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 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.