviate the computational costs and accelerate the solving process in handling large-scale 3D ECNDT problems, the KIT technique is applied for the matrix of BIE. In consideration of the exponential decay of Green’s function in the metal, the KT technique is applied to eliminate redundant information in the KI-compressed matrices to further improve the efficiency. The diagonal dominance properties in the interactions of the electric current-electric field and magnetic current-magnetic field in the BIE are analysed separately to determine appropriate KIT thresholds. To further enhance the solver’s efficiency in predicting the signals of B-scan with frequency sweeps, a Kriging interpolation model is studied to ensure overall performance. The results show that the proposed solver maintains remarkable accuracy comparable with experimental and numerical methods while achieving significantly improved computational efficiency.

Integral equations frequently arise in surface science, and in some cases, they must be treated as inverse problems. In our previous work on optical tweezers, atomic force microscopy, and surface force measurement apparatus, we performed inverse calculations to obtain the pressure between parallel plates from measured interaction forces. These inverse analyses were used to reconstruct solvation structures near solid surfaces and density distribution profiles of colloidal particles. In the course of these studies, we developed a method that enables inverse analyses through a unified and systematic procedure, hereafter referred to as the Probability Density Alternation (PDA) method. The central idea of this method is to reformulate a given integral equation in terms of probability density functions. In this letter, we demonstrate the validity of the PDA method both analytically and numerically. While the PDA method is less advantageous for single integral equations, it becomes a convenient and powerful approach for inverse analyses involving double or higher-order integral equations.

A multi-stage hybrid technique for 2D stochastic nonlinear itô-Volterra integral equations