Highly-accurate numerical scheme for a system of nonlinear Abel–Volterra integral equations

Dual Reciprocity Boundary Element Method analysis of MHD Brinkman–Rivlin–Ericksen viscoelastic flow past cylindrical obstacles in porous microchannel

plasmonX: An open-source code for nanoplasmonics

A Galerkin meshless boundary integral equation method for 2D exterior acoustics with arbitrary wavenumbers

Additive general integral equations in thermoelastic micromechanics of composites

Full-wave modeling of transcranial ultrasound using volume-surface integral equations and CT-derived heterogeneous skull data.

We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts.

In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results.

This paper examines an m-cyclic coupled system of sequential (k,ψ)-Hilfer and (k,ψ)-Caputo fractional differential equations with boundary conditions. The nonlinearities follow a cyclic pattern: for j=1,…,m−1, fj depends on xj and xj+1 and fm depends on xm and x1, forming a closed loop of interactions. We convert the system into an equivalent integral equation and establish existence and uniqueness results using four fixed-point theorems: the Banach contraction principle, Schaefer’s theorem, Krasnosel’skiĭ’s theorem, and the Leray–Schauder alternative. A thorough Ulam–Hyers stability analysis is presented with explicit stability constants. Numerical examples illustrate the applicability of the theoretical findings.

Abstract This paper systematically investigates the inverse scattering transform (IST) and its applications to the Davey–Stewartson (DS) type equations, focusing on the 3 × 3 matrix elliptic system. To clarify the theoretical framework of IST associated with this system, we decompose it into three core components: the direct problem, the inverse problem, and the reversibility of the S and S −1 . Furthermore, an in-depth analysis of the fundamental integral equation in z is conducted, and a series of critical properties of the Jost function μ ( z , k ) are established, establishing critical properties of the Jost function μ ( z , k ) and ν ( z , k ) , including solution uniqueness, global boundedness, regularity, asymptotic expansions of μ ( z , k ) and ν ( z , k ) , and L 2 -properties. By analyzing the Fourier transform of the fundamental integral equation in z , we find that the fundamental integral equation in ξ exhibits identical properties. Based on these direct and inverse scattering results, we establish various properties of the direct and inverse scattering maps, deepening the understanding of structural connections between the scattering data T and the potential function Q . Finally, using the above theoretical findings, we successfully derive the DS type equation, analyze the existence, uniqueness of its solution, and conservation laws. This work demonstrates the practical value of the 3 × 3 matrix-based IST theory in constructing and analyzing 2+1 dimensional nonlinear evolution equations, providing theoretical support for related studies.

Within the framework of Somigliana’s displacement and traction identities, we propose an extended equivalent elastic model that enables a BEM that is singularity-free in the primary solution stage for two-dimensional elastostatics. The central idea is to shift the integration boundary from the physical contour S1 to an auxiliary contour S2, introducing a geometric separation that removes boundary-source singularities from the discrete system. When the separation between S1 and S2 is sufficiently large, all integrals in the assembled algebraic equations become regular, and post-processing can be performed in a unified manner using the same nonsingular expressions for both boundary and interior evaluation. We introduce a practical separation measure, the dimensionless parameter δ, and verify that a moderate choice (e.g., δ≈0.5) is effective through a rigid-body displacement test. Benchmark examples demonstrate that, at lower computational cost, the proposed method improves accuracy and convergence compared with the conventional direct BEM (displacement boundary integral equation (BIE) with free-term coefficient c=1/2) and compares favorably with the finite element method (FEM). In particular, thin structures can be treated directly without invoking plate or shell theories.

A computational scheme for the approximate solution of an integral equation with the Grünwald–Letnikov fractional integral has been developed, based on the least squares method. A distinctive feature of this scheme is the use of a neural network to compute the coefficients for the least squares method. The relevance of the study is обусловлена by the fact that, at present, artificial intelligence is increasingly being applied to solve many practical problems related to various physical processes. An estimate of the convergence of approximate solutions to the exact solution has been obtained. Possible directions for the further application of artificial intelligence in solving physical problems are also considered.

ABSTRACT This paper takes the integrable Kuralay equations as the research object, aiming to derive various types of soliton solutions and explore the integrable motion of space curves induced by the equations, so as to support the research on nonlinear spin dynamics in the field of magnetic materials. In this paper, the unified ‐expansion method is used to systematically derive soliton solutions expressed by Jacobian elliptic functions. Through parameter degeneration (degenerating into hyperbolic function solutions when the modulus and trigonometric function solutions when ), the evolutionary relationship among different solutions is revealed. Eight types of soliton solutions are obtained in this paper, including periodic trigonometric function solutions, parabolic function solutions, singular solutions, and M‐shaped/W‐shaped solitons (corresponding to Figures 1–8). The parameter configurations and 2D/3D graphical characteristics of each solution are clarified (e.g., kink waves show unidirectional step‐like transitions, and M‐shaped bright waves possess symmetric double peaks). All solitons have clear boundaries without diffusion. For numerical verification, the fourth‐order Runge‐Kutta method combined with Richardson extrapolation is adopted, reducing the calculation error from to . In addition, phase portrait, bifurcation, and initial condition sensitivity analyses are supplemented, and the stability of equilibrium points is classified by the eigenvalues of the Jacobian matrix. In terms of physical implications, the soliton solutions are deeply associated with magnetic spin systems. For instance, kink waves correspond to the migration of spin domain walls, supporting the reading and writing operations of magnetic storage; M‐shaped/W‐shaped solitons contribute to the realization of multistate and high‐density storage. The quantitative influences of parameters on the low‐power consumption and high‐capacity performance of devices are clarified, providing theoretical support and practical guidance for the research on nonlinear spin dynamics and the design of magnetic storage and magneto‐optical modulation devices.

