Abstract
We use a new integral transform approach to solve the fractional Harry Dym equation and fractional Rosenau-Hyman equation in this work. The Elzaki transform and the integral transformation are combined in the suggested method (ET). To handle two nonlinear problems, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD). The ultimate purpose of this study is to find an error analysis that demonstrates that our final result converges to the exact and approximate result. The convergent series form solution demonstrates the method's efficiency in resolving several types of fractional differential equations. Furthermore, the solutions obtained in this study agree well with the exact solutions; thus, this strategy is powerful and efficient as an alternate way for obtaining approximate solutions to both linear and nonlinear fractional differential equations.