INVESTIGATING JERK PHENOMENON BY USING FRACTAL-FRACTIONAL ANALYSIS

Fractal-fractional calculus has attracted much attention in the last few years due to its wide range applications in various disciplines of science and technology. The class of differential equations with order lying in (2, 3] has significant importance in physics because the concerned equations are usually used to represent jerk or jolt-type phenomenon in real-world applications. Therefore, we consider a class of fractal-fractional differential equations with fractional order p ∈ (2, 3] and fractals dimension q ∈ (0, 1]. The mentioned problem involves proportional-type delay term called Pantograph. We also develop the existence and stability theory for the problem under our consideration using fractal-fractional Riemann-Liouville differential operator. On using the fixed point approach due to Schaefer's and Banach's, we investigate appropriate conditions for the existence of at least one solution and its unique. In addition, the Hyers-Ulam (HU) approach of stability is used for stability analysis. Keeping in mind the importance of numerical analysis, Haar wavelets procedure is used to compute some numerical results for the considered problem. Finally, the established results are applied to a numerical problem to explain its validity. Hence, various graphical results related to different fractals and fractional orders have been presented.