A robust technique to study fractional model describing economic and environmental mathematical system

Purpose The primary goal of this study is to create a more accurate and effective mathematical model for the economic and environmental system by utilizing a non-local derivative. Design/methodology/approach This study aims to produce results that better represent real-world complexity and dynamics. The arbitrary order of the economics and environmental mathematical model is categorised into three dynamics: the control achievement cost, the manufacturing element capability and the technical exclusion diagnostics cost. The proposed model includes a system of three equations which are studied via the Caputo fractional operator. The systems of nonlinear equations are evaluated by a semi-analytical approach called the q-homotopy analysis transform technique. Findings The behaviour of the model is analysed by 3D plots and graphs. The existence of the equilibrium points and their stability of the considered model is mathematically performed. The proposed scheme is more accurate and is a special case of q-HATM (i.e. n = 1), and we confirm that as the number of iterations increases, the q-HATM solutions converge to the exact solution. Originality/value The series solutions are achieved through the q-HATM method which converges rapidly. The convergence and uniqueness of the obtained solutions are evaluated for the studied fractional model. The results of this study demonstrate the importance and effectiveness of the projected derivative and technique in the analysis of time-dependent fractional mathematical models.