In this paper, extended complex variables method (ECVM) is presented in fluid flow problems for the first and second-order sensitivity analysis. The finite element method is used to solve the Navier-Stokes equations, and the complex variables method is implemented to it. In the complex variables method, a complex step that only includes the imaginary part is used, but in its development, it uses a complex step that includes both the imaginary part and the real part to achieve higher performance. In the first-order sensitivity calculation, the results are not dependent on the step size, but in the second-order sensitivity, the results of the sensitivity depending on the step size and inevitably the developed formulas should be used to obtain higher accuracy. The proposed method is first validated for a problem with a closed-form solution, and the convergence rate is investigated and, then, applied to a uniform flow past a cylindrical cylinder and, finally, the results are compared by finite difference method. The results show that the range of accuracy for second-order sensitivity in the extended complex variable method is doubled compared to the complex variable method and it can be reduced to 10-12. It means that the effectiveness of the proposed method has increased. The introduced method is applicable to a wide range of problems with simple and complex parameters.

Background: Asthma and allergic rhinitis are prevalent respiratory conditions influenced by genetic, environmental, and immunological factors. Understanding the distribution of these conditions and their associated allergens is essential for effective management and prevention strategies. This study aimed to investigate the prevalence of asthma, allergic rhinitis, and their co-occurrence in a specific population, with a focus on allergen sensitivity patterns. Methods: A retrospective analysis was conducted on patients diagnosed with asthma, allergic rhinitis, or both. Allergen sensitivity was assessed using skin prick tests for common environmental allergens, including weeds, mites, cockroaches, and mixed allergens. Data analysis included prevalence rates, allergen distribution, and a heatmap visualization of allergen-patient associations. Results: Males demonstrated a higher prevalence of asthma (68. 8%), allergic rhinitis (64. 4%), and combined conditions (67. 0%) compared to females. Weeds were the most common allergen among asthma patients (40. 0%), while mixed allergens were predominant among allergic rhinitis (45. 7%) and coexisting conditions (50. 0%). Heatmap analysis revealed strong associations between mites, cockroaches, and respiratory conditions, reinforcing their role as significant allergens. The high prevalence of mixed allergens in patients with both conditions suggests the need for comprehensive allergy management approaches. Conclusion: This study highlights a male predominance in asthma and allergic rhinitis cases and underscores the role of environmental allergens in disease manifestation. The findings support targeted allergen avoidance and personalized immunotherapy strategies for optimal respiratory disease management.