On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations

Abstract

Recently, fractals and fractional calculus have received much attention from researchers of various fields of science and engineering. Because the said area has been found applicable in modeling various real-world processes and phenomena. Hybrid differential equations (HDEs) play significant roles in mathematical modeling of various processes because the aforesaid equations incorporate different dynamical systems as specific cases. For instance, it is possible to model and describe non-homogeneous physical phenomena on using the said equations. Therefore, this research work is concerned with studying a class of nonlinear hybrid fractal–fractional differential equations. We develop the existence result for the qualitative study using a hybrid fixed point theorem. For the mentioned goal, a fixed point theory for the product of two operators is applied to deduce appropriate conditions for the existence of exactly one solution. Additionally, the stability result based on Ulam–Hyers is also deduced. The said stability results play an important role in numerical investigations. In addition, a numerical method based on Euler procedure is utilized to approximate the solution of the proposed problems. Various computational test problems are given to demonstrate the results. Also, using various fractal–fractional order values, several graphical presentations are given for the examples. The concerned analysis will help in investigating many real-world problems modeled using HDEs with fractal–fractional orders in the near future.

Keywords

• fractals and fractional calculus
• hybrid problem
• existence theory
• numerical results