In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments.

Nowadays, much of the world has a regional air pollution strategy to limit and decrease the pollution levels across governmental borders and control their impact on human health and ecological systems. Environmental protection is among the leading priorities worldwide. Many challenges in this research area exist since it is a painful subject for society and a fundamental topic for the healthcare system. Sensitivity analysis has a fundamental role during the process of validating a large-scale air pollution computational models to ensure their accuracy and reliability. We apply the best available stochastic algorithms for multidimensional sensitivity analysis of the UNI-DEM model, which plays a key role in the management of the many self-governed systems and data that form the basis for forecasting and analyzing the consequences of possible climate change. We develop two new highly convergent digital sequences with special generating matrices, which show significant improvement over the best available existing stochastic methods for measuring the sensitivity indices of the digital ecosystem. The results obtained through sensitivity analysis will play an extremely important multi-sided role.

Thorough examination of various aspects related to the distribution of air pollutants in a specific region and the factors contributing to high concentrations is essential, as these elevated levels can be detrimental. To accomplish this, the development and improvement of a digital twin that encompasses all relevant physical processes in the atmosphere is necessary. This tool, known as DIGITAL AIR, has been created, and it is now necessary to extend it with precise sensitivity analysis. DIGITAL AIR is gaining popularity due to its effectiveness in addressing complex problems that arise in intricate environments; this motivates our further investigations. In this paper, we focus on the preparation and further investigation of DIGITAL AIR through sensitivity analysis with improved stochastic approaches for investigating high-level air pollutants. We discuss and test the utilization of this digital tool in tackling the issue. The unified Danish Eulerian model (UNI-DEM) plays a crucial role within DIGITAL AIR. This mathematical model, UNI-DEM, is highly versatile and can be applied to various studies concerning the adverse effects caused by elevated air pollution levels.