The concepts of generalized higher derivations, Jordan generalized higher derivations, and Jordan generalized triple higher derivations on Γ-ring M into ΓM-modules X are presented. We prove that every Jordan generalized higher derivation of Γ-ring M into 2-torsion free ΓM-module X, such that aαbβc=aβbαc, for all a, b, c M and α,βΓ, is Jordan generalized triple higher derivation of M into X.

This paper introduces the Multistep Modified Reduced Differential Transform Method (MMRDTM). It is applied to approximate the solution for Nonlinear Schrodinger Equations (NLSEs) of power law nonlinearity. The proposed method has some advantages. An analytical approximation can be generated in a fast converging series by applying the proposed approach. On top of that, the number of computed terms is also significantly reduced. Compared to the RDTM, the nonlinear term in this method is replaced by related Adomian polynomials prior to the implementation of a multistep approach. As a consequence, only a smaller number of NLSE computed terms are required in the attained approximation. Moreover, the approximation also converges rapidly over a

In the present work, we use the Adomian Decomposition method to find the approximate solution for some cases of the Newell whitehead segel nonlinear differential equation which was solved previously with exact solution by the Homotopy perturbation and the Iteration methods, then we compared the results.

An approximate solution of the liner system of ntegral cquations fot both fredholm(SFIEs)and Volterra(SIES)types has been derived using taylor series expansion.The solusion is essentailly

     Let R be a commutative ring with 1 and M be a left unitary R-module. In this paper, we give a generalization for the notions of compressible (retractable) Modules. We study s-essentially compressible (s-essentially retractable). We give some of their advantages, properties, characterizations and examples. We also study the relation between s-essentially compressible   (s-essentially retractable modules) and some classes of modules.

Let M be ,-ring and X be ,M-module, Bresar and Vukman studied orthogonal
derivations on semiprime rings. Ashraf and Jamal defined the orthogonal derivations
on -rings M. This research defines and studies the concepts of orthogonal
derivation and orthogonal generalized derivations on ,M -Module X and introduces
the relation between the products of generalized derivations and orthogonality on
,M -module.

  Let L be a commutative ring with identity and let W be a unitary left L- module. A submodule D of an L- module W is called  s- closed submodule denoted by  D ≤sc W, if D has   no  proper s- essential extension in W, that is , whenever D ≤ W such that D ≤se H≤ W, then D = H. In  this  paper,  we study  modules which satisfies  the ascending chain  conditions (ACC) and descending chain conditions (DCC) on this kind of submodules.