System Of Fredholm Integral Equations Research Articles

A new approach based on semi-orthogonal B-spline wavelets for the numerical solutions of the system of nonlinear Fredholm integral equations of second kind

In this paper, a new approach based on linear B-spline wavelet method has been developed to approximate the solutions of system of nonlinear Fredholm integral equations of second kind. Compactly supported linear semi-orthogonal B-spline scaling functions and wavelet functions together with their dual functions are applied to approximate the solutions of nonlinear Fredholm integral equations system of second kind. This method reduces the system of integral equations to a system of algebraic equations that can be solved easily with any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method.

Nonlinear Fredholm-Volterra Integral Equation and its Numerical Solutions with Quadrature Methods

In this work, the existence and uniqueness of solution of the nonlinear Fredholm-Volterra integral equation ( NF - VIE ), with continuous kernels, are discussed and proved in the space L 2 ( 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 nonlinear system of Fredholm integral equations ( SFIEs ). This system of integral equations can be reduced, using quadrature methods, to a nonlinear algebraic system ( NAS ). Then, the NAS can be solved, using two numerical methods. These methods are: Trapezoidal rule method and Simpson's rule method. Finally, some numerical examples are considered and the error estimate, in each case, is computed.

Toeplitz Matrix Method and Volterra-Hammerstien Integral Equation with a Generalized Singular Kernel

In this work, the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind (V-HIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous kernel, while the Fredholm integral term (FIT) is considered in position with a generalized singular kernel. Using a numerical technique, V-HIESK is reduced to a nonlinear system of Fredholm integral equations (SFIEs). Using Toeplitz matrix method we have a nonlinear algebraic system of equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, some numerical examples when the kernel takes the logarithmic, Carleman, and Cauchy forms, are considered.

Exponentially Converging Nyström Methods in Oblique Diffraction of Arbitrarily Polarized Waves by Bianisotropic/Chiral Cylinders With Arbitrary Smooth Cross Section

We study oblique diffraction of arbitrarily polarized plane waves by a uniform homogeneous bianisotropic cylinder with arbitrarily shaped smooth boundary. Following a boundary integral equations formulation we develop a 4 × 4 system of combined field integral equations and another 4 × 4 system of Muller-type Fredholm integral equations. Both these systems, having the tangential to the surface of the cylinder components of the electric and magnetic fields as the unknowns, are of the second kind and uniquely solvable. Efficient discretization via simple and rapidly converging Nystrom algorithms enables one to obtain highly accurate results even for electrically large objects. The proposed solution technique: a) fully accounts for the singular nature of the kernels; b) yields simple closed form expressions for all matrix elements; and c) appears to converge exponentially versus matrix size. The analogous diffraction problem for an arbitrarily shaped smooth chiral cylinder may be treated as a simple special case of the present analysis.

Numerical treatment of a class of systems of Fredholm integral equations on the real line

In this paper the authors propose a Nystrom method based on a “truncated” Gaussian rule to solve systems of Fredholm integral equations on the real line. They prove that it is stable and convergent and that the matrices of the solved linear systems are well-conditioned. Moreover, they give error estimates in weighted uniform norm and show some numerical tests. Fredholm integral equations, Nystrom method, truncated Gaussian rule

Approximate Solutions of Linear Fredholm Integral Equations System with Variable Coefficients

In this paper, a new approximate method has been presented to solve the linear Fredholm integral equations system (FIEs). The technique is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Also, this method gives the analytic solution when the exact solutions are polynomials. So as to Show this capability and robustness, some systems of FIEs are solved by the presented method in order to obtain their approximate solutions.

Kernel iterative method for solving two-dimensional fuzzy Fredholm integral equations of the second kind

Using parametric form of fuzzy functions, we convert a linear two-dimensional fuzzy Fredholm integral equation of the second kind (2D-FFIE-2) to a linear system of Fredholm integral equations of the second kind with three variables in crisp case. In this paper an iterative method is presented to find the approximate solution of 2D-FFIE-2. Also a proof of convergence of this method is discussed in detail. Finally, the proposed method is illustrated by two numerical example.

Optimal Homotopy Asymptotic Method for Solving System of Fredholm Integral Equations

In this paper, optimal homotopy asymptotic method (OHAM) is applied to solve system of Fredholm integral equations. The effectiveness of optimal homotopy asymptotic method is presented. This method provides easy tools to control the convergence region of approximating solution series wherever necessary. The results of OHAM are compared with homotopy perturbation method (HPM) and Taylor series expansion method (TSEM).

Romberg Algorithm to Solve a System of Nonlinear Fredholm Integral Equations of Second Kind

Romberg Algorithm to Solve a System of Nonlinear Fredholm Integral Equations of Second Kind

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.

