Coupled Integral Equations Research Articles

Mathematical analysis on the propagation of Griffith crack in an initially stressed strip subjected to punch pressure

The purpose of this study is to analyze the features of a moving Griffith crack in an initially stressed infinitely long and finitely thick isotropic strip with moving parallel punches of constant load acting on its boundaries owing to plane wave propagation under point loading. Coupled singular integral equations and singularities of the Cauchy-type are used to formulate the present model, Dirac delta function is employed to analyze point load located at the moving crack edge, and Hilbert transformation properties are used for obtaining stress intensity factor (SIF) with constant point loading. Numerical simulations and graphical illustrations are performed to analyze the influences of the prevalent parameters, viz. initial stresses, punch pressure, distinct positions of point load, length and speed of the crack on the SIF for the considered isotropic material strip.

Locally Attractivity Solution of Coupled Fractional Integral Equation in Banach Algebras

In this paper we prove the Locally Attractivity solution of coupled hybrid functional integral equation of fractional order (CQHFIE) in Banach algebras. Our main result is based on the hybrid fixed point theory used theorem due to Dhage B. C.

Mathematical study on the propagation of Griffith crack in a dry sandy strip subjected to punch pressure

The present study investigates the features of a moving Griffith crack in a dry sandy infinitely long and finitely thick isotropic strip with moving parallel punches of constant load acting on both of its boundaries owing to plane wave propagation under point loading. Coupled singular integral equations and Cauchy-type singularities are used to formulate the present model, the Dirac delta function is employed to analyse the point load located at the moving crack edge, and the properties of Hilbert transformation are employed to obtain the closed-form expression of the stress intensity factor with constant point loading. For the sake of validation, the obtained results are matched with pre-established and standard results. Numerical simulations and graphical illustrations are performed to analyse the influences of the prevalent parameters, viz., sandiness parameter, punch pressure, distinct point load positions, length and speed of the crack corresponding to the plane wave on the stress intensity factor for the considered dry sandy isotropic material strip.

An improved lumped model for freezing of a freely suspended supercooled water droplet in air stream

This work deals with the mathematical modeling of the transient freezing process of a supercooled water droplet in a cold air stream. The aim is to develop a simple yet accurate lumped-differential model for the energy balance for a freely suspended water droplet undergoing solidification, that allows for cost effective computations of the temperatures and freezing front evolution along the whole process. The complete freezing process was described by four distinct stages, namely, supercooling, recalescence, solidification, and cooling. At each stage, the Coupled Integral Equations Approach (CIEA) is employed, which reduces the partial differential equation for the temperature distribution within the spherical droplet into coupled ordinary differential equations for dimensionless boundary temperatures and the moving interface position. The resulting lumped-differential model is expected to offer improved accuracy with respect to the classical lumped system analysis, since boundary conditions are accounted for in the averaging process through Hermite approximations for integrals. The results of the CIEA were verified using a recently advanced accurate hybrid numerical-analytical solution through the Generalized Integral Transform Technique (GITT), for the full partial differential formulation, and comparisons with numerical and experimental results from the literature. After verification and validation of the proposed model, a parametric analysis is implemented, for different conditions of airflow velocity and droplet radius, which lead to variations in the Biot numbers that allow to inspect for their influence on the accuracy of the improved lumped-differential formulation.

Scattering from an inhomogeneous dielectric cylinder partially buried in a dielectric half-space

A hybrid method is presented for computing the TM polarized plane wave scattered by an inhomogeneous penetrable cylinder embedded in the planar interface between two dielectric half-spaces. In this method, the infinite interface is represented by an unknown equivalent magnetic current, which is expressed in terms of the difference between the current with the cylinder present and the current of the known physical optics (PO) solution of the unperturbed (without cylinder) problem. Coupled integral equations for the equivalent surface electric and magnetic currents on the cylinder surface and the perturbed equivalent magnetic current on the finite section of the interface are obtained and solved by the Method of Moments (MoM). The half-space and the inhomogeneous cylinder configuration in this paper can be used in a wide range of settings.

