Fredholm System Research Articles

Solving linear systems of fractional integro-differential equations by Haar and Legendre wavelets techniques

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

A kernel-free boundary integral method for the nonlinear Poisson-Boltzmann equation

This work proposes a boundary integral formulation based Cartesian grid method for the nonlinear Poisson-Boltzmann (PB) interface problem in biophysics. The method (a) does not have the limitation associated with the standard finite difference method for complex interfaces, (b) avoids the generation of any unstructured volume or surface grids needed by the finite element method, and (c) solves the nonlinear and variable coefficient PDE in the framework of boundary integral equations. The method solves the nonlinear PB equation with the Newton iterative method. It first reformulates the linearized PB equation in each Newton iteration as a Fredholm system of two boundary integral (BI) equations of the second kind, which is well-conditioned, and then solves the BI system with a Krylov subspace method. The evaluation of boundary and volume integrals involved in the solution of the BI system is done with a kernel-free method, which does not need to know Green's function associated with the variable coefficient PB equation. The kernel-free method evaluates volume and boundary integrals by solving equivalent simple interface problems on Cartesian grids with either a fast Fourier transform based Poisson solver or a geometric multigrid preconditioned conjugate gradient solver, either of which involves computational work essentially linearly proportional to the number of nodes on the Cartesian grid. Numerical examples in both two and three space dimensions are presented to demonstrate the efficiency and accuracy of the proposed numerical method.

Banach orbifold structure on groupoids of morphisms of orbifolds

In this paper, we construct a natural Banach orbifold structure for the space of Wk,p maps between compact Riemannian orbifold groupoids. We then apply it to study the Fredholm systems associated with the moduli spaces of pseudo-holomorphic maps.

A New Route Choice Model for Urban Public Transit with Headway-based Service

A new approach to create the model of individual behavior in urban public transit systems with headway-based service that is built on the same principles of rational decision making as discrete choice models is presented. To describe a passenger's decision-making, the attractiveness function which reflects the difference between trip results and costs was used. The attractiveness of the transit route was determined by solving the system of Fredholm's integral equations based on observed frequencies of choosing the alternative routes by each individual. The frequencies were determined based on multi-day survey results and range between 0 and 1 but equal neither 0 nor 1. To estimate the significance level of the attractiveness function, the regression analysis was used and the accuracy of the choice probability forecast was evaluated. Computation of model coefficients was carried out based on the parameters of home-based work trips obtained via the travel survey that was conducted in the city of Kharkiv, Ukraine.

A short review on approximate analytical methods for non-linear problems

In this paper, some approximate analytical methods are reviewed, including the variational iteration method and the homotopy perturbation method. Based on the ideas of the two methods, a new method for solving a class of system of Fredholm integral equations of the second kind is proposed. Some examples are given to show the simple solution process and the accuracy of the solution for each method. The emphasis of this review paper focuses on Ji-Huan He?s unapproachable ideas and the mathematics beauty, so the references are not exhaustive.

Approximation methods for system of nonlinear Fredholm–Hammerstein integral equations

In this article, we apply projection methods and their iterated versions to approximate the solution of system of Fredholm–Hammerstein integral equations with both smooth and weakly singular kernels of algebraic and logarithmic type using the piecewise polynomial basis functions. We show that the iterated Galerkin approximate solution converges to the exact solution faster than the Galerkin approximate solution for both smooth and weakly singular algebraic and logarithmic type kernels. We improve these results further in iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results.

Numerical solution for system of nonlinear Fredholm–Hammerstein integral equations based on hybrid Bernstein Block-Pulse functions with the Gauss quadrature rule

This work provides an efficient method for solving system of nonlinear Fredholm–Hammerstein integral equations. The propose method approximates the unknown function with hybrid Bernstein Block-Pulse functions (HBPF). In order to do this, we apply these functions and then using the collocation method for the numerical solutions of this system. Furthermore, we implement this method in conjunction with the quadrature rule for converting the problem to a system of algebraic equations that can be solved easily by applying mathematical programming techniques. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations and on the other hand, the efficiency and accuracy of the hybrid Bernstein Block-Pulse functions with the Gauss quadrature rule (HBPF-GQR) for solving this system are remarkable. The existence and uniqueness of solution have been presented. Moreover, the convergence of this algorithm will be shown by preparing some theorems. Several numerical examples are presented to show the superiority and efficiency of current method in comparison with some other well-known methods.

