Fractional Extension of Nonlinear Fisher and Gardner Equations Using Adomian Decomposition Shehu Transform Method

This work presents a semi-analytical approach for solving the fractional Fisher equation and fractional Gardner equation using the Adomian decomposition Shehu transform method (ADShTM) with Caputo fractional derivatives. The proposed method efficiently handles the nonlinear fractional partial differential equations by combining the strengths of the Shehu transform and Adomian decomposition, offering a systematic and straightforward procedure for obtaining approximate solutions. To validate the effectiveness and accuracy of the method, we provide results for different fractional orders in both two dimensional and three dimensional plots, illustrating the behavior of the solutions as the fractional order varies. Furthermore, a comprehensive comparison with existing natural homotopy perturbation method (NHTM) is presented, highlighting the advantages of ADShTM in terms of simplicity, convergence, and accuracy. Nonlinear fractional partial differential equations pose significant analytical challenges due to the simultaneous presence of strong nonlinearity and fractional-order derivatives. The findings demonstrate that the proposed approach is a reliable and efficient tool for solving a broad class of nonlinear fractional differential equations.