A new framework for multivariate general conformable fractional calculus and potential applications

Some applications of fractional partial differential equations (FPDEs) based on conformable fractional derivative (CFD) are discussed recently. As an extension of CFD - power kernel is generalized to other normalized probability distribution kernels, general conformable fractional derivative (GCFD) provides a framework to explain the physical meaning of CFD and describe which phenomena or processes GCFD/CFD are suitable to model. We will establish the multivariate theory of GCFD in this paper. Compared with traditional hereditary fractional calculus like Riemann-Liouville type or Caputo type, the unified theory of GCFD vector calculus has well forms with simplicity and elegance. As applications, fractional Maxwell's equations of GCFD are given to describe electromagnetic fields of media that demonstrate inhomogeneous, complicated and local properties, and some fractional parabolic equations are presented as natural extensions of some nonlinear parabolic equations that cover several important famous equations.