On the eigenfrequencies of preloaded rotationally restrained extensible circular beams by Green’s functions

The article is devoted to the vibrations of heterogeneous curved beams with centerline extensibility accounted for. The end supports are rotationally restrained pins, and the effect of a central, either compressive or tensile, force (preload) is incorporated into the model. The coupled differential equations that govern the problem are derived from the principle of virtual work. These are replaced by a system of homogeneous Fredholm integral equations. The kernel of the integral equation system is the Green’s function matrix, which is given in closed form. The eigenvalue problem determined by the homogeneous Fredholm integral equation system is solved numerically, and the results obtained are presented graphically. If the spring stiffness tends to (zero) [infinity], the beam behaves as if it were (pinned–pinned) [fixed–fixed]. When the external force tends to zero, the model returns the eigenfrequencies of the free vibrations.

On General boundary value problem for an elliptic higher order equation in the plane with constant real coefficients

By means of a new approach, the general boundary value problem for a higher order elliptic equation with two independent variables, and a normal set of boundary conditions and simple complex characteristics is reduced to the Fredholm system of integral equations in a bounded region with a smooth boundary.

On the Performance of Haar Wavelet Approach for Boundary Value Problems and Systems of Fredholm Integral Equations

The Haar wavelet method applied to different kinds of integral equations (Fredholm integral equation, integro-differential equations and system of linear Fredholm integral equations) and boundary value problems (BVP) representation of integral equations. Three test problems whose exact solutions are known were considered to measure the performance of Haar wavelet. The calculations show that solving the problem as integral equation is more accurate than solving it as differential equation. Also the calculations show the efficiency of Haar wavelet in case of F. I. E. S and integro-differential equations comparing with other methods, especially when we increase the number of collocation points. All calculations are done by the Computer Algebra Facilities included in Mathematica 10.2.

A semi-analytical algorithm to solve systems of integro-differential equations under mixed boundary conditions

In this work we present a semi-analytical method to solve systems of integro-differential equations under mixed boundary conditions. The proposed method handles linear and nonlinear systems of Fredholm–Volterra integro-differential equations in a reliable manner. We present a convergence analysis for the proposed method to emphasize its reliability. Numerical examples are examined to show the efficiency and the accuracy of the scheme. We show that a few number of approximating terms gives results of high accuracy level, and by increasing the number of these terms, the error of the approximated solution decreases rapidly. The proposed method is powerful and reliable to solve other kinds of systems of integro-differential equations. Further, it has a small CPU time and its computational time is less than the other methods.

Solving system of Volterra–Fredholm integral equations with Bernstein polynomials and hybrid Bernstein Block-Pulse functions

This work approximates the unknown functions based on the Bernstein polynomials and hybrid Bernstein Block-Pulse functions, in conjunction with the collocation method for the numerical solution of system of Fredholm–Volterra integral equations. In both methods, the system of integral equations is approximated by the Gauss quadrature formula with respect to the Legendre weight function. The proposed methods reduce the system of integral equations to a system of algebraic equations that can be easily solved by any usual numerical methods. Moreover, the convergence analysis of these algorithms will be shown by preparing some theorems. Numerical experiments are presented to show the superiority and efficiency of proposed methods in comparison with some other well known methods.

Identification of thermal conductivity coefficient and volumetric heat capacity of functionally graded materials

Functionally graded materials are being widely implemented in many fields of technology. In the paper we present an approach to solving an inverse problem on a simultaneous identification of thermal conductivity coefficient and volumetric heat capacity of functionally graded materials. As an input data we use measurements of temperature and heat flux at body’s boundary. The inverse problem solving is reduced to an iterative procedure when each iteration provided solving of the system of Fredholm’s integral equations of the first kind. As an example, the problem for a rod is considered. The direct problem is solved by means of the Galerkin method. Examples of simultaneous reconstruction of different inhomogeneous laws of thermalphysic characteristics are presented.

