Grad-Shafranov Equation in Fractal Dimensions

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

“…To construct the frequency equation at the resonance case, we insert equation (33) into equation ( 25) and let ρ → 1, and the result gives…”

“…The small detaining parameter σ is defined in equation (25) and given by equation (33), measures the vibration of the resonance case, and is plotted in Figure 8. It is calculated for a system with a fixed value of…”

“…Theoretical applications of fractal analysis that have been of interest to various studies include the physical phenomena of hierarchical structure, thermal conductivity in micro-channels, diffusion, dynamics of the atmosphere and ocean, petroleum and nuclear engineering, meteorology, material design, and thermo-elasticity [25][26][27][28][29][30][31][32][33] . Through a porous structure, Sheng et al 34 investigated a fractal model, and Miao et al 35 looked into another fractal model of shale.…”

An innovative technique to solve a fractal damping Duffing-jerk oscillator

“…The fractal studying is an interesting idea that has distinctive applications in assorted areas, for instance, the physical phenomena of the hierarchical structure, diffusion, thermal conductivity within branched networks, meteorology, fluids flow resistance in micro-channels, ocean and atmosphere dynamics, nuclear and petroleum engineering, thermo elasticity, and materials design [30][31][32][33][34][35][36][37][38]. As a result of these important applications, the study of fractals was a fertile field for many scientists.…”