Numerical solutions of the fractal foam drainage equation

The flow of liquid relative to the bubbles is called drainage. This paper presents a study of the numerical solution of a non-linear foam drainage equation with time-fractional derivative. We use the two-scale approach which is formulated by combining the fractional complex transform and the homotopy perturbation method (HPM). With the aid of the fractional complex transform, first, we transform the problem into its differential partner and then HPM is applied to obtain the He's polynomials which are highly and powerful support for non-linear problems. Further, we put forward the theory of the two-scale approach which reveals the sketch between fractional complex transform and the solution of non-linear foam drainage equation. The significant results illustrate that this approach does not require any assumption while it reduces the heavy calculation without any restrictive variable. This approach also sheds a bright light on practical applications of fractal calculus.