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Aims to present a boundary integral equation method for solving Laplace's equation Δu=0 with nonlinear boundary conditions.

The nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation, with u on the boundary as the solution being sought. The integral equation is solved numerically by using the collocation method on smooth or nonsmooth boundary; the singularities of solution degrade the rates of convergence.

Variants of the methods for finding numerical solutions are suggested. So these methods have been compared with respect to number of iterations.

Numerical experiments show the efficiency of the proposed methods.

Provides new methods to solve nonlinear weakly singular integral equations and discusses difficulties that arise in particular cases.

Uses the continuous Legendre wavelets on the interval [0,1) in the manner of M. Razzaghi and S. Yousefi, to solve the linear second kind integral equations. We use quadrature formula for the calculation of inner products of any functions, which are required in the approximation for the integral equations. Then, we reduced the integral equation to the solution of linear algebraic equations.

The purpose of this paper is to discuss a numerical method for solving Fredholm integral equations of the first kind with degenerate kernels and convergence of this numerical method.

By using sinc collocation method in strip, the authors try to estimate a numerical solution for this kind of integral equation.

Some numerical results support the accuracy and efficiency of the stated method.

The paper presents a method for solving first kind integral equations which are ill‐posed.

The purpose of this paper is to use Alpert wavelet basis and modify the integrand function approximation coefficients to solve Fredholm‐Hammerstein integral equations.

L²[0, 1] was considered as solution space and the solution was projected to the subspaces of L²[0, 1] with finite dimension so that basis elements of these subspaces were orthonormal.

In this process, solution of Fredholm‐Hammerstein integral equation is found by solving the generated system of nonlinear equations.

Comparing the method with others shows that this system has less computation. In fact, decreasing of computations result from the modification.

The purpose of this paper is to explain the choice of Alpert multi‐wavelet as basis functions to discrete Fredholm integral equation of the second kind by using Petrov‐Galerkin method.

In this process, two kinds of matrices are obtained from inner product between basis of test space and trial space; some of them are diagonal with positive elements and some others are invertible. These matrices depend on type of selection of test and trial space basis.

In this process, solution of Fredholm integral equation of the second kind is found by solving the generated system of linear equations.

In previous work, convergence of Petrov‐Galerkin method has been discussed with some restrictions on degrees of chosen polynomial basis, but in this paper convergence is obtained for every degree. In point of computation, because of appearance of diagonal and invertible matrices, a small dimension system with a more accurate solution is obtained. The numerical examples illustrate these facts.

This paper sets out to introduce a numerical method to obtain solutions of Fredholm‐Volterra type linear integral equations.

The flow of the paper uses well‐known formulations, which are referenced at the end, and tries to construct a new approach for the numerical solutions of Fredholm‐Volterra type linear equations.

The approach and obtained method exhibit consummate efficiency in the numerical approximation to the solution. This fact is illustrated by means of examples and results are provided in tabular formats.

Although the method is suitable for linear equations, it may be possible to extend the approach to nonlinear, even to singular, equations which are the future objectives.

In many areas of mathematics, mathematical physics and engineering, integral equations arise and most of these equations are only solvable in terms of numerical methods. It is believed that the method is applicable to many problems in these areas such as loads in elastic plates, contact problems of two surfaces, and similar.

The paper is original in its contents, extends the available work on numerical methods in the solution of certain problems, and will prove useful in real‐life problems.

The purpose of this paper, with reference to compression of different images' portions with various qualities, is to obtain a high‐compression coefficient.

Usually, not all parts of a medical image have equal significance. Also, an image's background can be combined with noise. This method separates a part of the video which is moving from a part that is stationary.

This process results in the high‐quality compression of medical frames.

Separating parts of a frame using 2D and 3D wavelet transform makes a valuable contribution to biocybernetics.

The purpose of this study is to obtain a scheme for the numerical solution of Volterra integro-differential equations with time periodic coefficients deduced from the charged particle motion for certain configurations of oscillating magnetic fields.

The method reduces the solution of these types of integro-differential equations to the solution of two-dimensional Volterra integral equations of the second kind. The new method uses the discrete collocation method together with thin plate splines constructed on a set of scattered points as a basis.

The scheme can be easily implemented on a computer and has a computationally attractive algorithm. Numerical examples are included to show the validity and efficiency of the new technique.

The author uses thin plate splines as a type of free-shape parameter radial basis functions which establish an effective and stable method to solve electromagnetic integro-differential equations. As the scheme does not need any background meshes, it can be identified as a meshless method.

This paper aims to propose an efficient and convenient numerical algorithm for two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations (of Hammerstein and mixed types).

The main idea of the presented algorithm is to combine Bernoulli polynomials approximation with Caputo fractional derivative and numerical integral transformation to reduce the studied two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations to easily solved algebraic equations.

Without considering the integral operational matrix, this algorithm will adopt straightforward discrete data integral transformation, which can do good work to less computation and high precision. Besides, combining the convenient fractional differential operator of Bernoulli basis polynomials with the least-squares method, numerical solutions of the studied equations can be obtained quickly. Illustrative examples are given to show that the proposed technique has better precision than other numerical methods.

The proposed algorithm is efficient for the considered two-dimensional nonlinear Volterra-Fredholm integral equations and fractional integro-differential equations. As its convenience, the computation of numerical solutions is time-saving and more accurate.

The purpose of this paper is to develop rationalized Haar functions to approximate the solutions of the integro‐differential equations.

Properties of rationalized Haar functions are first presented, and the operational matrix of the product of two rationalized Haar functions vector is utilized to reduce the computation of integro‐differential equations to some algebraic equations.

Numerical results support the theoretical results.

Presents a method for solving integro‐differential equations.