Abstract
In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed. The new formulation is efficient in modeling a class of concentrations in the complex transport process.
•A new fractional operator of variable order is addressed for the first time.•The Laplace and Fourier transforms of the variable-order fractional operator are presented.•The anomalous diffusion equations of variable order are presented in detail.•The formulation is efficiently used to model the PDEs in the Mathematical physics.