Approximation error model (AEM) approach with hybrid methods in the forward-inverse analysis of the transesterification reaction in 3D-microreactors

This work advances the approximation error model approach for the inverse analysis of the biodiesel synthesis using soybean oil and methanol in 3D-microreactors. Two hybrid numerical-analytical approaches of reduced computational cost are considered to offer an approximate forward problem solution for a three-dimensional nonlinear coupled diffusive-convective-reactive model. First, the Generalized Integral Transform Technique (GITT) is applied using approximate non-converged solutions of the 3D model, by adopting low truncation orders in the eigenfunction expansions. Second, the Coupled Integral Equations Approach (CIEA) provides a reduced mathematical model for the average concentrations, which leads to inherently approximate solutions. The AEM approach through the Bayesian framework is illustrated in the simultaneous estimation of kinetic and diffusion coefficients of the transesterification reaction. For this purpose, the fully converged GITT results with higher truncation orders for the 3D partial differential model are employed as reference results to define the approximations errors. The results highlight that either the non-converged solutions via GITT or the reduced model solution obtained via CIEA, when taking into account the model error, are robust and cost-effective alternatives for the inverse analysis of nonlinear convection–diffusion-reaction problems.

A Multilevel Iteration Method for Solving a Coupled Integral Equation Model in Image Restoration

The problem of out-of-focus image restoration can be modeled as an ill-posed integral equation, which can be regularized as a second kind of equation using the Tikhonov method. The multiscale collocation method with the compression strategy has already been developed to discretize this well-posed equation. However, the integral computation and solution of the large multiscale collocation integral equation are two time-consuming processes. To overcome these difficulties, we propose a fully discrete multiscale collocation method using an integral approximation strategy to compute the integral, which efficiently converts the integral operation to the matrix operation and reduces costs. In addition, we also propose a multilevel iteration method (MIM) to solve the fully discrete integral equation obtained from the integral approximation strategy. Herein, the stopping criterion and the computation complexity that correspond to the MIM are shown. Furthermore, a posteriori parameter choice strategy is developed for this method, and the final convergence order is evaluated. We present three numerical experiments to display the performance and computation efficiency of our proposed methods.

Scattering of surface waves by a pair of asymmetric thin elliptic arc shaped plates with variable permeability

Abstract We consider the interaction between incident surface water waves and two asymmetric curved porous plates which assume the shape of ellipse. The permeability of the plates is not uniform. This is assumed to vary along the circumference of the elliptical arc. Two coupled integral equations are derived for the problem and are solved by employing an appropriate algorithm. Using these solutions, the numerical estimates for the reflection and the transmission coefficients, amplitudes of the hydrodynamic forces and energy dissipation coefficient are computed. The accuracy of the numerical results is established through an energy identity valid for permeable plates, and by comparing them with the published ones. A selection of new results is also presented for illustrative purposes. Importantly, in the present paper a mathematical method is developed to solve the problem of wave interaction with non-vertical (elliptic) plates with variable permeability. The derived numerical results indicate that the system of dual elliptic plates serves as a suitable model for a wave lens.

Improved lumped analysis of Graetz problems with axial diffusion

This paper proposes analytical approximations for solving extended Graetz problems with axial diffusion in infinite domains. A general formulation, valid for both parallel-plates channels and circular ducts, is employed, and a discontinuous Dirichlet condition is applied at the wall. The adopted methodology consists of transforming the 2D convection equation into simpler one-dimensional forms, using approximation rules provided by the Coupled Integral Equations Approach. This technique is employed for producing expressions for the bulk temperature, and different levels of approximations are analyzed. The results are compared with an exact analytical solution to the problem and an expression obtained with the classical lumped system analysis (CLSA). While the results obtained with the CLSA are demonstrated to be substantially discrepant compared to the exact solution, the proposed improved formulas are equivalently simple and are shown to provide very good estimates for calculating the bulk temperature distribution. In addition, estimates for the length through which heat is diffused backward are also calculated with the derived approximate formulations and a practically perfect agreement is obtained with the higher-order approximation and the exact solution data.

A MoM solution for TM scattering by dielectric cylinders above an infinite flat surface

A simple solution method is presented for transverse magnetic plane wave scattering by a dielectric cylinder of arbitrary cross-section in front of an infinite flat dielectric surface. The interface between two semi-infinite, dissimilar half-spaces is considered by a perturbed equivalent magnetic current. Coupled integral equations are obtained from a special form of surface equivalence principles and are solved numerically by the Method of Moments. The scattered tangential electric field at interface is displayed and far-zone fields are compared to the results available in the literature.

