Fractional mathematical modeling and beyond

Abstract

The theory of derivatives of non-integer order goes back to the Leibniz’s note in his list to L’Hospital, Sep 30, 1695, in which the meaning of the derivative of order one half is discussed (Fractional-order). Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Due to this reason, Fractional Mathematical Modeling(FMM) or Fractional-order (partial) differential equations(FPDEs) have been successfully applied in physics, biology, applied sciences, and engineering. In this talk, I discuss several difficulties in finding numerical approximations for Frac- tional Mathematical Modeling, such as an expensive computational cost. Also, I introduce recent research improvements to overcome these difficulties and new engineering applica- tions in nanofluids. In addition, I introduce Fractional Physics-informed neural networks (fPINNs), an extended variant of PINNs that utilize standard feedforward neural networks (NN) while explicitly incorporating partial differential equations (PDEs) into the neural network architecture via automatic differentiation.