Multivariable functions represented as fractional power series and generalized Taylor’s formula

Fractional power series in several variables are investigated within the framework of fractional calculus. By employing the Riemann-Liouville fractional integral and Caputo derivative, a generalized Taylor’s formula for multivariable functions is established, extending previously known results for the single-variable case. A sufficient condition for representing a multivariable function as a fractional power series is stated and a framework for obtaining approximate solutions of fractional partial differential equations is provided. In addition, the paper presents a method for determining restricted local extrema of multivariable functions. Illustrative examples are included.