Two-dimensional Fredholm Integral Equations Research Articles

A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains

This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good conditioning of the method are proved in suitable subspaces of continuous functions of Sobolev type. The cubature formula, on which the Nyström method is constructed, has an error that behaves like the best polynomial approximation of the integrand function. Consequently, it is also shown how the Nyström method inherits this property and, hence, the proposed numerical strategy is fast when the involved known functions are smooth. Some numerical examples illustrate the efficiency of the method, also in comparison with other methods known in the literature.

The Use of Homotopy Regularization Method for Linere and Nonlinner Fredholm Integral Equations of the First Kind

Recently, Wazwaz has studied the regularization method to the one-dimensional linear Fredholm integral equations of the first kind [Wazwaz, 2011]. In this work, we develop this method for the linear and nonlinear two-dimensional Fred-holm integral equations of the first kind. Indeed, the regularization method is used for linear integral equations directly. But nonlinear integral equations of the first kind are transformed to linearintegral equations of the first kind by a change of variable; then, The Regularization-Homotopy Method is applied. The combination of the regularization method and the homotopy perturbation method, or shortly, the regularization-homotopy method, is used to find a solution to the equation. Some examples will be used to highlight the reliability of the generalized of Regularization-Homotopy Method.

Combined Legendre Spectral-Finite Element Methods for Two-Dimensional Fredholm Integral Equations of the Second Kind

In this paper, we will discuss on the Combined Legendre spectral-Finite element methods (CLSFEM) for the two-dimensional Fredholm integral equations with smooth kernel on the Banach spaces and the corresponding eigenvalue problem. In these methods, the approximated finite dimensional space is the cartesian product of spline space and Legendre polynomial space. The problem is approximated by the CLSFEM using orthogonal projection, which projects from the Banach space into the finite dimensional space. The convergence analysis for both Fredholm integral equations and the corresponding eigenvalue problem will be discussed in both L 2 and norms. The numerical results will be shown to validate the theoretical estimate.

A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular domain with the right-hand side being representable in the form of a Fourier series. A number of general solutions of the contact problem, which were published in the literature, are discussed. Based on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of the contact pressure at the contact center and the contact stress-intensity factor at the contour of the contact area. Since the named general solution does not employ the operation of differentiation of a double integral with respect to the coordinates that enter it as parameters, the form of the general solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when the contact pressure beneath the indenter is bounded, is recommended for use as the most simple closed-form general solution of the non-axisymmetric Boussinesq contact problem.

Numerical Treating of Mixed Integral Equation Two-Dimensional in Surface Cracks in Finite Layers of Materials

The goal of this paper is study the mixed integral equation with singular kernel in two-dimensional adding to the time in the Volterra integral term numerically. We established the problem from the plane strain problem for the bounded layer medium composed of different materials that contains a crack on one of the interface. Also, the existence of a unique solution of the equation proved. Therefore, a numerical method is used to translate our problem to a system of two-dimensional Fredholm integral equations (STDFIEs). Then, Toeplitz matrix (TMM) and the Nystrom product methods (NPM) are used to solve the STDFIEs with Cauchy kernel. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the methods. The codes were written in Maple.

Extrapolation of the Galerkin solution for two-dimensional nonlinear Fredholm integral equations with orthogonal basis of Boubaker polynomials

Extrapolation of the Galerkin solution for two-dimensional nonlinear Fredholm integral equations with orthogonal basis of Boubaker polynomials

Mixed Fourier Legendre spectral Galerkin methods for two-dimensional Fredholm integral equations of the second kind

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations (fies) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems (evps) associated with the two-dimensional fies. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fies in both L2 and L∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L2 and L∞ norms. The numerical illustrations are introduced for the error of these methods.

Chebyshev Spectral Projection Methods for Two-Dimensional Fredholm Integral Equations of Second Kind

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.

A new generalized definition of singular kernel for two-dimensional Fredholm integral equation

A new generalized definition of singular kernel for two-dimensional Fredholm integral equation

Solving two-dimensional fuzzy Fredholm integral equations via sinc collocation method

In this paper, we present a numerical method to solve two-dimensional fuzzy Fredholm integral equations (2D-FFIE) by combing the sinc method with double exponential (DE) transformation. Using this method, the fuzzy Fredholm integral equations are converted into dual fuzzy linear systems. Convergence analysis is performed in terms of the modulus of continuity. Numerical experiments demonstrate the efficiency of the proposed method.

Numerical reconstruction of two-dimensional particle size distributions from laser diffraction data

In this paper, we consider the two-dimensional Fredholm integral equation of the first kind. The kernel of this equation models the scattering of a laser beam by spheroids having the same fixed orientation. The unknown function under the integral describes the distribution of spheroids along two semi-axes. The input data are the diffraction pattern corresponding to the scattering of the laser beam by the particles. We show that the Tikhonov regularization method allows one to reconstruct two-dimensional distributions in the case when the diffraction pattern is modulated by white noise with relative amplitude up to 1%. In applications, this means novel possibility of obtaining two-dimensional particle size distributions, rather than one-dimensional ones, as in the classical version of the method. This significantly expands the capabilities of particle sizing via static laser diffraction technique. We show that the solution of the corresponding equation is unique in space and exists when the right-hand-side function is in a set dense in . We provide test experimental results in the framework of laser ektacytometry of red blood cells, which serves as the main application of the proposed approach presently.

