Capturing complexities with composite operator and differential operators with non-singular kernel.

The composite operator has been used in functional analysis with a clear application in real life. Nevertheless, a pure mathematical concept becomes very useful if one can apply it to solve real world problems. Modeling chaotic phenomena, for example, has been a concern of many researchers, and several methods have been suggested to capture some of them. The concept of fractional differentiation has also been used to capture more natural phenomena. Now, in elementary school, when composing two functions, we obtain a new function with different properties. We now ask when we compose two equations, could we a get new dynamics? Could we capture new natural problems? In this work, we make use of the composite operator to create a new kind of chaotic attractors built from two different attractors. In the linear case, we obtain integro-differential equations (classical and fractional) in the Caputo-Fabrizio case. We suggested a new numerical scheme to solve these new equations using finite difference, Simpson, and Lagrange polynomial approximations. Without loss of generality, we solve some examples with exact solutions and compare them with our proposed numerical scheme. The results of the comparison leave no doubt to believe that the proposed method is highly accurate as the error is of the order of 10.