The existeness and uniqueness result of solutions to initial value problems of nonlinear diffusion equations involving with the conformable variable derivative

Abstract

In this paper, based on the conformable derivative, we introduce the concept of conformable variable order derivative. The conformable derivative is a local operator, it has many properties like usual integer derivative’s ones. Similar to the conformable derivative, we study some properties of the conformable variable order derivative. We investigate the fundamental solutions to initial value problem for linear homogeneous and inhomogeneous diffusion differential equations with the conformable variable order derivative. In addition, using upper and lower solutions and monotone iterative method, we consider the existence and uniqueness of solutions to an initial value problem for nonlinear diffusion differential equations involving with the conformable variable order derivative.