ABSTRACT In this article, we introduce and investigate a new class of integral operators whose kernels are expressed through the generalized Mittag–Leffler matrix function and the confluent hypergeometric matrix function. We establish the boundedness of these operators in the Lebesgue space and the continuous space . Furthermore, we analyze the composition of these new operators with standard Riemann–Liouville fractional integrals. Subsequently, we formulate a Cauchy‐type problem for a fractional integro‐differential equation with matrix arguments, elegantly reducing it to a Volterra integral equation of the second kind and obtaining its solution via the method of successive approximations. In addition, we present effective computational strategies for the numerical evaluation of these integral operators.

Abstract In this paper, we study best proximity point results and the existence of solutions for a class of nonlinear functional and integral equations in strictly convex Banach spaces and Banach algebras. By employing measures of weak noncompactness, weak sequential continuity, and Lipschitz-type conditions, we extend classical fixed point theorems to the setting of non-weakly compact operators. We establish general conditions under which proximal condensing operators admit best proximity points, even in non-reflexive Banach spaces. These results are further applied to nonlinear functional equations in $$L^1[0,1]$$ , where the operators involved are Nemytskii-type or integral operators that are not necessarily weakly compact. We also consider product-type operators in Banach algebras satisfying property $$(\textbf{P})$$ , demonstrating the applicability of our main theorems to nonlinear integral equations with an explicit example. The results provide a unified framework for analyzing solvability of nonlinear equations, extending existing approaches based on the Schauder–Tychonoff and O’Regan fixed point theorems, and illustrate the role of weak noncompactness measures in obtaining best proximity solutions.

We consider the second initial boundary value problem for a spatially multidimensional second-order parabolic equation with Dini continuous coefficients in a semi- bounded domain with a nonsmooth with respect to time variable lateral boundary. Using the boundary integral equations method we construct a regular solution to this problem. Estimate for the solution is obtained.

The characteristics of the scattering of a stationary sound signal from a set of concentrated sources are obtained, simulating the pulsations of the "volumetric" component of the sound of rotation of the propulsion model of an uninhabited underwater vehicle (UUV) in the nozzle. An algorithm is used for the numerical solution of the integral equation for the displacement vector of an elastic shell of a non-analytical shape approximating the surface of the casing and Kirchhoff's integral equation for diffuse pressure. The external surfaces of the UUV housings and nozzles are modeled by a set of isotropic gas-filled shells (finite cylindrical, hemispherical and conical). The total sound field at the observation point takes into account the contribution from the radiation of point sources directly into the liquid medium and through the wall of the nozzle, as well as the sound field of point sources scattered on the body of the UUV model. The considered approach can also be extended to account for the "power" component (dipole radiators) by introducing additional sources of false signs. A comparative computational evaluation of the effectiveness of several variants of the propulsion nozzle of the UUV model for reducing the levels of the sound signal scattered by the body has been performed. The features of scattering characteristics related to the corresponding ranges of the wave dimensions of the model and the viewing angles are noted.

In this paper, we introduce a new class of generalized rational Suzuki-type contractions in the setting of double controlled metric spaces. This framework extends controlled metric spaces by incorporating two independent control functions, thereby providing a more flexible structure for the analysis of nonlinear mappings. We establish an existence and uniqueness theorem for fixed points under this new contractive condition, which unifies and generalizes several known results in metric, $b$-metric, and controlled metric spaces. In addition, we investigate fundamental properties of the proposed framework, including a characterization via Picard iteration, Ulam--Hyers stability, convergence rate estimates, and data dependence of fixed points. As an application, we study the existence and uniqueness of solutions to a nonlinear Fredholm integral equation, showing that the proposed approach accommodates nonlinear kernels under weaker conditions than classical methods. The results contribute to the development of fixed point theory in generalized metric structures and provide a robust analytical tool for nonlinear problems arising in applied mathematics.

Abstract This study introduces a novel five-unknown hyperbolic Higher-Order Shear Deformation Theory (HSDT) for analyzing the mechanical stability and vibration behavior of functionally graded (FG) plates under dynamic loading across various boundary conditions. By incorporating the transverse shear effects without correction factors, the proposed theory employs the virtual work principle and Hamilton model to derive the equilibrium and motion equations, along with analytical solutions for the buckling and vibration responses. A new shape function is proposed to enhance the modeling accuracy, complemented by a Quasi-3D theory with five unknown terms using integral equations. This study validated the efficiency and accuracy of this approach by comparing it with existing literature, demonstrating its robustness. Comprehensive parametric studies revealed the intricate effects of the material index, homogeneity ratio, aspect ratio, and thickness ratio on the critical buckling loads, natural frequencies, and dynamic responses for diverse loading scenarios. The findings highlight the significant influence of boundary conditions and material gradation on structural performance, offering valuable insights for optimizing FG plate designs in engineering applications in which stability and vibration control are critical. This study advances the understanding of FG plate mechanics using a refined theoretical framework and detailed analytical investigation.

In this paper, we investigate a (k,Υ) fractional quadratic integral equation in the Banach space of real-valued continuous functions on [0,1]. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point theorem, we provide sufficient conditions for the existence of at least one monotonic solution and analyze its stability. Finally, an illustrative example is presented to demonstrate the theoretical results, including several particular cases.