Fractional Longitudinal Wave Dynamics in Magneto-Electro- Elastic Materials: A Neural Network-Based Approach

Fractional derivatives introduce an effective mathematical structure to describe memory effects, long-range interactions, and anomalous transport processes that are not well represented by the traditional integer-order models. This paper presents the unidirectional fractional longitudinal wave equation as a governing equation where a model is proposed to explain the steady wave propagation of solitary waves in a magneto-electro-elastic circular rod. Magneto-electro-elastic substances are a groundbreaking category of advanced functional materials with tremendous nanotechnology and biomedical engineering prospects because of their effective multi-field energy conversion and temperature responsiveness. In order to solve this complicated fractional nonlinear equation, we introduce a new computation-analysis approach: the Riccati subequation neural network method. This hybrid solution is a synergistic combination of an analytical solution structure and a neural network structure consisting of input, hidden, and output layers, with interconnection between neurons through weighted connections and activation functions. It is important to note that every neuron in the first hidden layer is coupled to the solutions of the Riccati equation, and this allows the systematic use of the new trial functions. With the suggested method, analytical solutions are obtained for the spacetime fractional partial differential equations of the unidirectional fractional longitudinal wave equation in the exact form of trigonometric, hyperbolic, and rational functions. This paper is the first attempt to combine the Riccati subequation method with a neural network model, which has given rise to new types of solitary wave solutions. The three-dimensional, two-dimensional, and contour plots are used to visualize the dynamic nature of these solutions and to display the rich nonlinear wave behavior. The effectiveness and the robustness of the implemented technique is not only proven through our findings but also provides more profound information about the nonlinear wave phenomena in the advanced multifunctional materials, which can inform future developments in energy harvesting and the design of biomedical devices.