Mathematical Modeling and Numerical Simulation of Musical Melodies’ Effects on Brain Waves

Brain waves—delta, theta, alpha, beta, and gamma—can be modulated by musical stimuli. The objective of this study is to investigate, through mathematical modeling and numerical simulation, how melodies influence the relative power distribution of these EEG bands. Five formulations were implemented: the first-order linear equation, the linear hyperbolic system, the Burgers equation, the Korteweg–de Vries equation, and a fully nonlinear equation. In all cases, the melody was introduced as a boundary condition or external forcing, and the response was recorded at an interior point of the domain. The results indicate that linear models reproduce the uniform propagation of the stimulus, whereas quasilinear and nonlinear models generate additional modulations associated with dispersion, diffusion, and nonlinear effects. This framework provides a conceptual foundation for future applications in computational neuroscience and music therapy.