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The paper addresses a computational method to simulate more accurate models for population growth with immigration, focusing on integral equations (IEs) featuring a delay parameter in the time variable. The proposed method utilizes Legendre wavelets within the Galerkin scheme as a orthonormal basis. Legendre wavelets are known for their localized functions, offering suitable precision and stability in simulating time-delay biological models. This approach employs the composite Gauss-Legendre (CGL) quadrature rule to compute integrals appeared in the scheme. An error bound analysis demonstrates a convergence rate of order 2 − M k. Additionally, various numerical examples are presented to show the efficiency, accuracy and validate the theoretical error estimate of the novel technique. [ABSTRACT FROM AUTHOR]

Motivated by applications in population models, we consider S -asymptotically periodic solution of fractional differential equations with periodic environment forces or asymptotically periodic ones. The system is quasi-monotone, and the existence of positive S -asymptotically periodic solution is established by using upper and lower solutions. The sufficient conditions that ensure the uniqueness of positive S -asymptotically periodic solution are also established on the basis of theory of sublinear operator. The applications of the general conclusions to classical population models yield the global convergence of positive S -asymptotically periodic solution in logistic equation with or without weak Allee effect, and the model of two cooperative populations. • Existence of asymptotically periodic solution. • Uniqueness of asymptotically periodic solution. • Upper and lower solutions. • Monotonic operator. • Ecological applications. [ABSTRACT FROM AUTHOR]

• A series solution for an interface crack under remote mixed-moded loading is given. • Fracture mode mixity based on the non-oscillatory cohesive-spring interface solution is presented. • Machine-learning model was developed for fast evaluation of phase angle. • Can be applied to interface fracture characterization for IC interconnect structure. The oscillatory crack-tip field in the classical interface fracture mechanics solution often leads to complications in identifying the mode mixity in bimaterial interface fracture problems. In this paper, a non-singular series solution for the continuum problem of a crack on the cohesive-spring interface under mixed-mode loading is presented and compared to the classical oscillatory solution. A Gaussian process regression model was also developed to enable quick evaluation of the phase angle for interface delamination characterization by using mixed-mode bending fracture test. [ABSTRACT FROM AUTHOR]

This paper is concerned with water wave scattering of an obliquely incident surface wave train on a new type of breakwater model, which is called inverse T-type bottom-standing breakwater. Based on linear water wave theory, the physical problem reduces to a BVP (boundary value problem) which is solved by using the methodology of Havelock's expansion and Galerkin's approximation technique. In the solution process, a set of first kind Fredholm-type integral equations is solved approximately by choosing Gegenbauer polynomial as basis function in the multi-term Galerkin's technique. To address the singularities of order one-third that appears at the corners of the inverse T-type breakwater, appropriate basis polynomial with suitable weight functions is considered for solving the system of integral equations which consist cube root singularity. This study validates its findings by comparing the results with prior research in the field. The efficiency of the breakwater system is analyzed by evaluating the reflection coefficient and horizontal drift force for a broad variety of structural and wave parameters which are found to feature significant effects on the structures. The identification of Bragg resonance and the analogy to the Doppler effect in this study provide a remarkable benchmark for analyzing the design of water wave breakwaters. • The physical problem involving inverse T-type breakwater reduces to a BVP. • Havelock's expansion and Galerkin technique are used to solve the physical problem. • Cube root singularities are tackled using Gegenbauer polynomial as basis function. • Drift force, reflection and transmission coefficients are presented graphically. • Bragg resonance and Doppler effect have been observed in different graphs. [ABSTRACT FROM AUTHOR]

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point. [ABSTRACT FROM AUTHOR]

