This paper spotlights whether gross domestic product (GDP), energy consumption and urbanization affect Carbon dioxide (CO₂) emissions covering the time span 1972-2014 in Pakistan. The empirical estimates of autoregressive distributed lag (ARDL) affirmed that energy consumption and GDP are the main drivers of the pollutant environment in Pakistan, and also quantitatively determine the definite impact percentage of each dynamic force in Pakistan. Specifically, the empirical findings determine that in the long run, energy consumption and GDP intensify CO₂ emissions showing significant impact; however, industrialization and urbanization are found to be insignificant. Based on empirical findings, this suggests that the constricting economic gap implementation between urban and rural areas, amendments to the energy erection and mechanical novelty will offer a treasured reference for Pakistan in exertions to diminish carbon emissions.

In this paper, we study the HIV infection model based on fractional derivative with particular focus on the degree of T-cell depletion that can be caused by viral cytopathicity. The arbitrary order of the fractional derivatives gives an additional degree of freedom to fit more realistic levels of CD4⁺ cell depletion seen in many AIDS patients. We propose an implicit numerical scheme for the fractional-order HIV model using a finite difference approximation of the Caputo derivative. The fractional system has two equilibrium points, namely the uninfected equilibrium point and the infected equilibrium point. We investigate the stability of both equilibrium points. Further we examine the dynamical behavior of the system by finding a bifurcation point based on the viral death rate and the number of new virions produced by infected CD4⁺ T-cells to investigate the influence of the fractional derivative on the HIV dynamics. Finally numerical simulations are carried out to illustrate the analytical results.

In this article, a numerical scheme is formulated and analysed to solve the time-space fractional advection–diffusion equation, where the Riesz derivative and the Caputo derivative are considered in spatial and temporal directions, respectively. The Riesz space derivative is approximated by the second-order fractional weighted and shifted Grünwald–Letnikov formula. Based on the equivalence between the fractional differential equation and the integral equation, we have transformed the fractional differential equation into an equivalent integral equation. Then, the integral is approximated by the trapezoidal formula. Further, the stability and convergence analysis are discussed rigorously. The resulting scheme is formally proved with the second order accuracy both in space and time. Numerical experiments are also presented to verify the theoretical analysis.