On Uncertain-Fractional Modeling: The Future Way of Modeling Real-World Problems

Abstract

It has been a long time a challenge for many researchers to give a real interpretation of derivatives with fractional order. Some researchers said, fractional derivative is the shadow on the wall. This interpretation was wrong since the shadow of any object does not provide the real properties of the real object, for instance a black man has the same shadow with a white man. Using the definition and applications of a convolution, we gave new interpretation of derivative with fractional order. We gave specific interpretation for Caputo and Caputo–Fabrizio types as the fractional order changes. It was long believed that, the derivative with fractional order portray the effect of memory, this was only proved to be true in theory of elasticity and nowhere else. In this chapter, we introduced a new operator called uncertain derivative capable or portraying the memory effect in almost all situation. In order to include into mathematical formulation, the real rate of change and also the effect of memory, we introduced a new way of modeling real-world problem called uncertain-fractional modeling (UFM) and applied it to advection dispersion model. Numerical simulations of the new model show that real-world observation. This method will be the future way of modeling real-world problem efficiently.