The System of Ax=b in which A is a Toeplitz and symmetric positive definite (SPD) is derived from Convolution-type integral equations. At normal state, the system contains in appropriate eigenvalues not clustering around 1. In the present paper, the preconditioned CG method is used to solve the above-mentioned system. The using of CG method with suitable preconditioners causes clustering eigenvalues of the new system around 1. As a result, the stability and convergence rate are guaranteed and at most we reach the answer in a steps.

In this paper, we present an approximate method to solve the solution of the second kind Volterra integral equations. This method is based on a previous scheme, applied by Maleknejad et al., [K. Maleknejad and Aghazadeh, Numerical solution of Volterra integral equations of the second kind with Convolution kernel by using Taylor-series expansion method, Appl. Math. Comput. (2005)] to gain the approximate solution of the second kind Volterra integral equations with Convolution kernel and Maleknejad et al. [K. Maleknejad and T. Damercheli, Improving the accuracy of solutions of the linear second kind volterra integral equations system by using the Taylor expansion method, Indian J. Pure Appl. Math. (2014)] to gain the approximate solutions of systems of second kind Volterra integral equations with the help of Taylor expansion method. The Taylor expansion method transforms the integral equation into a linear ordinary di erential equation (ODE) which, in this case, requires speci ed boundary conditions. Boundary conditions can be determined using the integration technique instead of di erentiation technique. This method is more stable than derivative method and can be implemented to obtain an approximate solution of the Volterra integral equation with smooth and weakly singular kernels. An error analysis for the method is provided. A comparison between our obtained results and the previous results is made which shows that the suggested method is accurate enough and more stable.

The aim of this paper is to solve a class of  auto-Convolution Volterra integral equations by the well-known differential transform method. The analytic property of solution and convergence of the method under some assumptions are discussed and some illustrative examples are given to clarify the theoretical results, accuracy and performance of the proposed method.

The proposed study is focused to introduce a novel integral transform op-erator, called Generalized Bivariate (GB) transform. The proposed trans-form includes the features of the recently introduced Shehu transform, ARA transform, and Formable transform. It expands the repertoire of existing Laplace-type bivariate transforms. The primary focus of the present work is to elaborate fashionable properties and Convolution theorems for the proposed transform operator. The existence, inversion, and duality of the proposed transform have been established with other existing transforms. Implementation of the proposed transform has been demonstrated by ap-plying it to different types of differential and integral equations. It validates the potential and trustworthiness of the GB transform as a mathematical tool. Furthermore, weighted norm inequalities for integral Convolutions have been constructed for the proposed transform operator.

Using the mean-value theorem for integrals we tried to solved the nonlinear integral equations of Hammerstein type. The mean approach is to obtain an initial guess with unknown coefficients for unknown function y(x). The procedure of this method is so fast and don’t need high cpu and complicated programming. The advantages of this method is that we can applied for those integral equations which have not the unique solution too.