Solution and Error Analysis of Two Dimensional Fredholm-Volterra Integral Equations Using Piecewise Constant Functions

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations.Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems.Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields.The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs).In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively.We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind.We introduced efficient method, constructed on modified Bernstein polynomials.The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind.This method is tested for some examples from nonlinear integral equations.Maple software was used to obtain the solutions of these examples.The results demonstrate reliability of the proposed method.Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

The key task of constructing stationary ground stations for information exchange via an ionosphere channel is to develop antenna systems which provide high spatial-frequency selectivity of transmitting and receiving signals with seasonal and daily fluctuations of electrophysical parameters of the propagation medium. An effective solution to this problem is based on the use of vertical asymmetric vibrators placed on the earth's surface on concentric circles as transmission and reception structures of arrays. The selected array configuration allows for a circular view of the space with a high directional coefficient under deployment area constraints within the transceiver station. The annular structure, which preserves the invariance of the structure when rotating in the azimuth plane around the central longitudinal axis at an angular distance between the vibrators, provides a circular view of the space with a high coefficient of directional action. The placement of antenna elements on several rings with a common center contributes to the expansion of the frequency range for scanning space compared to a single ring system with restrictions on the occupied area. Rational versions of arrays are established on the basis of electrodynamic analysis of relationships between parameters of structures and characteristics of signal transmission and reception. The purpose of the article is to establish patterns of change in directional properties of arrays with various electrical dimensions, the number of antenna elements and concentric circles. Using the apparatus of Hallen integral equations and the Krylov–Bogolyubov numerical method, electrodynamic analysis of ring arrays of vertical asymmetric vibrators for stationary terrestrial transmission and reception posts has been carried out to solve them. Setting of the boundary problem has been performed with application of principle of mirror images from flat platform, chosen for array rotation on earth surface, which has properties of ideally conducting material in the range of ionosphere propagation wavelengths. A system of integral equations with nuclei in the form of Green functions for point emitters in a homogeneous isotropic infinitely long medium relative to equivalent array currents has been obtained from the condition of turning to zero the tangential components of the superposition of intrinsic electric fields on the surfaces of vibrators and their equality to vertical projections of fields created by generators of electromotive forces in gaps for connecting outputs of a distribution line. According to results of numerical solution of integral equations by conversion to system of linear algebraic equations for calculation of complex amplitudes of equivalent currents of antenna elements at the approximation by sequences of piecewise constant basic functions, spatial-frequency selectivity of transmission-reception by arrays with certain parameters of structures has been investigated. Distribution of array currents is restored by summation of sequences of piecewise constant basis functions with specified weight coefficients found at partial inversion of matrix operator and control of achievable accuracy of calculations by residual error of boundary conditions of a boundary problem. Laws of change of array directivities at various electric dimensions, number of vibrators and density of their arrangement on concentric circles have been revealed. On the basis of electrodynamic analysis, relationships of characteristics and parameters of structures of ring arrays of vertical asymmetrical vibrators located on infinitely extended perfectly conducting flat surface have been established. Possibilities of increase of spatial-frequency selectivity of signals transmission-reception when propagating along ionosphere channel have been investigated. When the generators of external electromotive forces are included in the final axial breaks of the elements, the effective receiving-emitting surfaces of the vibrators decrease, the absolute values of currents decrease in comparison with the values achieved when excited by point sources of high-frequency oscillations. To increase the directivity of the array with a distribution line to a value achievable when excited by ideal sources in infinitesimal axial gaps, it is necessary to increase the lengths of the vibrators by the width of the gap. The obtained results make it possible to justify rational design alternatives of antenna arrays when simultaneously fulfilling tactical and technical requirements for functional properties and maximum permissible dimensions for deployment with stationary ground transceiver stations.

The paper consists of two parts. In the first part we investigate a Nystrom- or product integration method for second kind singular integral equations. We prove an asymptotically optimal error estimate in the scale of Sobolev Hilbert spaces. Although the result can also be obtained as a special case of a discrete iterated collocation method our proof is more direct and uses the Nystrom interpolation. In the second part of this paper we consider the Dirichlet problem for thin elastic plates with transverse shear deformation. The boundary value problem is transformed into a 3 x 3 system of singular Fredholm integral equations of second kind. After discussing existence and uniqueness of the solution to the integral equations in a Sobolev space setting, we apply the Nystrom method to solve the integral equations numerically.

In this paper, we approximate the two-dimensional linear Fredholm integral equations (fies) with smooth kernels using Chebyshev spectral Galerkin and collocation methods. The existence and convergence analysis for the problem have been discussed. The errors between approximated solutions with exact solution have been evaluated in $$L^2_\omega $$ norm applying both these methods. We also solve the associated eigenvalue problems (evps) by using above methods and obtain the errors between approximated eigenelements with the exact eigenelements in $$L^2_\omega $$ and $$L^\infty _\omega $$ norms. Numerical examples are presented to illustrate the theoretical results.

