New investigation on the conformable version (CoV) of multivariable calculus is proposed. The conformable derivative (CoD) of a real-valued function (RVF) of several variables (SVs) and all related properties are investigated. An extension to vector-valued functions (VVFs) of several real variables (SRVs) is studied in this work. The CoV of chain rule (CR) for functions of SVs is also introduced. At the end, the CoV of implicit function theorem (IFThm) for SVs is established. All results in this work can be potentially applied in studying various modeling scenarios in physical oceanography such as Stommel’s box model of thermohaline circulation and other related models where all our results can provide a new analysis and computational tool to investigate these models or their modified formulations.
In the present paper, we implement a novel numerical method for solving differential equations with fractional variable-order in the Caputo sense to research the dynamics of a circulant Halvorsen system. Control laws are derived analytically to make synchronization of two identical commensurate Halvorsen systems with fractional variable-order time derivatives. The chaotic dynamics of the Halvorsen system with variable-order fractional derivatives are investigated and the identical synchronization between two systems is achieved. Moreover, graph simulations are provided to validate the theoretical analysis.