Fredholm Integral Equation Research Articles

Optimal design of electrodes for functional electrical stimulation applications to single layer isotropic tissues

PurposeThe purpose of this paper is to investigate the design of skin surface electrodes for functional electrical stimulation using an isotropic single layered model of the skin and underlying tissue. A concentric ring electrode geometry was analysed and compared with a conventional configuration, specifically to localise and maximise the activation at depth and minimise the peak current density at the skin surface.Design/methodology/approachThe mathematical formulation determines the spatial electric potential distribution in the tissue, using the solution to the Laplace equation in the lower half space subject to boundary conditions given by the complete electrode model and appropriate asymptotic decay. Hence, it is shown that the electric potential satisfies a weakly singular Fredholm integral equation of the second kind which is then solved numerically in MATLAB for a novel concentric ring electrode configuration and the conventional two disk side-by-side electrode configuration.FindingsIn both models, the electrode geometry can be optimised to obtain a higher activation and lower maximum current density. The concentric ring electrode configuration, however, provides improved performance over the traditional two disk side-by-side electrode configuration.Research limitations/implicationsIn this study, only a single layer of medium was investigated. A comparison with multilayer tissue models and in vivo validation of numerical simulations are required.Originality/valueThe developed mathematical approaches and simulations revealed the parameters that influence nerve activation and facilitated the theoretical comparison of the two electrode configurations. The concentric ring configuration potentially may have significant clinical advantages.

Common fixed point theorem for pair of quasi triangular α-orbital admissible mappings in complete metric space with application

In this paper, we establish a common fixed point theorem for pair of quasi triangular α-orbital admissible with an interpolative (φ, ψ)- Banach-Kannan-Chatterjea type Z-contraction mappings with respect to simulation function in complete metric space. An illustrative example is furnished to validate our main result. Our result extends the result of M. S. Khan et al. [15]. As an application, we provide the existence of a solution for a nonlinear Fredholm integral equations.

Optimization of the regularization parameter in the Dual Annealing method used for the reconstruction of energy spectrum of electron beam generated by the AQURE mobile accelerator

IntroductionThe shape of the energy spectrum is an essential component of any electron beam Monte Carlo model. Due to specialized equipment and the long measurement time for the direct methods for determining the energy spectrum, attractive alternatives are backward spectrum reconstructions from the measured data. One such approach is solving the first-degree Fredholm integral equation with appropriate regularization. It makes it possible to calculate the depth distribution as the sum of the distributions from monoenergetic beams. This study aims to determine the optimal value of the regularization parameter for the problem of determining the spectrum of the electron beam produced by a mobile accelerator used during intraoperative radiotherapy. Material and methodsThe Geant4 package was used to generate the distributions of deep doses for monoenergetic beams for two models with different degrees of complexity, i.e. simple (theoretical) and full (for the mobile accelerator). The dose distributions for four different shapes of energy spectrum (for each model) were obtained similarly. They were established as the reference data for further calculations. The Dual Annealing optimization method was used to obtain the reconstructed spectrum. The multiple optimizations that differ by the regularization parameter (ranging from 0 to 1) were performed. For each reconstruction, similarity indicators of the energy spectrum and the dose distribution to the referenced data were calculated to determine the optimal regularization parameters. ResultsOptimal regularization parameters determined by similarity indicators for the spectrum and the dose distribution differ for geometry models considered in the study. The regularization parameter for the simple geometry ranged from 0.03 to 0.05, while for full geometry, they were from 0.05 to 0.06. The results for conventional linear accelerators found in the literature range from 0.5 to 1.1. ConclusionThe Dual Annealing optimization method can be effectively used to solve the Fredholm equation with Tikhonov regularization to reconstruct an electron beam's energy spectrum. The regularization parameter value depends on the beam-forming system. Its value for the mobile accelerator considered in the study ranges from 0.05 to 0.06, depending on the nominal beam energy value.

