Abstract

Fractional calculus has emerged as a hot research topic soon after Mandelbrot presented the fractal theory. Fractal theory is mathematically beautiful, but natural fractals fail to fulfil the rigid mathematical definition. Many hierarchical structures in nature are being solved as fractals. However, we lose useful information in this approach. The two-scale fractal theory is given to deal with natural fractals, a mathematical framework is  provided, and many valuable results have been achieved in fluid dynamics, MEMS, biology, thermodynamics, and engineering problems. Perhaps nothing in mathematics is more important than differential equations. In addition to two-scale, numerical and analytical methods have been proposed to deal with differential equations of different kinds. Two-scale, along with proposed methods, has the ability to solve a variety of nonlinear models.