A visco-acoustic medium is an approximate representation of the absorption and attenuation characteristics of the actual earth medium, which will cause energy attenuation and velocity dispersion of seismic waves. The propagation law of scattered waves in viscoelastic media can be revealed through seismic wave forward simulation. However, the numerical simulation of scattered waves in a visco-acoustic medium using the De Wolf approximation method based on the Lippmann-Schwingr (L-S) equation has not been studied. Therefore, based on the Futterman dispersion relation, the mathematical expression of the integral equation of acoustic wave propagation and scattering in the visco-acoustic medium is derived. We combine the De Wolf approximation with the quality factor Q. The thin slab approximation and screen approximation methods are two realizations of the De Wolf approximation. The thin slab approximation and the screen approximation continuation operator in the viscous-acoustic medium are constructed and applied to the numerical simulation of the scattered wave field. The research results of the concave model and complex model indicate that due to the influence of medium viscosity, the energy of the deep reflected waves is weaker in the forward modeling records of the thin slab approximation method and screen approximation method, and the dominant frequency of seismic waves moves towards the low-frequency direction; Compared with the finite difference results, they only calculate one reflection, and there are no multiple wave signals in the seismic records. Finally, We discuss the thin slab approximation and the screen approximation method for detail around the division of the thickness of the thin slab, the selection of the reference frequency, and the non-uniformity of the medium. • Based on the De Wolf approximation, we proposed a numerical simulation method for scattered waves in visco-acoustic media. • The proposed method can reveal the propagation law of scattered waves in viscous acoustic media. • We combined the De Wolf approximation with the quality factor Q , and used two methods to achieve it. • We analyzed the influence of different thin slab thicknesses on the forward modeling of scattered waves. [ABSTRACT FROM AUTHOR]

We present a computational method for reconstructing the refractive index of an unknown complex-shaped two-dimensional medium with embedded metal inclusions from its transverse magnetic electromagnetic scattering properties. We present a novel hybrid surface-volume integral equation that generalises the Lippmann–Schwinger equation to media containing embedded scatterers. Using this hybrid equation, we show that the Frechet derivative of the forward-mapping satisfies a particular inhomogeneous wave scattering problem. We solve the inhomogeneous wave scattering problem using a novel coupled FEM-BEM formulation. Our numerical scheme is based on a thin plate spline ansatz for the refractive index, which can be constructed using a relatively small number of control points, and can be efficiently constructed and evaluated even for the complex-shaped multiply-connected domains of interest. Numerical experiments demonstrate the effectiveness of our method by reconstructing several challenging media. [ABSTRACT FROM AUTHOR]

In this paper, integral equation formulations and field-dependent micromechanical modeling of global nonlinear magneto-mechanical behavior of magneto-elastic composites under high strain and magnetic field are elaborated. The modeling is obtained based on an extension of micro–macro transition inclusion problem to nonlinear behavior using field-dependent and highly nonlinear localization tensors. A methodological procedure is elaborated based on newly introduced strain and magnetic field-dependent Green tensors, strain and magnetic field-dependent integral equations linked to tensors of Eshelby and micromechanical approaches. The field-dependent global moduli are predicted based on the Mori–Tanaka approach and self-consistent method. Iterative incremental schemes based on the Newton–Raphson and fixed-point algorithms are examined and established precise semi-analytic equations of the effective magneto-elastic properties of composites that are dependent on the magnetic-strain field for different types of inclusions. A numerical code is elaborated for numerical predictions and the obtained field-dependent effective magneto-elastic coefficients are obtained and presented for various volume fractions of inclusion, types and shapes of the reinforced nonlinear composites. • Nonlinear micromechanical modeling of fully coupled magneto-elastic under large deformation and high magnetic field. • Magnetic-Strain dependent Green tensors and associated localization tensors. • Iterative incremental schemes for semi-analytical magnetic-strain field dependent effective magneto-elastic properties. • Magneto-mechanical field dependent effective properties for several types of matrix and inclusions phases. [ABSTRACT FROM AUTHOR]

Induced currents in the bulk superconducting cylinder of finite length placed in a transverse magnetic field are calculated by the partial and complete separation of variables. The Fourier transform over the angular and axial coordinates reduces the problem to the solution of one-dimensional integral equations. The harmonic balance method is applied to account for the highly nonlinear constitutive relation between the voltage and current. The current and field distribution is compared with the results of FEA. The approximation of the current distribution by a set of cylindrical and disc layers is presented. The magnetic moment of induced currents is analyzed in the increasing applied field when the cylinder is magnetized up to the state of saturation. [ABSTRACT FROM AUTHOR]

In this work, we study the inverse problem of determining the convolution kernel in the Cauchy problem for the time-fractional diffusion equation by a single observation at the point x = 0 of the diffusion process. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation is constructed. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function which will be used for the study inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. By the contracted mapping principle, the local existence and global uniqueness results are proven. Also the stability estimate is obtained. [ABSTRACT FROM AUTHOR]