Exploring the Modified Generalized Mapping Neural Network Technique: Diversity of Wave Profiles to the Nonlinear Dynamical Models

ABSTRACT This work investigates the nonlinear wave dynamics of the nonlinear Murray equation and linear quadratic model. The proposed models have various applications in medical theory such as the nonlinear Murray equation models the blood flow dynamics where the linear quadratic model describes the population growth of the tumour cells. Researchers are captivated by solitary wave solutions because they significantly clarify nonlinear phenomena, which have many practical implications. Their remarkable characteristics and stability exemplify innovative nonlinear models spanning several domains, including physical, biological, and medical modeling. This work combines modified generalized Riccati equation mapping method and neural networks to compute different wave structures for the proposed nonlinear models. The suggested technique integrates the solutions of the Riccati problem into neural networks. Neural networks are multi‐layer computational representations consisting of activation and weights functions connecting neurons across input, hidden, and output layers. In this method, each neuron in the first hidden layer is allocated to the solutions of the Riccati equation. Thus, the new trial functions are derived. To verify the mathematical framework of this method, this study offers innovative solutions in the different solitary waves as the technique is used in the neural networks model for the first time. The dynamic characteristics of some wave‐related solutions have been shown using different visuals. The analysis demonstrates that the suggested technique is more effective, powerful, concise, and practical for investigating complex nonlinear partial and fractional differential equations.