The use of Sherman–Morrison formula in the solution of Fredholm integral equation of second kind

In this article, we apply the Single Term Walsh Series (STWS) method to solve the systems of Volterra integral equations of first kind (SVIEF). Because of the ill-posedness of the first kind problem, the SVIEF is converted into system of Volterra integral equations of second kind (SVIES). The STWS method for the SVIES with variable coefficient is presented. Numerical examples for linear and non-linear problems are experimented to show the efficiency and simplicity of the STWS method. The obtained numerical solutions (results) have been compared with the existing method.

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

ABSTRACTThis article proposes two novel numerical techniques for solving two‐dimensional stochastic Itô–Volterra Fredholm integral equations. Many two‐dimensional stochastic integral equations present significant challenges for analytical solutions. Consequently, possessing an efficient method to get very precise numerical solutions for these stochastic integral equations is of paramount importance. In this article, two novel numerical techniques based on two‐dimensional wavelets have been successfully introduced to acquire the numerical solutions of the two‐dimensional stochastic Itô–Volterra Fredholm integral equations. This efficient and noteworthy method transforms the stochastic integral equation into a system of algebraic equations, which is then solved using an appropriate numerical technique. Moreover, convergence analysis and the error analysis have also been well established successfully. At the end, some test problems have been presented to illustrate the accuracy, efficiency, simplicity, and plausibility of the proposed techniques.

Superconvergent methods based on quasi-interpolating operators for fredholm integral equations of the second kind.

In this paper, a new technique is offered for solving three types of linear integral equations of the 2nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods is produced. Finally, for more explanation, an algorithm is proposed and applied for testing examples to illustrate the effectiveness of the new technique.

In this paper, the existence and uniqueness of solution of the Fredholm‐Volterra integral equation (F‐VIE), with a generalized singular kernel, are discussed and proved in the spaceL2(Ω)×C(0,T). The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced to a linear algebraic system (LAS) of equations by using two different methods. These methods are: Toeplitz matrix method and Product Nyström method. A numerical examples are considered when the generalized kernel takes the following forms: Carleman function, logarithmic form, Cauchy kernel, and Hilbert kernel.

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

The paper is focused on the effect of a sudden impact of a torsional load on a penny-shaped crack sandwiched between two elastic layers embedded in an elastic medium. The axisymmetric mixed boundary value problem is reduced to the problem of solving a pair of dual integral equations by using Hankel and Laplace transforms. Further, the integral equations are then reduced to a Fredholm integral equation of second kind which is solved numerically. Expression for the stress intensity factor at the tip of the crack is obtained and plotted for different parameters and materials.

We reduce the solution of contact problems in the interaction of rigid bodies (dies) with thin-walled elements (one-dimensional problems) to Volterra integral equations. We study the effect of the model describing the stress-strain state of plates on the type of integral equations and the structure of their solutions. It is shown that taking account of reducing turns the problem into a Volterra integral equation of second kind, which has a unique solution that is continuous and agrees quite well with the results obtained from the three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra three-dimensional theory. In the case of a theory of Timoshenko type the problem is reduced to a Volterra integral equation of first kind that has a unique continuous solution; but for dies without corners the Herz condition does not hold (p(a) ≠ 0), and the contact pressure assumes its maximal value at the end of the zone of contact. For thin-walled elements, whose state can be described by the classical Kirchhoff-Love theory, the integral equation of the problem (a Volterra equation of first kind) has a solution in the class of distributions. The contact pressure is reduced to concentrated reactions at the extreme points of the contact zone. We give a comparative analysis of the solutions in all the cases just listed (forces, normal displacements, contact pressures). Three figures, 1 table. Bibliography: 5 titles.

Single and multiple integral problems are often applied in pure analysis, mechanical vibration, and related engineering and physics mathematics fields. Further, this study aims to solve two integral equations, i.e., Volterra–Fredholm integral equations of the second kind (2VFIE’s). To produce corresponding collocation approximation equations, we carry out the collocation discretization process by applying the Bernstein approximation function in various orders, namely seventh, ninth and eleventh. Furthermore, this study also aims to determine the accuracy of high-order Bernstein collocation solutions for solving Volterra–Fredholm integral equations of the second kind. For the sake of implementing the discretization process, trapezoidal and collocation schemes were used to discretize 2VFIE’s and then generate three systems of linear Bernstein approximation equations, namely seventh, ninth and eleventh orders. In getting the approximate solution of each linear system, the direct method is applied to solve three linear systems. Next, based on the analysis of numerical results, it can be pointed out that the approximate solutions of the seventh, ninth and eleventh-order Bernstein approximation functions as well the direct method applied to 2VFIE’s are more accurate based on root mean square error (RMSE).KeywordsHigh-order Bernstein polynomialsVolterra–Fredholm integral equationsCollocation approachDirect methodRMSE