Two-dimensional Integral Equations Research Articles

On the numerical solution of nonlinear integral equations on non-rectangular domains utilizing thin plate spline collocation method

This article investigates an approximate scheme to solve nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The integral equations considered in the current paper are considered together with either smooth or weakly singular kernels. The offered method utilizes thin plate splines as a basis in the discrete collocation method. We can regard thin plate splines as a type of the free shape parameter radial basis functions. These basis functions establish an accurate and stable technique to estimate an unknown function by using a set of scattered points on the solution domains. Since the thin plate splines have limited smoothness, the integrals appeared in the scheme cannot be estimated by classical integration rules. Therefore, we introduce a special precise quadrature formula on non-rectangular domains to compute these integrals. The proposed scheme does not require any mesh generations, so it is meshless and does not depend on the domain form. Error analysis is also provided for the method. The performance and convergence of the new approach are tested on four two-dimensional integral equations given on the wing, mushroom, pentagon and fish-like domains.

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.

LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations

This paper is devoted to extend an iterative algorithm for sparse least-squares problems for solving one- and two-dimensional linear Fredholm integral equations. We consider the operator form of these equations and then develop the LSMR method for solving them in an appropriate manner. The proposed method is based on bidiagonalization process of Golub–Kahan and reducing the linear operator $${\mathcal {L}}$$ to the lower bidiagonal matrix form. Convergence property of the numerical solution associated with the suggested scheme is also provided. The method is fast and very accurate in solving two-dimensional equations. Other advantage of the proposed method is its high accuracy for solving Fredholm integral equations with non-smooth solutions. Solving such problems with other numerical methods, for instance, spectral methods, are not accurate as our proposed LSMR method. Numerical results confirm this claim.

Two-step collocation methods for two-dimensional Volterra integral equations of the second kind

Abstract In this paper, we develop two-step collocation (2-SC) methods to solve two-dimensional nonlinear Volterra integral equations (2D-NVIEs) of the second kind. Here we convert a 2D-NVIE of the second kind to a one-dimensional case, and then we solve the resulting equation numerically by two-step collocation methods. We also study the convergence and stability analysis of the method. At the end, the accuracy and efficiency of the method is verified by solving two test equations which are stiff. In examples, we use the well-known differential transform method to obtain starting values.

A New Computational Method Based on Bernstein Operational Matrices for Solving Two-Dimensional Linear Stochastic Volterra Integral Equations

This paper is concerned with obtaining the approximate numerical solution of two-dimensional linear stochastic Volterra integral equation by using two-dimensional Bernstein polynomials as basis. Properties of these polynomials and operational matrix of integration together with the product operational matrix are utilized to transform the integral equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Bernstein coefficients. Some theorems are included to show the convergence and advantage of the proposed method. The numerical example illustrates the efficiency and accuracy of the method.

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.

Solving two-dimensional integral equations of the second kind on non-rectangular domains with error estimate

In this paper, the collocation method is applied on two-dimensional integral equations of the second kind on non-rectangular domains. Since the domains of these equations are non-rectangular and so directly applying the collocation method for them is difficult, at first, the integral equations are converted to equivalent integral equations on rectangular domains. Then, two-dimensional Jacobi collocation method is applied. Furthermore, an error estimate for the method is investigated and several examples demonstrate the accuracy and efficiency of the method.

Validation of a volume integral equation method for indoor propagation modelling

A full-wave indoor propagation model based on the two-dimensional (2D) volume electric field integral equation (VEFIE) is presented. The model is discretised with the method of moments (MoM). The linear system arising from the MoM discretisation is solved with an iterative solver accelerated by using the fast Fourier transform and a reduced operator. The 2D VEFIE is validated against the analytical Mie series solution for scattering from a dielectric cylinder before being applied to a typical indoor propagation environment and compared against measurements. It is also compared against current popular empirical models and ray tracing. The ability of the frequency domain VEFIE method to produce time domain information is demonstrated with an investigation of numerical considerations that must be made.

Effective wear coefficient and wearing-in period for a functionally graded wear-resisting punch

A two-dimensional wear contact problem for an elastic layer and a wear-resisting punch is considered. The contact area and the contact load are assumed to be fixed, whereas the punch’s shape changes according to Archard’s law of wear with variable wear coefficient. By neglecting the effect of tangential tractions, the problem of determining the normal contact pressure is reduced to a two-dimensional integral equation containing a Fredholm coordinate operator and a Volterra time operator. By the method of separation of variables, the transient contact pressure distribution has been constructed in terms of the solutions of some eigenvalue problem. A special attention is paid to quantities of practical interest, such as the wearing-in period and the transient effective wear coefficient.

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.

A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations

This article describes a numerical scheme to solve two-dimensional nonlinear Volterra integral equations of the second kind. The method estimates the solution by the Galerkin method based on the use of moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin method results from the numerical integration of all integrals associated with the scheme. In the current work, we employ the composite Gauss-Legendre integration rule to approximate the integrals appearing in the method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The algorithm of the described scheme is computationally attractive and easy to implement on computers. The error bound and the convergence rate of the presented method are obtained. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.