A Kernel-Free Boundary Integral Method for 2-D Magnetostatics Analysis

Performing magnetostatic analysis accurately and efficiently is very important for multi-objective optimization of electromagnetic device designs. In this research, a kernel-free boundary integral method (KFBIM) has been introduced for solving magnetic computations in a toroidal core geometry in 2-D. This study is very relevant in the design and optimization of toroidal inductors or transformers used in electrical systems, where lighter weight, higher inductance, higher efficiency, and lower leakage flux are required. The governing partial differential equations (PDEs) have been formulated as a system of the Fredholm integral equations of the second kind. Unlike traditional boundary integral methods or boundary element methods, KFBIM does not require an analytical form of Green’s function for evaluating integrals via numerical quadrature. Instead, KFBIM computes integrals by solving an equivalent interface problem on a Cartesian mesh. Compared with traditional finite difference methods for solving the governing PDEs directly, KFBIM produces a well-conditioned linear system. Therefore, the KFBIM requires only a fixed number of iterations when an iterative method [e.g., generalized minimal residual method (GMRES)] is applied for solving the linear system, and the numerical solution is not sensitive to computer round-off errors. The obtained results are then compared with a commercial finite element solver (ANSYS), which shows excellent agreement. It should be noted that, compared with FEM, the KFBIM does not require a body-fit mesh and can achieve high accuracy with a coarse mesh. In particular, the calculations of the magnetic potential and the tangential field intensity on the boundaries are more stable and exhibit almost no oscillations.

New classes of control functions for nonlinear contractions and applications

We initiate the use of sub and super homogeneous control functions for nonlinear contractions in complete metric spaces and establish new fixed point theorems. Moreover, we develop other variants of control functions for the fixed point theorems of Boyd-Wong [2] and Matkowski [3]. As application, we present new sufficient conditions ensuring the existence of solutions to some classes of integral equations of Fredholm and Volterra type.

On the solvability of a nonlinear optimization problem with boundary vector control of oscillatory processes

In the paper, the solvability of the nonlinear boundary optimization problem has been investigated for the oscillation processes, described by the integro-differential equation in partial derivatives with Fredholm integral operator. It has been established that the components of the boundary vector control are defined as a solution to a system of nonlinear integral equations of a specific form, and the equations of this system have the property of equal relations. An algorithm for constructing a solution to the problem of nonlinear optimization has been developed.

Mathematical models of spheroidal spiral-frame radiating structures

The article considers mathematical models of two spheroidal spiral-frame emitters, built on the basis of a general approach involving the use of an integral representation electromagnetic field. The internal problem of electrodynamics is reduced to a system of Fredholm integral equations of the 1st kind. The resulting system was solved by the method of moments with piecewise constant basis functions and delta functions as test functions. In this case, the local linearization of the generating conductors of the structures under consideration was carried out. The dependences of current distributions, input resistance, and radiation characteristics of structures on frequency have been studied. It is shown that standing, traveling, and mixed current waves can exist in the structures under consideration. The current regime is determined by the wave sizes and the geometry of the structures and determines the behavior of the wave resistance in the frequency range. Despite the similar geometry, the characteristics of the considered structures have certain differences.

Spectral characteristics of the integral operator of the internal problem of electrodynamics for elliptical frame structure

The article is devoted to the analysis of electrodynamic properties elliptical frame structure. Taking into account double symmetry internal problem of electrodynamics for the structure under consideration in the framework of the thin-wire approximation is reduced to four integral Fredholm equations of the first kind, written with respect to independent current functions. A study of spectral characteristics of the integral operators of the corresponding integral equations for various values of the electrical length and ellipticity of the frame. It is shown that the eigenfunctions of integral operators for close values of these parameters have a high degree of correlation, with In this case, the eigenfunctions are close in form to trigonometric functions. Features of the frequency dependence of the eigenvalues integral operators. The conclusion is made about the resonant nature of these dependences, what makes an elliptical frame structure in many respects similar to the previously considered tubular vibrator and spherical spiral particle. The results presented in the article form an in-depth understanding of the processes occurring in the structure under consideration, and also serve as a guideline in the construction of approximation models for solving the internal tasks.

