This paper is concerned with the solution of system of nonlinear fractional and integer order ordinary and partial differential equations. To achieve that aim, a method of solution is proposed which is developed from an integral transform and the well-known Adomian decomposition method. The Shehu transform Adomian decomposition method (STADM) proposed leverage on the unique advantage that Shehu transform, unlike Laplace transform, is applicable to both constant and variable coefficient initial value problems. The nonlinearity in all its forms is handled by developing corresponding Adomian polynomials, while the fractional order derivatives are interpreted in Caputo sense. STADM, when applied to the class of fractional order problems considered in the present work reduces the computational volume and time. The proposed method is applied to selected problems from the literature, and in most cases gives the exact solutions. The results of the problems solved are equally presented in 3D graphs for ease of visualization.