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.

In this paper,  a generalized version of the auto-convolution Volterra Integral equation of the first kind as an ill-posed problem is studied. We apply the piecewise polynomial collocation method to reduce the numerical solution of this equation to a system of algebraic equations. According to the proposed numerical method, for $n=0$ and  $n=1,\ldots, N-1$, we obtain a  nonlinear and linear system, respectively. We have to distinguish between two cases, nonlinear and linear systems of algebraic equations. A double iteration process based on the modified Tikhonov regularization method is considered to solve the nonlinear algebraic equations. In this process, the outer iteration controls the evolution path of the unknown vector $U_0^{\delta}$ in the selected direction $\tilde{u}_0$, which is determined from the inner iteration process. For the linear case, we apply the Lavrentiev $\tilde{m}$ times iterated regularization method to deal with the ill-posed linear system. The validity and efficiency of the proposed method are demonstrated by several numerical experiments.

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 weighted semigroup algebra Mb (S, w) is studied via its identification with Mb (S) together with a weighted algebra product *w so that (Mb (S, w), *) is isometrically isomorphic to (Mb (S), *w). This identification enables us to study the relation between regularity and amenability of Mb (S, w) and Mb (S), and improve some old results from discrete to general case.

Let G be a locally compact Hausdorff topological group and H be a compact subgroup of G. Then, the homogeneous space G/H possesses a specific Radon measure, which is called a relatively invariant measure. We show that the concepts of convolution and involution can be extended to the integrable functions defined on this homogeneous space. We study the properties of convolution and prove that the space of integrable functions is an involutive Banach algebra with an approximate identity. We also find a necessary and sufficient condition on a closed subspace of this Banach algebra to make it a left ideal.

We show that the second-order Hochschild cohomology groups of measure algebra M(G) and group algebra L-1(G) with coefficients in C-phi are Banach spaces where G is a locally compact group and phi is the augmentation character.