Cauchy Problem Approach to Biharmonic Models in Fractal Time and Space

Abstract

This paper pioneers the application of fractal calculus to higher alpha-order differential models defined on non-Euclidean spaces. We establish and solve the fractal Cauchy problem for the biharmonic equation, providing detailed visualizations that demonstrate the unique influence of fractal geometry on solution behavior. The methodology is subsequently validated through applications to critical physical scenarios, namely the cooling of a clamped thin beam and the vibration of a thin elastic plate. These case studies reveal how the fractal dimensions of time and space fundamentally modify the dynamics of classical systems. Overall, this study underscores the effectiveness and necessity of fractal calculus for accurately capturing complex, scale-dependent phenomena in non-standard frameworks.