Application of Hilbert–Schmidt SVD approach to solve linear two-dimensional Fredholm integral equations of the second kind

Meshfree techniques based on infinitely smooth radial kernels have the great potential to provide spectrally accurate function approximations with irregular domain in high dimensions. The maximum accuracy can mostly be found when the RBF shape parameter is small, i.e., when the radial kernel is relatively smooth. However, as the shape parameter goes to zero, the standard RBF interpolant matrix will be very ill-conditioned. The ill-conditioning can be alleviated using alternate bases. One of these alternative bases is the Hilbert–Schmidt SVD basis. The Hilbert–Schmidt SVD approach suggests a stable mechanism for replacing a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space. In this work, the Gaussian Hilbert–Schmidt SVD basis functions method is presented to numerically solve the linear two-dimensional Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert–Schmidt SVD basis functions constructed on a set of scattered points. The emerged integrals in the scheme are approximately computed by the Gauss–Legendre quadrature rule. This approach reduces the problem under study to a linear system of algebraic equations which can be solved easily via applying an appropriate numerical technique. Also, the convergence of the proposed approach is established. Finally, numerical results are compared with standard RBF method to indicate the accuracy and efficiency of the suggested approach.

On the numerical treatment and analysis of two-dimensional Fredholm integral equations using quasi-interpolant

In this paper, we study the quadratic rule for the numerical solution of linear and nonlinear two-dimensional Fredholm integral equations based on spline quasi-interpolant. Also the convergence analysis of the method is given. We show that the order of the method is $$O(h_{x}^{m+1})+O(h_{y}^{m^{\prime }+1})$$. The theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical part.

Approximate Solution of Two-dimensional Fredholm Integral Equation of the First Kind Using Wavelet Base Method

A numerical direct method for solving two-dimensional linear and nonlinear Fredholm integral equations of the first kind based on Haar wavelet is introduced. The main characteristic of the method is that, unlike several other methods, it does not involve numerical integration, which leads to higher accuracy and quick computations as well. Further more an estimation of error bound for the present method is proved. Finally several test problems have been solved and compared with existing recent methods in order to demonstrate the effectiveness and applicability of the proposed technique.

A novel technique based on Bernoulli wavelets for numerical solutions of two-dimensional Fredholm integral equation of second kind

PurposeA novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials.Design/methodology/approachTo solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations.FindingsNumerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate.Originality/valueThe efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

Solving Two-Dimensional Nonlinear Fredholm Integral Equations Using Rationalized Haar Functions in the Complex Plane

As far as we are aware, no research has been published about two-dimensional integral equations in the complex plane by using Haar bases or any other kinds of wavelets. We introduce a numerical method to solve two-dimensional Fredholm integral equations, using Haar wavelet bases. To attain this purpose, first, an operator and then an orthogonal projection should be defined. Regarding the characteristics of Haar wavelet, we solve an integral equation without using common mathematical methods. We prove the convergence and an upper bound that mentioned in the method by employing the Banach fixed point theorem. Moreover, the rate of convergence our method is $$O(n(2q)^n)$$ . We present several examples of different kinds of functions and solve them by this method in this study.

Numerical solution of two-dimensional nonlinear fuzzy Fredholm integral equations via quadrature iterative method

Numerical solution of two-dimensional nonlinear fuzzy Fredholm integral equations via quadrature iterative method

New approach to solve two-dimensional Fredholm integral equations

In this paper, we present an efficient iterative method based on quadrature formula to solve two-dimensional nonlinear Fredholm integral equations. We investigate the convergence analysis of the proposed method. The error estimation is given in terms of uniform and partial modulus of continuity. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.

Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation

In this study, at first, we propose a new approach based on the two-dimensional fuzzy Lagrange interpolation and iterative method to approximate the solution of two-dimensional linear fuzzy Fredholm integral equation (2DLFFIE). Then, we prove convergence analysis and numerical stability analysis for the proposed numerical algorithm by two theorems. Finally, by some examples, we show the efficiency of the proposed method.

On the numerical solution of two-dimensional integral equations using a meshless local discrete Galerkin scheme with error analysis

This investigation presents a simple and effective method for numerically solving two-dimensional Fredholm integral equations of the second kind on non-rectangular domains. The general framework of the current scheme is based on the Galerkin method together with moving least squares (MLS) technique constructed on scattered points in which all integrals are approximated by a suitable quadrature formula. The MLS approach estimates an unknown function utilizing a locally weighted least square polynomial fitting. The method does not require any cell structures, so it is meshless and consequently independent of the geometry of the domain. The algorithm of the presented scheme is attractive and easy to implement on computers. Furthermore, the error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and the efficiency of the new method and confirm the theoretical error estimates provided in the current paper.