New method for solving fractional differential equations: H K transform

In this study, a transformation solution of high-order linear fractional differential equations is provided using Riemann-Liouville and Caputo fractional derivatives. To achieve this objective, HK transformed fundamental mathematical functions before outlining its salient characteristics were developed. Afterward, the fractional formula of HK transform was derived for both fractional calculus fractional integral and fractional derivative. Then the exact solution of an example of a fractional differential equation was found. This paper provided several numerical issues and solved them by strict stepwise solutions to illustrate the effectiveness of HK transform. The results demonstrate that the original HK Transform has been introduced offers accurate solutions for higher-order fractional differential equations while reducing computational complexity because the HK operator is very simply applied to denominators, complements, and fractions, the inverse is not complicated to obtain an exact solution without complicated algebraic equations.