Utilizing feed-back neural network approach for solving linear Fredholm integral equations system

This paper intended to offer an architecture of artificial neural networks (NNs) for finding approximate solution of a second kind linear Fredholm integral equations system. For this purpose, first we substitute the N-th truncation of the Taylor expansion for unknown functions in the origin system. By applying the suggested neural network for adjusting the real coefficients of given expansions in resulting system. The proposed NN is a two-layer feed-back neural network such that it can get a initial vector and then calculates it’s corresponding output vector. In continuance, a cost function is defined by using output vector and the target outputs. Consequently, the reported NN using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. Eventually, we have showed this method in comparison with existing numerical methods such as trapezoidal quadrature rule provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples with computer simulations.

Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations II

AbstractIn this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^3$. This representation leads to a coupled system of Fredholm integral equations of the second kind for four scalar densities supported on the material interface. Like the classical Müller equation, it has no spurious resonances. Unlike the classical approach, however, the representation does not suffer from low‐frequency breakdown. We illustrate the performance of the method with numerical examples. © 2012 Wiley Periodicals, Inc.

Thermostressed state of an infinite cylindrical shell with coordinate-dependent heat exchange coefficients and ambient temperature

We investigate the stress-strain state of an infinite cylindrical shell caused by a difference between the temperatures of the surrounding medium on the faces and the dependence of the coefficients of heat exchange from them on the coordinate. The contour of the shell at infinity is free from forces and moments, and the influence of the curvature of the shell on the temperature distribution and stress distribution is also taken into account. The problem of heat conduction is reduced to a system of Fredholm integral equations of the second kind, which is solved numerically by the method of quadrature formulas. We present results of a numerical analysis of distributions of temperature characteristics, the deflection, stresses, force, and bending moments.

Numerical treatment of boundary value problems by solving a system of second kind Fredholm integral equations

A Nystrom method is proposed for solving systems of Fredholm integral equations equivalent to special boundary value problems of order 2s . The stability and the convergence of the proposed procedure is proved. The GMRes method is applied to solve the involved systems of linear equations. Some numerical examples are provided in order to illustrate the accuracy of the method.

Triangular functions method for the solution of Fredholm integral equations system

A numerical method based on orthogonal triangular functions (TFs) is proposed to approximate the solution of Fredholm integral equations systems. The powerful properties of orthogonal triangular functions are utilized in a direct method to reduce a system of Fredholm integral equations to a system of mere algebraic equations. The proposed method does not need any integration for obtaining the constant coefficients hence, it can be applied in a simple and fast technique. Convergence analysis and associated theorems are considered. Some numerical examples illustrate the accuracy and computational efficiency of the proposed method.

Two Dimensional Fredholm Integral Equation with Time

In this paper, the existence and uniqueness of solution of the twodimensional Fredholm integral equation with time, with singular kernel, arediscussed and proved. The Fredholm integral term (FIT) is considered inposition while the Volterra integral term (VIT) is considered in time. Usinga numerical technique we have a system of Fredholm integral equations(SFIEs) in two dimensions . This system of integral equations can be reducedto a linear algebraic system (LAS) of equations by using two differentmethods. These methods are: Toeplitz matrix method and Product Nystrommethod. A numerical examples are considered when the kernel takes thefollowing forms: Carleman function, logarithmic form, Cauchy kernel.

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.

Identification of the properties of an inhomogeneous electroelastic medium

Abstract A method for the effective solution of inverse coefficients problems on establishing the laws of variation of the properties of an inhomogeneous electroelastic medium in a rectangle is proposed. Two types of loads are considered – electrical and mechanical, which enable three inverse coefficient problems for finding the required functions to be formulated. An iteration process is formulated for solving the inverse problem, at each step of which the corrections to the unknown functions with respect to the initial approximations are determined from the solution of a system of Fredholm integral equations of the first and second kind. Computational experiments to establish the inhomogeneous elastic and piezoelectric characteristics are carried out.

A Comparison Study Between a Chebyshev Collocation Method and the Adomian Decomposition Method for Solving Linear System of Fredholm Integral Equations of the Second Kind

In this paper, we obtain the approximate solutions of a system of linear integral equations by using both Chebyshev polynomials method and Adomian decomposition method. Also, a comparison between the two methods is discussed.

Spectral Ratios for Crack Detection Using P and Rayleigh Waves

We obtain numerical results to help the detection and characterization of subsurface cracks in solids by the application of P and Rayleigh elastic waves. The response is obtained from boundary integral equations, which belongs to the field of elastodynamics. Once the implementation of the boundary conditions has been done, a system of Fredholm integral equations of the second kind and order zero is found. This system is solved using the method of Gaussian elimination. Resonance peaks in the frequency domain allow us to infer the presence of cracks using spectral ratios. Several models of cracked media were analyzed, where effects due to different crack orientations and locations were observed. The results obtained are in good agreement with those published in the references.