New method for solving fractional differential equations: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> </mml:math> transform

Abstract 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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">HK</mml:mi> </mml:math> transformed fundamental mathematical functions before outlining its salient characteristics were developed. Afterward, the fractional formula of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">HK</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">HK</mml:mi> </mml:math> transform. The results demonstrate that the original <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">HK</mml:mi> </mml:math> Transform has been introduced offers accurate solutions for higher-order fractional differential equations while reducing computational complexity because the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">HK</mml:mi> </mml:math> operator is very simply applied to denominators, complements, and fractions, the inverse is not complicated to obtain an exact solution without complicated algebraic equations.