A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique

In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the system corresponding to the Galerkin method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.     

Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p,where p is a prime number.As a consequence we prove if |G|=2δp,δ=0,1,2 and p prime,then Γ=Cay(G,S) is a connected normal 1/2 arc-transitive Cayley graph only if G=F4p,where S is an inverse closed generating subset of G which does not contain the identity element of G and F 4p is a group with presentation F4p = a,b|ap=b4=1,b-1ab=aλ,where λ2≡-1(mod p).