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

The Fredholm integral equations of the first kind with two regular and hypersingular kernels are considered on [−1, 1]. The hypersingular kernel is considered smooth enough with no additional conditions. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to obtain high accurate approximations. The proposed method reduces the underlying integral equation to a system of algebraic equations. Several illustrative examples are provided to show the efficiency of the method.

Приближенное решение нелинейного интегрального уравнения Фредгольма второго рода

{Статья посвящена численному решению нелинейных интегральных уравнений Фредгольма второго рода. Рассматриваемое уравнение имеет специальное ядро в том смысле, что представляет собой произведения двух частей: слабо сингулярной части, не зависящей от решения, и нелинейной дифференцируемой по Фреше части, зависящей от решения. Приближенное решение, предложенное в статье, определяется как итерационная последовательность типа Ньютона --- Канторовича. При этом используются три численных метода: метод Ньютона --- Канторовича для линеаризации задачи, метод регуляризации с конволюцией и разложением в ряд Фурье. Это необходимо, чтобы получить конечную последовательность, и <<Hat functions projection>> для работы с нелинейным членом, возникающим в конструкции Ньютона --- Канторовича. Доказано, что такая специальная последовательность типа Ньютона --- Канторовича сходится к точному решению. Кроме того, приведен численный пример, демонстрирующий практическую эффективность численного метода и подтверждающий точность теоретических результатов.

New method for unfolding neutron source spectra

For solving the Fredholm equation of the first kind, the domain can be discretized in M spheres and N energy bins using a Bonner multisphere spectrometer, in which the neutron flux is to be deduced from the response matrix Rij(E), and the count rate obtained for each sphere Ci. For this study we use an approach that introduces certain neutron energy indeterminations, which are difficult to account for if some absolute calculations are required to be performed. For M different from N, a problem arises after obtaining the deconvolution of this equation, because the extracted solution is not unique, even if a host neutron energy spectrum is used. We propose a different method for extracting neutron energy spectra from the count rate obtained using a Bonner spectrometer. The method consists of partitioning the response matrix into a number of matrices such that M is equal to N, and depends on the response matrix and the number of spheres under consideration. Thus, the overall solution of the Fredholm equation can be divided and applied to N/M segments with M equations, and M unknowns, which can be solved in every energy segment for the neutron energy spectra. With this, the neutron spectra can be recovered by combining successive energy intervals as will be thoroughly described in the paper.

Convergence Analysis for Generalized Yosida Inclusion Problem with Applications

A new generalized Yosida inclusion problem, involving A-relaxed co-accretive mapping, is introduced. The resolvent and associated generalized Yosida approximation operator is construed and a few of its characteristics are discussed. The existence result is quantified in q-uniformly smooth Banach spaces. A four-step iterative scheme is proposed and its convergence analysis is discussed. Our theoretical assertions are illustrated by a numerical example. In addition, we confirm that the developed method is almost stable for contractions. Further, an equivalent generalized resolvent equation problem is established. Finally, by utilizing the Yosida inclusion problem, we investigate a resolvent equation problem and by employing our proposed method, a Volterra–Fredholm integral equation is examined.

Approximate solutions to several classes of Volterra and Fredholm integral equations using the neural network algorithm based on the sine-cosine basis function and extreme learning machine.

In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra-Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability.

