In this work, we analyze porous elastic system with microtemperature from second spectrum viewpoint. Indeed, by using the classical Faedo-Galerkin method combined with the a priori estimates, we prove the existence and uniqueness of a global solution of this problem. Then we prove that this solution is exponentially stable without assuming the condition of equal wave speeds. Then, we introduce a finite element approximation and we prove that the associated discrete energy decays. Finally, we obtain some a priori error estimates assuming additional regularity on the solution and we present some numerical results which demonstrate the accuracy of the approximation and the behaviour of the solution

The generalized group of units of the ring modulo n was first introduced by El-Kassar and Chehade, written as U k (Zn). This allows us to formulate a new generalization to the Euler phi function ϕ(n), that represents the order of U k (Zn) and it is denoted by ϕ k (n). In this paper, we introduce this newly defined function, where we compute its explicit form and examine some of its properties similar to that of ϕ(n). In addition, we study some generalized equations involving ϕ k (n) where complete solution is given for some equations by considering the general case and others for some particular cases.