Comparative Study of the Nonlinear Fractional Generalized Burger-Fisher Equations Using the Homotopy Perturbation Transform Method and New Iterative Transform Method

The time-fractional generalized Burger–Fisher equation (TF-GBFE) is utilized in many physical applications and applied sciences, including nonlinear phenomena in plasma physics, gas dynamics, ocean engineering, fluid mechanics, and the simulation of financial mathematics. This mathematical expression explains the idea of dissipation and shows how advection and reaction systems can work together. We compare the homotopy perturbation transform method and the new iterative method in the current study. The suggested approaches are evaluated on nonlinear TF-GBFE. Two-dimensional (2D) and three-dimensional (3D) figures are displayed to show the dynamics and physical properties of some of the derived solutions. A comparison was made between the approximate and accurate solutions of the TF-GBFE. Simple tables are also given to compare the integer-order and fractional-order findings. It has been verified that the solution generated by the techniques given converges to the precise solution at an appropriate rate. In terms of absolute errors, the results obtained have been compared with those of alternative methods, including the Haar wavelet, OHAM, and q-HATM. The fundamental benefit of the offered approaches is the minimal amount of calculations required. In this research, we focus on managing the recurrence relation that yields the series solutions after a limited number of repetitions. The comparison table shows how well the methods work for different fractional orders, with results getting closer to precision as the fractional-order numbers get closer to integer values. The accuracy of the suggested techniques is greatly increased by obtaining numerical results in the form of a fast-convergent series. Maple is used to derive the approximate series solution’s behavior, which is graphically displayed for a number of fractional orders. The computational stability and versatility of the suggested approaches for examining a variety of phenomena in a broad range of physical science and engineering fields are highlighted in this work.