Numerical Simulation of the Heat Transfer for a Three-Phase Reactor of Fixed Bed

Heat transfer plays an important role in several applications of fixed bed reactors with concurrent down-flow of liquid and gas. Because of the possibility of forming hot spots on the bed of this reactor, which can inhibit the process catalyst and impair the proper functioning of the equipment. Thus, the present paper presents a mathematical model to simulate the process of heat transfer in these reactors. The mathematical modeling was performed by the technique known as Coupled Integral Equation Approach (CIEA). It is a new formulation, based on Hermite integral approximations (H0,0/H0,0 and H0,0/H1,1), present more accurate results with a shorter CPU time, than the more traditional techniques and transforms the partial differential equations (PDEs) into ordinary differential equations (ODEs). From this formulation, the results of the ODEs were obtained through the Euler method and the variation of the temperature over time with different values of the heat transfer coefficient was perceptible, being clear the influence of this parameter for the good application of the equipment the desired processes, such as steam reforming of hydrocarbons. In addition, the paper contains a comparison of the mathematical techniques of the approximations (H0,0/H0,0 and H0,0/H1,1), making it clear that H0,0/H1,1 is the best technique to be used in the simulation.

Estimation of kinetic coefficients in micro-reactors for biodiesel synthesis: Bayesian inference with reduced mass transfer model

High yields in the production of biodiesel at very short residence times can be obtained by means of micro-reactors technology. The theoretical study of the reaction mechanisms involved in biodiesel synthesis in micro-reactors is crucial in achieving adequate design conditions towards maximizing biodiesel production. Such a physico-chemical phenomenon involves complex liquid–liquid flow and mass transfer processes, besides the transesterification reaction, and the determination of the kinetic coefficients is essential in using the mathematical models for the design of the micro-reactors. In this work, a nonlinear coupled mathematical model of first-order ordinary differential equations has been developed, using the Coupled Integral Equations Approach (CIEA) for model reduction, starting from a diffusive-convective-reactive three-dimensional mathematical model that describes the local species concentrations involved in the biodiesel synthesis. The ODEs system is numerically solved using the NDSolve routine of the Mathematica platform. The Markov Chain Monte Carlo method (MCMC) is employed in solving the inverse problem to estimate the kinetic coefficients, using synthetic experimental data with low conversion rates, which maximize the presence of intermediate species and increase the sensitivity of the problem to the desired parameters. The results presented indicate that the CIEA in combination with the MCMC statistical inference yield an efficient and robust combination for the direct - inverse analysis of such reactive mass transfer problems.

A Perturbation Method for Scattering by a Dielectric Cylinder Buried in a Half-Space

A simple method is presented for transverse magnetic plane-wave scattering by a dielectric cylinder of arbitrary cross section buried in a half-space. The interface between two semi-infinite dissimilar half-spaces is considered as an infinite aperture, which is represented by a perturbed equivalent magnetic current. Coupled integral equations for the aperture electric field (or equivalent magnetic current) and the equivalent currents on the buried cylinder are obtained from surface equivalence principles and are solved numerically by the method of moments. Data for the electric field at the interface and far field are presented for cases of interest and are compared to the results available in the literature.

Coupled Volterra integral equations with blowing up solutions

In this paper, a system of nonlinear integral equations related to combustion problems is considered. Necessary and sufficient conditions for the existence and explosion of positive solutions are given. In addition, the uniqueness of the positive solutions is shown. The main results are obtained by monotonicity methods.

Scattering of waves by a half-space of periodic scatterers using broadband Green's function.

An efficient scatterer-free full-wave solution for plane wave scattering from a half-space of two-dimensional (2D) periodic scatterers is derived using broadband Green's function. The Green's function is constructed using band solutions of the infinite periodic structure, and it satisfies boundary conditions on all the scatterers. A low wavenumber extraction technique is applied to the Green's function to accelerate the convergence of the modal expansion. This facilitates the Green's function with low wavenumber extraction (BBGFL) to be evaluated over a broadband as the modal solutions are independent of wavenumber. Coupled surface integral equations (SIE) are constructed using the BBGFL and the free-space Green's function respectively for the two half-spaces with unknowns only on the interface. The method is distinct from the effective medium approach which represents the periodic scatters with an effective medium. This new approach provides accurate near-field solutions around the interface with localized field patterns useful for surface plasmon polaritons and topological edge states examinations.

