Solving weakly singular integral equations utilizing the meshless local discrete collocation technique

The current work presents a computational scheme to solve weakly singular integral equations of the second kind. The discrete collocation method in addition to the moving least squares (MLS) technique established on scattered points is utilized to estimate the solution of integral equations. The MLS scheme approximates a function without mesh refinement over the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation technique for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize an accurate quadrature formula based on the use of non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the approach. The proposed scheme does not require any meshes, so it can be called as the meshless local discrete collocation (MLDC) method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates. Keywords: Discrete collocation method, Weakly singular integral equation, Meshless method, Moving least squares (MLS), Error analysi

Chebyshev Wavelet Method for Numerical Solution of Fredholm Integral Equations of the First Kind

A computational method for solving Fredholm integral equations of the first kind is presented. The method utilizes Chebyshev wavelets constructed on the unit interval as basis in Galerkin method and reduces solving the integral equation to solving a system of algebraic equations. The properties of Chebyshev wavelets are used to make the wavelet coefficient matrices sparse which eventually leads to the sparsity of the coefficients matrix of obtained system. Finally, numerical examples are presented to show the validity and efficiency of the technique

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates

A meshless local Galerkin integral equation method for solving a type of Darboux problems based on the radial basis functions

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions. doi:10.1017/S144618112100037