Fractional Dual-Phase-Lag Non-Fourier Heat Transfer in a Bimaterial with a Circular Interface Insulator

The transient temperature response of a bimaterial with a circular insulated interface region is studied under sudden heating or cooling. The time-fractional dual-phase-lag heat conduction model is adopted to simulate the non-Fourier effect. The problem is reduced to an initial-boundary value problem. The Laplace transform is applied to convert the problem to a mixed boundary value problem, and then the Hankel transform reduces it to a Fredholm integral equation. Special situations for asymptotic thermal behavior near the insulated circular edge and for the steady-state cases are discussed, respectively. The dynamic intensity factors of heat flux and temperature gradient near the insulated circular edge are computed numerically through Stehfest’s Laplace inversion transform technique. The influences of fractional order and relaxation times on the instantaneous temperature change are analyzed. The exact solution of temperature fields for the steady-state case is derived and displayed graphically. The wave-like diffusion behavior of the fractional dual-phase-lag model is interpreted.

Collocation method for solving two-dimensional nonlinear Volterra–Fredholm integral equations with convergence analysis

This paper presents a method for solving the two-dimensional nonlinear Volterra–Fredholm integral equation. The main idea of the method is to use the Lagrange interpolation function to approximate the unknown solution and the Legendre–Gauss quadrature formula to approximate the integral. The advantage of the method is that it requires relatively few collocation points to obtain a relatively small error and does not require the calculation of integrals. Under certain sufficient conditions, the existence and uniqueness of the original equation are given. In addition, the existence and uniqueness of the solutions of the discrete equations are given using the theory of compact operators. The convergence analysis and error estimates of the method are also derived. Finally, several numerical examples are used to demonstrate its efficiency and accuracy.

Solution Reduction of a Dynamic Plates Bending Problem to the Sequential Solution of the First Kind Fredholm Integral Equations

Solution Reduction of a Dynamic Plates Bending Problem to the Sequential Solution of the First Kind Fredholm Integral Equations

Solving Fredholm integral equation based on Padé approximants using Particle swarm optimization

Several scientific and engineering applications are usually described as integral equations. A new approach for solving the type of linear and nonlinear Fredholm integral equation of the second kind is proposed. Although many methods provide an analytic solution, there are different types of integral equations are difficult to solve. Therefore, the numerical approach for solving integral equations is used. Fredholm integral equations of the second kind have been converted to unconstrained optimization problems to find their approximate solutions. This work employs particle swarm optimization combined with padé expansion to find an approximate solution of the Fredholm integral equation. This is applied by minimizing the fitness function value. The fitness function is calculated using the discrete least squares weighted function. The proposed algorithm is applied to solve linear and non-linear FIE. The results are compared to exact solutions. The stability of the proposed algorithm is also presented. The results are promising in terms of convergence , stability and accuracy of the approximate solution.

Application of Hybrid Bernstein Polynomials and Block-Pulse Functions for Solving Nonlinear Fuzzy Fredholm Integral Equations

In this paper, hybrid Bernstein polynomials and block-pulse functions based on the method of successive approximations are applied to obtain the approximate solution of nonlinear fuzzy Fredholm integral equations. The main idea of using the proposed method is that fuzzy integral in any iterative process will be reduced to the crisp integration. Some results concerning the error estimate and stability of the numerical method are presented. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.

Piecewise Polynomial in solving Fredholm Integral Equation of Second Kind by using Successive over Relaxation method

Piecewise Polynomial in solving Fredholm Integral Equation of Second Kind by using Successive over Relaxation method

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

It is our main objective to find the critical load for three beams with cross sectional heterogeneity. Each beam has three supports, of which the intermediate one is a spring support. Determination of the critical load for these beams leads to three point boundary value problems (BVPs) associated with homogeneous boundary conditions—the mentioned BVPs constitute three eigenvalue problems. They are solved by using a novel solution strategy based on the Green functions that belong to these BVPs: the eigenvalue problems established for the critical load are transformed into eigenvalue problems governed by homogeneous Fredholm integral equations with kernels that can be given in closed forms provided that the Green function of each BVP is known. Then the eigenvalue problems governed by the Fredholm integral equations can be manipulated into algebraic eigenvalue problems solved numerically by using effective algorithms. It is an advantage of the way we attack these problems that the formalism established and the results obtained remain valid for homogeneous beams as well. The numerical results for the critical forces can be applied to solve some stability problems in the engineering practice.