Bicomplex Caputo Derivative: A Comparative Study with Bicomplex Riemann–Liouville Operators and Applications

Abstract

The aim of this article is to define the Caputo derivative of bicomplex order for the functions of a bicomplex variable, which we refer to as the Bicomplex Caputo Derivative (BCD) throughout the paper. We achieve BCD via the development of the Caputo derivative of bicomplex order for the bicomplex-valued functions of real variable and discuss some of its significant properties. We also compare BCD with the bicomplex Riemann–Liouville derivative and integral. We demonstrate the advantages of the properties of BCD by finding the derivatives of some elementary bicomplex functions. The useful applications of BCD are found in constructing bicomplex fractional Maxwell’s equations in the vacuum and source-free domains. The solutions of bicomplex fractional Maxwell’s equations are obtained by considering a bicomplex vector field, and it is proved that in this way the number of equations is reduced by half.