Moment method analysis of periodic array of thin conducting strips using waveguide simulator

In this paper, analysis of free-standing periodic array of thin conducting strips using waveguide simulator is presented. The method of moment is applied to numerically obtain its scattering parameters. With only two elements inside the waveguide simulator, scattering response of the infinite array illuminated by plane waves is accurately achieved. Coupled integral equations, due to the presence of two elements, are solved using pulse expansion and point matching technique. Numerically obtained results are compared with waveguide and plane wave simulations in CST MWS and measurement results for validation.

Fracture analysis of a penny-shaped dielectric crack in a piezoelectric cylinder

Electroelastic behavior induced by a penny-shaped dielectric crack in a piezoelectric cylinder (i.e., a finite piezoelectric body in the radius direction) is considered in this paper. Two coupled Fredholm integral equations are derived by employing the Hankel transform technique with the introduction of certain auxiliary functions. The intensity factors of stress, electrical displacement and crack opening displacement (COD) are obtained in closed forms. The effects of the radius of the piezoelectric cylinder, the applied electrical field and permittivity of the crack interior on the COD intensity factor are illustrated numerically. The results of both the electrically permeable and impermeable boundary conditions are simultaneously displayed as special cases.

Magnetization in a Finite Superconducting Hollow Cylinder in the Presence of the Radial and Azimuthal Transport Currents

The main aim of this study is to investigate the magnetization of a finite superconducting hollow cylinder in an axial applied magnetic field and in presence of different values of the radial and azimuthal transport currents in the framework of the three critical states models, namely, Bean, Kim, and exponential models. A set of self-consistency coupled-integral equations for the magnetization hysteresis loops have been solved by using Newton’s numerical method. Throughout the calculations, it has been assumed that the flux penetration starts from the inner lateral surface of the sample. A comparison has been made between the magnetic responses of the sample in applying the radial and azimuthal transport currents. It was found that when the transport current increases, the saturation value, enclosed area of the hysteresis loops, and the initial slopes in the virgin curve decreases. However, given that the flux penetration and demagnetizing effect did not affect by the azimuthal angle, the rate of the mentioned changes as the transport current is in azimuthal direction was slower than when that is in the radial direction. In addition, the radial transport current destroyed the azimuthal symmetry of the shielding current distribution and induced the nonuniform magnetic field in the finite sample. The obtained results have shown the importance of the magnetic field dependence of the critical current density on the magnetic response of the finite superconducting hollow cylinder.

Fano-resonant aluminum and gold nanostructures created with a tunable, up-scalable process

An up-scalable approach for creating Fano-resonant nanostructures on large surfaces at visible wavelengths is demonstrated. The use of processes suitable for high throughput fabrication and the choice of aluminum as a cost-efficient plasmonic material ensure that the presented insights are valuable even in consideration of typical industrial constraints. In particular, wafer-scale fabrication and the process compatibility with roll-to-roll embossing are demonstrated. It is shown that through adjustment of readily accessible evaporation parameters, the shape and position of the optical resonance can be tuned within a spectral band of more than 70 nm. The experimental data are complemented with rigorous coupled wave analysis and surface integral equation simulations. Calculated electric fields as well as surface charges shed light onto the physics behind the present resonances. In particular, a surface plasmon polariton is found to couple to a localized plasmonic mode with a hexapolar charge distribution, leading to a Fano-like resonance. Further understanding of the interactions at hand is gained by considering both aluminum and gold nanostructures.

Physics of backscattering in terms of mode coupling applied to measured seabed roughness spectra in shallow water

Energy conserving coupled integral equations for the forward and backward propagating modal components have been previously developed [J. Acoust. Soc. Am. 130, 2673–2680 (2011)]. A rough seabed surface leads to a backscattered field and modifies the interference structure of the forward propagating field. Perturbation theory applied to the basic coupled integral equations allows for physical insight into the correlation of the roughness spectrum to the forward and backward modal intensities and cross mode coherence. This study applies the Nx2D integral equation coupled-mode approach to 3-D roughness measurements and examines the physics of the coupling of the forward and backward field components and computes the modal intensities as a function of azimuth. The roughness measurements were made in about 20 m of water off Panama City, Florida. [Work supported by ONR Code 322 OA.]