A meshless method based on boundary integral equations and radial basis functions for biharmonic-type problems

This paper presents a meshless method, which replaces the inhomogeneous biharmonic equation by two Poisson equations in terms of an intermediate function. The solution of the Poisson equation with the intermediate function as the right-hand term may be written as a sum of a particular solution and a homogeneous solution of a Laplace equation. The intermediate function is approximated by a series of radial basis functions. Then the particular solution is obtained via employing Kansa’s method, while the homogeneous solution is approximated by using the boundary radial point interpolation method by means of boundary integral equations. Besides, the proposed meshless method, in conjunction with the analog equation method, is further developed for solving generalized biharmonic-type problems. Some numerical tests illustrate the efficiency of the method proposed.     

Analytical formulation of meshless local integral equation method

In this paper, the exact forms of integrals in the meshless local boundary integral equation method are derived and implemented for elastostatic problems. A weak form for a set of governing equations with a unit test function or polynomial test functions is transformed into local integral equations. Each node has its own support domain and is surrounded by a local integral domain with different shapes of boundaries. The meshless approximation based on the radial basis function (RBF) is employed for the implementation of displacements. A completed set of closed forms of the local boundary integrals are obtained. As there are no numerical integrations to be carried out the computational time is significantly reduced. Three examples are presented to demonstrate the application of this approach in solid mechanics.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

Numerical solution of the two-dimensional Fredholm integral equations using Gaussian radial basis function

In this paper, we introduce a numerical method for the solution of two-dimensional Fredholm integral equations. The method is based on interpolation by Gaussian radial basis function based on Legendre-Gauss-Lobatto nodes and weights. Numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the method.     

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

In this work, we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels. Particularly, we consider the case when the underlying solutions are sufficiently smooth. In this case, the proposed method leads to a fully discrete linear system. We show that the fully discrete integral operator is stable in both infinite and weighted square norms. Furthermore, we establish that the approximate solution arrives at an optimal convergence order under the two norms. Finally, we give some numerical examples, which confirm the theoretical prediction of the exponential rate of convergence.     

On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation

In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods.The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström’s method with the trapezoidal rule. For this, a convergence analysis is carried out.     

Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets

In this work, we present a computational method for solving nonlinear Fredholm integral equations of the second kind which is based on the use of Haar wavelets. Error analysis is worked out that shows efficiency of the method. Finally, we also give some numerical examples.     

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.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

Note on the integral mean value method for Fredholm integral equations of the second kind

We study the paper of Avazzadeh et al. [Z. Avazzadeh, M. Heydari, G.B., Loghmani, Numerical solution of Fedholm integral equations of the second kind by using integral mean value theorem, Appl. Math. Model. 35 (2011) 2374–2383] with the integral mean value method for Fredholm integral equations of the second kind. The objective of the note is threefold. First, we point out a basic error in the paper. Second, we find that the given numerical examples are only related to the special cases of Fredholm integral equations of the second kind with the degenerate kernels, which can be solved simply. Third, due to the basic error, our observations reveal that generally the suggested method should not be considered for a Fredholm integral equation of the second kind.     

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.     

On the numerical solution of nonlinear Burgers’-type equations using meshless method of lines

In this paper, a meshless method of lines (MMOL) is proposed for the numerical solution of nonlinear Burgers’-type equations. This technique does not require a mesh in the problem domain, and only a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions (RBFs). The scheme is tested for various examples. The results obtained by this method are compared with the exact solutions and some earlier work.     

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem

In this paper, we present a new semi-analytical method for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind and the systems including them. The main idea in this method is applying the mean value theorem for integrals. Some examples are presented to show the ability of the model. The results confirm that the method is very effective and simple.     

Best estimates of RBF-based meshless Galerkin methods for Dirichlet problem

In this paper, Galerkin methods based on the radial basis functions to deal with the partial differential equations are discussed. The best error estimates for this method are obtained.     

A meshless method based on the moving least squares (MLS) approximation for the numerical solution of two-dimensional nonlinear integral equations of the second kind on non-rectangular domains

This paper investigates a numerical method for solving two-dimensional nonlinear Fredholm integral equations of the second kind on non-rectangular domains. The scheme utilizes the shape functions of the moving least squares (MLS) approximation constructed on scattered points as a basis in the discrete collocation method. The MLS methodology is an effective technique for approximating unknown functions which involves a locally weighted least square polynomial fitting. The proposed method is meshless, since it does not need any background mesh or cell structures and so it is independent of the geometry of the domain. The scheme reduces the solution of two-dimensional nonlinear integral equations to the solution of nonlinear systems of algebraic equations. The error analysis of the proposed method is provided. The efficiency and accuracy of the new technique are illustrated by several numerical examples.     

第二类Fredholm积分方程的泰勒展开解法

本进一步发展了用Taylor公式求解第二类Fredholm积分方程的方法，并给出了近似解的误差精度分析．     

A new numerical approach for the solution of nonlinear Fredholm integral equations system of second kind by using Bernstein collocation method

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B‐spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind

Periodic harmonic wavelets (PHW) were applied as basis functions in solution of the Fredholm integral equations of the second kind. Two equations were solved in order to find out advantages and disadvantages of such choice of the basis functions. It is proved that PHW satisfy the properties of the multiresolution analysis.     

A numerical method for solving the hyperbolic telegraph equation

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this article, we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation using collocation points and approximating the solution using thin plate splines radial basis function. The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Meshless method of lines for the numerical solution of generalized Kuramoto-Sivashinsky equation

In this paper, the numerical solution of the generalized Kuramoto-Sivashinsky equation is presented by meshless method of lines (MOL). In this method the spatial derivatives are approximated by radial basis functions (RBFs) giving an edge over finite difference method (FDM) and finite element method (FEM) because no mesh is required for discretization of the problem domain. Only a set of scattered nodes is required to approximate the solution. The numerical results in comparison with exact solution using different radial basis functions (RBFs) prove the efficiency and accuracy of the method.