Study on convergence and error of a numerical method for solving systems of nonlinear Fredholm–Volterra integral equations of Hammerstein type

The aim of this paper is to present an efficient analytical and numerical procedure for solving systems of nonlinear Fredholm–Volterra integral equations of the Hammerstein type with the aid of fixed point techniques and the usual Schauder basis in an adequate Banach space.

Hybrid Legendre Block-Pulse functions for the numerical solutions of system of nonlinear Fredholm–Hammerstein integral equations

In this paper, the numerical technique based on hybrid Legendre-Block-Pulse function has been developed to approximate the solution of system of nonlinear Fredholm–Hammerstein integral equations. These functions are formed by the hybridization of Legendre polynomials and Block-Pulse functions. These functions are orthonormal and have compact support on [0,1]. The proposed method reduces the system of integral equations to a system of nonlinear algebraic equations that can be solved easily by any usual numerical method. The numerical results obtained by the presented method have been compared with those obtained by Legendre wavelet method (LWM). Numerical examples are presented to illustrate the accuracy of the method.

Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

This paper aims to construct a general formulation for the shifted Jacobi operational matrices of integration and product. The main aim is to generalize the Jacobi integral and product operational matrices to the solving system of Fredholm and Volterra equations. These matrices together with the collocation method are applied to reduce the solution of these problems to the solution of a system of algebraic equations. The method is applied to solve system of linear and nonlinear Fredholm and Volterra equations. Illustrative examples are included to demonstrate the validity and efficiency of the presented method. Also, several theorems, which are related to the convergence of the proposed method, will be presented.

Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

Shifted Jacobi Collocation Method Based on Operational Matrix for Solving the Systems of Fredholm and Volterra Integral Equations

Electromagnetic scattering by a plane angular sector: I. Diffraction coefficients of the spherical wave from the vertex

Electromagnetic problem by a plane angular sector is studied. The perfectly conducting sector is illuminated by a plane electromagnetic wave. The scattered wave field consists of several components at large distances from the vertex of the sector. In the framework of the consequent and mathematically justified procedure, based on the Watson–Bessel and Sommerfeld integral representations of the electromagnetic Debye potentials, we develop expressions for the electromagnetic diffraction coefficients of the spherical wave from the vertex. To that end, we carefully study analytic properties of the Sommerfeld transformants and give a new procedure of analytic continuation for them. Like in the acoustic problem, the singularities of the Sommerfeld transformants on the real axis are responsible for the different components in the far field asymptotics.The expressions for the diffraction coefficients are specified in the form of integrals depending on the so called spectral functions, i.e. solutions of the boundary value problem for the Laplace–Beltrami operator on the unit sphere with the cut along a segment of big circle. The spectral functions are found in terms of series of “spherical harmonics”, whereas the coefficients of the series solve a Fredholm system of linear summation equations of the second kind. The properties of the spectral functions are also discussed and their interconnections with the “nonstationary” Green’s functions for the problems for some hyperbolic operator on the unit sphere with the cut are also established. The latter Green’s functions are exploited in the formulas for the analytic continuation of the Sommerfeld transformants. The results of the work form a theoretical basis for the numerical evaluation of the diffraction coefficients and for the description of the other components in the far field asymptotics to be given in the second part of the work.

Numerical solutions of system of linear Fredholm–Volterra integro-differential equations by the Bessel collocation method and error estimation

In this study, the Bessel collocation method is presented for the solutions of system of linear Fredholm–Volterra integro-differential equations which includes the derivatives of unknown functions in integral parts. The Bessel collocation method transforms the problem into a system of linear algebraic equations by means of the Bessel functions of first kind, the collocation points and the matrix relations. Also, an error estimation is given for the considered problem and the method. Illustrative examples are presented to show efficiency of method and the comparisons are made with the results of other methods. All of numerical calculations have been made on a computer using a program written in Matlab.

Spline Collocation for Solving System of Fredholm and Volterra Integral Equations

Spline Collocation for Solving System of Fredholm and Volterra Integral Equations