Theoretical Analysis for Bioengineering Mathematical Model of Time‐Fractional‐Order Using New Effective Method

Water pollution poses a serious threat to public health, particularly in underdeveloped countries. In the current work, we solve very important applications related to nonlinear dynamic systems, such as the nonlinear time‐fractional water pollution (FWP) model and the time‐fractional‐order Bloch equations model in bioengineering, employing an effective natural Adomian decomposition method (NADM), a combination of the natural transform method (NTM) and the Adomian decomposition method (ADM). We provide proofs for the existence and uniqueness theorems as they correspond to nonlinear fractional differential equations based on the Banach fixed‐point theory. The fractional Bloch equation system under investigation plays a crucial role in the implementation of novel techniques in the physical sciences. We established analytical solutions for the Bloch equations in bioengineering models and approximate solutions for time‐fractional‐order water pollution, which are crucial for both the bioengineering and ocean engineering communities. The Caputo–Liouville fractional derivative is often preferred because its initial conditions align with those of integers. The results indicate that the NADM is highly efficient and useful. Some of the symbolic and numerical computations were performed using Mathematica 13 software.