A MESHLESS LOCAL GALERKIN METHOD FOR THE NUMERICAL SOLUTION OF HAMMERSTEIN INTEGRAL EQUATIONS BASED ON THE MOVING LEAST SQUARES TECHNIQUE

In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the system corresponding to the Galerkin method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.

Numerical Solution of Two-Dimensional Nonlinear Stochastic Itô-Volterra Integral Equations by Applying Block Pulse Functions

This paper investigates the numerical solution of two-dimensional nonlinear stochastic Ito-Volterra integral equations based on block pulse functions. The nonlinear stochastic integral equation is transformed into a set of algebraic equations by operational matrix of block pulse functions. Then, we give error analysis and prove that the rate of convergence of this method is efficient. Lastly, a numerical example is given to confirm the method.

The method for calculating singular integrals in problems of axially symmetric Stokes flows

The flow of a viscous fluid at small Reynolds numbers (Stokes flow) in a three-dimensional formulation is investigated. In this case, the inertial terms in the equations of motion can be neglected. Such flows can occur in nanotubes that can be considered as inclusions in representative volume elements of nanomaterials. By using the fundamental solution of Ossen, an integral representation of the velocity is proposed. This representation is used to receive an integral equation for an unknown density. The solution of the resulting equation makes it possible to calculate the fluid pressure on the walls of the shell. The case of axially symmetric flows is investigated. For this, an integral representation of the unknown velocity in cylindrical coordinates is obtained. By integrating over the circumferential coordinate, the two-dimensional singular integral equation is reduced to one-dimensional one. It has been proved that the components of the kernels in singular operators are expressed in terms of elliptic integrals of the first and second kind. It has been proved that the singularities of the kernels of one-dimensional singular integral equations have a logarithmic character. To calculate elliptic integrals, the Gaussian algorithm based on the use of the arithmetic-geometric mean value is proposed. This procedure allows us to obtain logarithmic singular components with high accuracy, which makes it possible to use special quadrature formulas to calculate such integrals. An algorithm with usage of the boundary element method for the numerical solution of the obtained singular integral equations is proposed. The method for solving one-dimensional singular equations, where the kernels contain elliptic integrals with logarithmic singularities (i.e logarithmic singularity is not expressed explicitly) has been tested. The obtained numerical results have been compared with the well-known analytical solutions. The data obtained indicate the high efficiency of the proposed numerical method.

Кругове теплоактивне міжфазне включення в кусково-однорідному трансверсального-ізотропному просторі

An exact solution of the stationary thermoelasticity problem about interfacial circular absolutely rigid inclusion, which is under conditions of complete adhesion and under conditions of smooth contact with transversely homogeneous spaces, is constructed. The task with the help of the constructed discontinuous solution, by the method of singular integral relations, is reduced to a system of singular integral equations (SIE). An exact solution has been built for the specified systems of two-dimensional singular integral equations. As a result, dependences jumps of stresses and displacement on temperature, equivalent load, main moments and thermomechanical characteristics of transversally isotropic materials. The influence of the type of contact interaction on the behavior of the solutions is established. In particular, it has been shown that the stresses in the neighborhood of the inclusion with a smooth contact have a root singularity, and with complete coupling, the root singularity, which is amplified by oscillation. The behavior of the generalized intensity coefficient (GCIN) was studied for the combination of various transversely isotropic materials at different power and temperature loads.

To the Theory of Non-Model Two-Dimensional Integral Equations of Volterra Type with a Strongly Singular and Weakly Singular Line on a Strip

To the Theory of Non-Model Two-Dimensional Integral Equations of Volterra Type with a Strongly Singular and Weakly Singular Line on a Strip

An Efficient Operational Matrix Method for Solving a Class of Two-Dimensional Singular Volterra Integral Equations

In this paper, we consider a spectral method to solve a class of two-dimensional singular Volterra integral equations using some basic concepts of fractional calculus. This method uses a modification of hat functions for finding a numerical solution of the considered equation. Some properties of the modification of hat functions are presented. The main contribution of this work is to introduce the fractional order operational matrix of integration for the considered basis functions. Making use of the Riemann-Liouville fractional integral operator helps us to reduce the main problem to a system of linear algebraic equations which can be solved easily. After that, error analysis of the method is discussed. Finally, numerical examples are included to confirm the accuracy and applicability of the suggested 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.

Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order

In this paper, we develop a numerical scheme based on two-dimensional orthonormal Bernstein polynomials (2D-OBPs) to solve two-dimensional nonlinear integral equations of fractional order. The fractional integral considered here is in the Riemann–Liouville sense. By using definition of Riemann–Liouville fractional integral, two-dimensional nonlinear fractional integral equations is transformed into two-dimensional nonlinear ordinary integral equations. Operational matrices method based on 2D-OBPs are applied to obtain an approximate solution with high accuracy for these equations. In addition, error analysis of the proposed method is discussed and an upper error bound is provided under weak assumptions. Some linear and nonlinear examples are given to demonstrate the accuracy, efficiency and speed of the suggested method.