A difference equation is an equation that contains a difference in one or more than one variable. It has significant applications in the mathematical models of the applications of engineering and science. In this paper, we proposed a method for solving ordinary difference equations (OΔE) using discrete Sumudu transform (DSDM). Some properties of the discrete Sumudu transform have been introduced. The implementations for the test examples of initial value problems of difference equations examined the efficiency of the proposed method. The approximated solutions of initial value problems of (OΔE) in discrete domain with different orders have been evaluated and compared with exact solutions of these problems. Discrete fractional calculus (DFC) is continuously spreading in the neural networks, chaotic maps, engineering practice, and image encryption, which is appropriately assumed for discrete-time modelling in continuum problems. For solving problems including difference operators (classic and fractional), we employ a discrete version of the Adomian decomposition method (ADM). This method helps to find the solutions of linear and nonlinear classic and fractional difference problems (CDPs and FDPs). Examples are given to clarify and confirm the obtained results and some of particular cases of CDPs and FDPs are highlighted. we have used some methods to solve a class of fractional difference equations(FΔE) of Riemann type. The first method is a combination method based on the successive approximation method (SAM) with Sumudu transformation. Also, the second one is a combination method consisting of the Homotopy perturbation method (HPM) with Sumudu transformation. The obtained results from both methods are compared. Also, we have presented theories and illustrative examples that support the research findings
