We investigate the criteria for both the existence and uniqueness of solutions within a nonlinear coupled system of Hilfer-Hadamard sequential fractional differential equations featuring varying orders. This system is complemented by nonlocal coupled Hadamard fractional integral boundary conditions. The desired outcomes are attained through the application of well-established fixed-point theorems. It is underscored that the fixed-point approach serves as an effective method for establishing both the existence and uniqueness of solutions to boundary value problems. The results obtained are further demonstrated and validated through illustrative examples.

Tax evasion remains a significant challenge that undermines government revenue and economic stability. In this paper, we propose proposes a new fractional-order three compartment tax evasion model that highlights the transition from susceptible individuals to honest taxpayers, capturing voluntary compliance driven by awareness and behavioral reinforcement mechanisms. The fundamental properties such as positivity, boundedness, and the existence and uniqueness of solutions, were established for the model. In this study, basic reproduction number $(R_0)$ was derived to characterize the persistence of tax evasion. Stability analysis showed that the evader-free equilibrium was locally and globally asymptotically stable when the threshold parameter was less than one, and forward bifurcation occurred when the threshold parameter exceeded one, indicating the persistence of tax evasion. Sensitivity analysis using the Partial Rank Correlation Coefficient (PRCC) method identified that the influence rate $(\beta )$ and the transition rate $(\alpha )$ are the dominant factors driving tax evasion, whereas audit effectiveness and reformation rate significantly reduce evasion. The model was numerically investigated using the Modified Fractional Euler Method (MFEM) for various combinations of parameters. Two control measures, namely media awareness campaigns ( $u_1$) and penalties ( $u_2$), are incorporated to mitigate tax evasion. Pareto and efficiency analyses revealed that penalties are effective when implemented individually, whereas the combined intervention strategy is the most cost-effective and economically viable approach for enhancing tax compliance. The results are presented in the form of figures and tables.

This study aimed to investigate the existence, uniqueness, and Ulam-Hyers stability of solutions in a nonlinear coupled system of Hilfer-Hadamard sequential fractional integrodifferential equations, which were further enhanced by nonlocal coupled Hadamard fractional integrodifferential multipoint boundary conditions. The desired conclusions were obtained by using well-known fixed-point theorems. It was emphasized that the fixed-point technique was useful in determining the existence and uniqueness of solutions to boundary value problems. In addition, we examined the solution's Ulam-Hyers stability for the suggested system. The resulting results were further demonstrated and validated using demonstration instances.