In this paper, we will present a new method for a Volterra integral equation with Weakly singular kernel in the reproducing kernel space. Firstly the equation is transformed into a new equivalent equation. Its exact solution is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximation un(t) to the exact solution u (t) is obtained. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.

Compact finite difference scheme is applied for a partial integro-differential equation with a Weakly singular kernel. The product trapezoidal method is applied for discretization of the integral term. The order of accuracy in space and time is O (h4, k2-a), where 0<a<1. Stability and convergence in 2 L norm are discussed through energy method. Numerical examples are provided to confirm the theoretical prediction and to show that the combination of the compact finite difference approximation and product trapezoidal method give an efficient method for solving a partial integro-differential equation.

In this work, the convection-diffusion integro-differential equation with a Weakly singular kernel is discussed. The Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices. Also the convergence analysis for shifted Legendre polynomials and error estimation for tau method have been discussed and approved with the exact solution. Finally, several numerical examples are given to demonstrate the high accuracy of the method.

The presented paper investigates a new numerical method based on the characteristics of Legendre wavelet for solving Volterra Integral equations in this method,With the help of block-pulse functions and their characteristics, we obtain the fractional integral operational matrices corresponding to these wavelets. Then, by introducing collocation points, we use them to convert the desired equations into a system of algebraic equations. After solving the device, the approximate solution of the equation is easily calculated. Finally, we provide some numerical examples to demonstrate the accuracy and efficiency of the proposed methods.

There are many methods for numerical solutions of integral equations. In various branches of science and engineering, chemistry and biology, and physics applications integral equation is provided by many other authors. In this paper, a simple numerical method using a fuzzy, for the numerical solution of the integral equation with the weak singular kernel is provided. Finally, by providing three examples of the effectiveness of the proposed method was evaluated. In all the calculations and diagrams of the software, Mathematica is used.The advantage of the proposed method is that the algorithm is simple, appropriate and consistent with the exact solution provides a numerical answer. Examples and results presented in the previous section suggest this claim. A fast algorithm for the numerical solution for converting n fuzzy partition is also provided. The speed and simplicity of the algorithm allow any author to obtain a numerical answer with the help of this method it used to take.

In the present paper, we propose a method to solve a class of Weakly singular Fredholm integral equations of the second kind in reproducing kernel spaces. The Taylor series of the unknown function is used to remove the singularity and bases of reproducing kernel spaces are used to solve this equation. Efficiency of the proposed method is investigated through various examples.

In this paper, we will present a new method for solving a class of two-dimensional linear Volterra integral equation of the second kind with Weakly singular kernel from Abel type in the reproducing kernel space. The reproducing kernel function is discussed in detail. Weak singularity of problem is removed by applying integration by parts. Further, improper integral belongs to L_2 (Ω ). In our method the exact solution ϕ (x, t) is represented in the form of series in the reproducing kernel space W(ω ), and the approximate solution ϕ _n (x, t) is constructed via truncating the series to n terms. Convergence analysis of the method is proved in detail. Some numerical examples are also studied to demonstrate the efficiency and accuracy of the presented method. The obtained results show that the error of the approximate solution is monotone decreasing in the sense of the norm of W(ω ), when increasing the number of the nodes. Also, that indicate the method is simple and effective. It turns out that this method is valid.

This paper describes and compares application of wavelet basis and Block-Pulse functions (BPFs) for solving fractional integrodi erential equation (FIDE) with a Weakly singular kernel. First, a collocation method based on Haar wavelets (HW), Legendre wavelet (LW), Chebyshev wavelets (CHW), second kind Chebyshev wavelets (SKCHW), Cos and Sin wavelets (CASW) and BPFs are presented for driving approximate solution FIDEs with a Weakly singular kernel. Error estimates of all proposed numerical methods are given to test the convergence and accuracy of the method. A comparative study of accuracy and computational time for the presented techniques is given.